# ON THE DISCRETE FUNCTIONAL $L_{p}$ MINKOWSKI PROBLEM

Tuo Wang  Institut für Diskrete Mathematik und Geometrie
TU Wien
Wiedner Hauptstrasse 8-10, 1040 Wien
Austria<EMAIL_ADDRESS>
###### Abstract.

The discrete functional $L_{p}$ Minkowski problem is posed and solved. As a
consequence, the general affine Pólya-Szegö principle and the general affine
Sobolev inequalities are established.

2000 AMS subject classification: 46B20 (46E35, 52A21, 52B45)
Keywords: affine Pólya-Szegö principle, functional Minkowski problem, convex
symmetrization.

##  1\. Introduction

The even functional Minkowski problem was first introduced by Lutwak, Yang and
Zhang in [20] to answer the following question:

Given a function $f\in W^{1,1}(\mathbb{R}^{n})$, where
$W^{1,1}(\mathbb{R}^{n})$ is the usual $L^{1}$ Sobolev space on
$\mathbb{R}^{n}$, which norm $\|\cdot\|_{o}$ on $\mathbb{R}^{n}$ minimizes the
quantity $\int_{\mathbb{R}^{n}}\|\nabla f\|_{o}dx,$ if the unit ball of the
dual norm of $\|\cdot\|_{o}$ has the same volume as the Euclidean unit ball in
$\mathbb{R}^{n}$?

The solution to this problem turns out to be unique and is called the optimal
Sobolev norm.

To answer the question, Lutwak, Yang and Zhang introduced the following even
functional Minkowski problem on $W^{1,1}(\mathbb{R}^{n}):$

Given $f\in W^{1,1}(\mathbb{R}^{n})$, find an $o$-symmetric convex body
$\langle f\rangle_{e}$ such that

(1) $\int_{S^{n-1}}\Psi(u)dS(\langle
f\rangle_{e},u)=\int_{\mathbb{R}^{n}}\Psi(\nabla f(x))dx$

for all continuous functions $\Psi$ on $\mathbb{R}^{n}$ that are even and
positively homogeneous of degree $1$. Here $S(\langle f\rangle_{e},\cdot)$ is
the Alexandrov-Fenchel-Jessen surface area measure of $\langle f\rangle_{e}$
and $o$-symmetric stands for origin-symmetric.

The optimal Sobolev body $\langle\cdot\rangle_{e}$ defined through equation
(1) provides the solution of the optimal Sobolev norm problem. The even
functional Minkowski problem is shown to play an important role in the theory
of affine Sobolev inequalities on $\mathbb{R}^{n}$ (see [13][20][27][29]).
Using valuations on Sobolev spaces, Ludwig obtained a natural description of
the operator $f\mapsto\langle f\rangle_{e}$ (see [13]). By extending the even
functional Minkowski problem to $BV(\mathbb{R}^{n}),$ the space of functions
of bounded variations, the author was able to extend the Zhang-Sobolev
inequality to $BV(\mathbb{R}^{n})$ and got a characterization of the equality
cases (see [27]). By combining the even functional Minkowski problem with the
convex symmetrization technique introduced in [1], the author got a Brothers-
Ziemer type theorem for the affine Pólya-Szegö principle, which clarified the
equality condition of the affine Pólya-Szegö principle (see [29]). Based on
these facts, it is natural to ask the following question:

Is there always a hidden convex body that plays an important role in each
affine Sobolev type inequality?

While for the symmetric optimal $L_{p}$ affine Sobolev inequality, the answer
was obtained by the Lutwak, Yang and Zhang [20]. The aim of this paper is to
define and solve the general functional $L_{p}$ Minkowski problems on Sobolev
spaces. By making use of the solutions of the general functional $L_{p}$
Minkowski problems and the general $L_{p}$ affine isoperimetric inequalities
[7], we arrive at the general affine Pólya-Szegö principle and general affine
Sobolev type inequalities, which include the results in [8][9][22] as special
cases. This shows that the crucial convex body behind the discrete version
general affine Sobolev is the solution of the discrete functional Minkowski
problem.

To to specific, we study the discrete versions of functional $L_{p}$ Minkowski
Problems. Instead of $W^{1,1}(\mathbb{R}^{n})$, we consider the functional
$L_{p}$ Minkowski problem on $L^{1,0}(\mathbb{R}^{n})$, a dense subspace of
$W^{1,p}(\mathbb{R}^{n})$.

######  Definition 1.

The space of piecewise affine functions on $\mathbb{R}^{n}$, where a function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is called piecewise affine, if $f$ is
continuous and there are finitely many $n$-dimensional simplices
$T_{1},...,T_{m}\subset\mathbb{R}^{n}$ with pairwise disjoint interiors such
that the restriction of $f$ to each $T_{i}$ is affine and $0$ outside
$T1\cup\cdots\cup T_{m}.$

In particular, we get the following result:

######  Theorem 1.

Given a nontrivial function $f\in L^{1,0}(\mathbb{R}^{n})$, there exists a
unique polytope $\langle f\rangle_{p}\in\mathcal{P}_{0}^{n}$ such that

(2) $n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla
f(x))dx=\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f\rangle_{p},u),$

for every continuous function $\Psi:\mathbb{R}^{n}\rightarrow[0,\infty)$ that
is homogeneous of degree $1$.

The new functional $L_{p}$ Minkowski problem enjoys some natural affine
invariance property and valuation property. By studying this new functional
Minkowski problem on $L^{1,0}(\mathbb{R}^{n})$, we are led to the general
affine Pólya-Szegö principle and general affine Sobolev inequalities.

We remark that, although the discovery of the discrete $L^{p}$ Minkowski
problem on $W^{1,p}(\mathbb{R}^{n})$ is the main issue of this piece of work,
a very important tool that we make use in this piece of work is the Haberl-
Schuster version of the general $L_{p}$ affine isoperimetric inequality in
[7].

##  2\. Background material

Background on the $L_{p}$ Brunn-Minkowski theory. For quick reference we
recall in this section some background material from the $L_{p}$ Brunn-
Minkowski theory of convex bodies. This theory has its origin in the work of
Firey from the 1960’s and has expanded rapidly over the last two decades since
the work of Lutwak [17] (see, e.g.,[2, 3, 4, 7, 9, 10, 11, 12, 14, 16, 17, 18,
19, 20, 21, 22, 23, 24, 27, 28]).

A convex body is a compact convex subset of $\mathbb{R}^{n}$ with non-empty
interior. We write $\mathcal{K}^{n}$ for the set of convex bodies in
$\mathbb{R}^{n}$ endowed with the Hausdorff metric and we denote by
$\mathcal{K}^{n}_{e}$ the set of origin symmetric convex bodies, by
$\mathcal{K}^{n}_{0}$ the set of convex bodies containing origin in the
interior and by $\mathcal{P}^{n}_{0}$ the set of polytopes containing origin
in the interior. Each non-empty compact convex set $K$ is uniquely determined
by its support function $h(K,\cdot)$, defined by

$h(K,x)=\max\\{x\cdot y:y\in K\\},~{}~{}x\in\mathbb{R}^{n},$

where $x\cdot y$ denotes the Euclidean inner product of $x$ and $y$ in
$\mathbb{R}^{n}$. Note that $h(K,\cdot)$ is positively homogeneous of degree
$1$ and sub-additive. Conversely, every function with these two properties is
the support function of a unique compact convex set.

For later use, we include the following lemma from [26]:

######  Lemma 1.

Let $K_{i}$ and $K$ be convex bodies in $\mathbb{R}^{n}$. If
$h(K_{i},\cdot)\rightarrow h(K,\cdot)$ pointwise, then
$h(K_{i},\cdot)\rightarrow h(K,\cdot)$ uniformly on $S^{n-1}$.

If $K\in\mathcal{K}^{n}_{0}$ , then the polar body $K^{\circ}$ of $K$ is
defined by

$K^{\circ}=\\{x\in\mathbb{R}^{n}:x\cdot y\leq 1~{}~{}~{}for~{}all~{}y\in
K\\}.$

From the polar formula for volume it follows that the $n$-dimensional Lebesgue
measure $|K^{\circ}|$ of the polar body $K^{\circ}$ can be computed by

$|K^{\circ}|=\frac{1}{n}\int_{S^{n-1}}h(K,u)^{-n}du,$

where integration is with respect to spherical Lebesgue measure.

For real $p\geq 1$ and $\alpha,\beta>0$, the $L_{p}$ Minkowski combination of
$K,L\in\mathcal{K}_{0}^{n}$ is the convex body $\alpha\cdot K+_{p}\beta\cdot
L$ defined by

$h^{p}(\alpha\cdot K+_{p}\beta\cdot L,u)=h^{p}(K,u)+h^{p}(L,u),$

and the $L_{p}$ mixed volume of $K,L$ is defined by

$V_{p}(K,L)=\frac{p}{n}\lim_{\epsilon\rightarrow
0^{+}}\frac{|K+_{p}\epsilon\cdot L|-|K|}{\epsilon}.$

Clearly, the diagonal form of $V_{p}$ reduces to ordinary volume, i.e., for
$K\in\mathcal{K}_{0}^{n},$

$V_{p}(K,K)=|K|.$

It was also shown in [17] that for all convex bodies
$K,L\in\mathcal{K}_{0}^{n},$

$V_{p}(K,L)=\frac{1}{n}\int_{S^{n-1}}h(L,u)^{p}dS_{p}(K,u),$

where $dS_{p}(K,u)=h(K,u)^{1-p}dS(K,u)$. Recall that for a Borel set
$\omega\subset S^{n-1}$, $S(K,\omega)$ is the $(n-1)$-dimensional Hausdorff
measure $H^{n-1}$ of the set of all boundary points of $K$ for which there
exists a normal vector of $K$ belonging to $\omega.$

A fundamental inequality that we need is the $L_{p}$ Minkowski inequality
[17].

######  Theorem 2 ([17]).

If $1<p<\infty$ and $K,L\in\mathcal{K}_{0}^{n},$ then

(3) $V_{p}(K,L)\geq|K|^{1-p/n}|L|^{p/n}.$

Equality holds if and only if $L=tK$ for some $t>0.$

Projection bodies have become a central notion within the Brunn-Minkowski
theory. They arise naturally in a number of different areas such as functional
analysis, stochastic geometry and geometric tomography. The fundamental affine
isoperimetric inequality which connects the volume of a convex body with that
of its polar projection body is the Petty projection inequality [25]. This
inequality turned out to be stronger than the classical isoperimetric
inequality and its functional form is Zhang’s discovery, namely the affine
Zhang-Sobolev inequality [30].

The $L_{p}$ projection body $\Pi_{p}K$ of $K\in\mathcal{K}_{0}^{n},$ was
introduced in [18] as the convex body such that

$h(\Pi_{p}K,u)^{p}=\int_{S^{n-1}}|u\cdot v|^{p}dS_{p}(K,v),~{}u\in S^{n-1}.$

The $L_{p}$ analog of Petty’s projection inequality plays a key role in the
$L_{p}$ Brunn-Minkowski theory. It was first proved by Lutwak, Yang, and Zhang
[21] (see also Campi and Gronchi [3] for an independent approach): If
$K\in\mathcal{K}_{0}^{n}$, then

$|K|^{n/p-1}|\Pi_{p}K|\leq|B_{2}|^{n/p-1}|\Pi_{p}B_{2}|,$

where $B_{2}$ is the Euclidean unit ball and equality is attained if and only
if $K$ is an ellipsoid centered at the origin.

For a finite Borel measure $\mu$ on $S^{n-1}$, we define a continuous function
$C_{p}^{+}(\mu)$ on $S^{n-1}$, the asymmetric $L_{p}$ cosine transform of
$\mu$, by

$(C_{P}^{+}\mu)(u)=\int_{S^{n-1}}(u\cdot v)^{p}_{+}d\mu(v),u\in S^{n-1},$

where $(u\cdot v)_{+}=\max\\{u\cdot v,0\\}$. The asymmetric $L_{p}$ projection
body $\Pi^{p}_{+}(K)$ of $K\in\mathcal{K}_{0}^{n},$ first considered in [12],
is the convex body defined by

$h(\Pi^{p}_{+}(K),\cdot)^{p}=C_{p}^{+}(S_{p}(K,\cdot)).$

For $p>1$, by using valuations, Ludwig [12] established the $L_{p}$ analogue
of her classification of the projection operator: She showed that the convex
bodies

$(1-\lambda)\cdot\Pi_{p}^{+}(K)+_{p}\lambda\cdot\Pi_{p}^{-}(K),K\in\mathcal{K}_{0}^{n}$

where $0\leq\lambda\leq 1$ and $\Pi_{p}^{-}(K)=\Pi_{p}^{+}(-K)$, constitute
all natural $L_{p}$ extensions of projection bodies.

Haberl and Schuster established the following general $L_{p}$ Petty projection
inequalities in [7].

######  Theorem 3 ([7]).

Let $K\in\mathcal{K}_{0}^{n}$ and $p>1$. If $\Phi_{\lambda,p}K$ is the convex
body defined by

(4)
$\Phi_{\lambda,p}(K)=(1-\lambda)\cdot\Pi_{p}^{+}(K)+_{p}\lambda\cdot\Pi^{-}_{p}(K),$

where $0\leq\lambda\leq 1$, then

$|K|^{n/p-1}|\Phi_{\lambda,p}K|\leq|B_{2}|^{n/p-1}|\Phi_{\lambda,p}B_{2}|,$

with equality if and only if $K$ is an ellipsoid centered at the origin.

The classical Minkowski problem has a counterpart in the $L_{p}$ Brunn-
Minkowski theory. The $L_{p}$ Minkowski problem asks the following: Given a
Borel measure $\mu$ on $S^{n-1}$, does there exist a convex body $K$ such that
$\mu=S_{p}(K,\cdot)$?

If the data $\mu$ is an even measure, Lutwak [17] gave an affirmative answer
to this problem when $p\neq n$. Later on, Lutwak, Yang and Zhang [24]
introduced the volume-normalized even $L_{p}$ Minkowski problem, for which the
case $p=n$ can be handled as well. What we need in this work is the following
discrete version of the $L_{p}$ Minkowski problem.

######  Theorem 4 ([10]).

Let vectors $u_{1},...,u_{m}\in S^{n-1}$ that are not contained in a closed
hemisphere and real numbers $\alpha_{1},...,\alpha_{m}>0$ be given. Then, for
any $p>1$ with $p\neq n$, there exists a unique polytope
$P\in\mathcal{P}_{0}^{n}$ such that

$\sum_{j=1}^{m}\alpha_{j}\delta_{u_{j}}=S_{p}(P,\cdot).$

For treating the case $p=n$, we need the normalized version of the above
theorem.

######  Theorem 5 ([10]).

Let vectors $u_{1},...,u_{m}\in S^{n-1}$ that are not contained in a closed
hemisphere and real numbers $\alpha_{1},...,\alpha_{m}>0$ be given. Then there
exists a unique polytope $P\in\mathcal{P}_{0}^{n}$ such that

$\sum_{j=1}^{m}\alpha_{j}\delta_{u_{j}}=\frac{S_{n}(P,\cdot)}{|P|}.$

##  3\. The discrete functional $L_{p}$ Minkowski problem

We assume $1<p<\infty$ and $p\neq n$ throughout this section.

###  3.1. The discrete case.

######  Theorem 6.

Given a nontrivial function $f\in L^{1,0}(\mathbb{R}^{n})$, there exists a
unique polytope $\langle f\rangle_{p}\in\mathcal{P}_{0}^{n}$ such that

(5) $n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla
f(x))dx=\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f\rangle_{p},u),$

for every continuous function $\Psi:\mathbb{R}^{n}\rightarrow[0,\infty)$ that
is homogeneous of degree $1$.

###### Proof.

Let $\Sigma=\\{x:\nabla f(x)=0\\}.$ Then the map

$\Upsilon\mapsto n\int_{\mathbb{R}^{n}\backslash\Sigma}\Upsilon(-\frac{\nabla
f(x)}{|\nabla f(x)|})|\nabla f(x)|^{p}dx$

defines a nonnegative bounded linear functional on the space of continuous
functions on $S^{n-1}$. It follows from the Riesz representation theorem that
there exists a unique Borel measure $S_{p}(f,\cdot)$ on $S^{n-1}$ such that

$n\int_{\mathbb{R}^{n}\backslash\Sigma}\Upsilon(-\frac{\nabla f(x)}{|\nabla
f(x)|})|\nabla f(x)|^{p}dx=\int_{S^{n-1}}\Upsilon(u)dS_{p}(f,u),$

for each continuous function $\Upsilon:S^{n-1}\rightarrow\mathbb{R}.$

Choose $\Upsilon=\Psi^{p}$ and radially extend $\Psi$ to $\mathbb{R}^{n}$ by
requiring $\Psi$ to be of homogeneity degree $1$. Then we have

$\displaystyle n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla f(x))dx$
$\displaystyle=$ $\displaystyle
n\int_{\mathbb{R}^{n}\backslash\Sigma}\Psi^{p}(-\nabla f(x))dx$
$\displaystyle=$ $\displaystyle
n\int_{\mathbb{R}^{n}\backslash\Sigma}\Psi^{p}(-\frac{\nabla f(x)}{|\nabla
f(x)|})|\nabla f(x)|^{p}dx$ $\displaystyle=$
$\displaystyle\int_{S^{n-1}}\Psi^{p}(u)dS_{p}(f,u).$

We claim that $S_{p}(f,u)$ is a finite combination of Dirac measures on the
sphere for any $f\in L^{1,0}(\mathbb{R}^{n})$.

Notice from the definition, we have

$S_{p}(f(x),\cdot)=S_{p}(f(x+x_{0}),\cdot),$

for any $x_{0}\in\mathbb{R}^{n}.$

Next we consider a triangulation of the support of $f$, i.e. on each simplex
$M_{i}$, $f$ is affine, where $i=1,...,m.$

Without loss of generality, we assume

$f(x)=x\cdot u_{i}+c_{i}$

for $x\in M_{i}$. Then we have

$\displaystyle n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla f(x))dx$
$\displaystyle=$ $\displaystyle n\sum_{i=1}^{m}\int_{M_{i}}\Psi^{p}(-\nabla
f(x))dx$ $\displaystyle=$ $\displaystyle
n\sum_{i=1}^{m}|M_{i}|\Psi^{p}(-u_{i})dx.$

Therefore we have

$S_{p}(\langle f\rangle_{p},\cdot)=n\sum_{i=1}^{m}|M_{i}|\delta_{-u_{i}},$

where $\delta_{-u_{i}}$ is the Dirac measure concentrated on $-u_{i}.$

We remark that the fact that $S_{p}(f,u)$ does not concentrate on any great
sub-sphere on $S^{n-1}$ follows basically from the same argument of Lemma 4.1
in [20]. In fact, by tracing back to the definition, one gets

$S_{p}(f,u)(S^{n-2})\neq 0,$

where $S^{n-2}$ is any big sphere of $S^{n-1}$.

Therefore, for any $f\in L^{1,0}(\mathbb{R}^{n})$, $S_{p}(f,u)$ is a linear
combination of the Dirac measures on the sphere which do not concentrate on
any great subsphere. By using Theorem 4, we prove the existence of the
polytope $\langle f\rangle_{p}\in\mathcal{P}_{0}^{n}.$

The uniqueness is a consequence of the $L_{p}$ Minkowski inequality (Theorem
2). ∎

For $P\in\mathcal{P}_{0}^{n}$, define the piecewise affine function $l_{P}$ by
requiring that $l_{P}(0)=1,$ that $l_{P}(x)=0$ for $x\not\in P$, and that
$l_{P}$ is affine on each simplex with apex at the origin and base equal to a
facet of $P$. Define $\mathcal{P}^{1,p}\subset L^{1,0}(\mathbb{R}^{n})$ as the
set of all $l_{P}$ for $P\in P_{0}^{n}$. Note that for $\phi\in GL(n),$

$l_{\phi P}=l_{P}\circ\phi^{-1}.$

We remark that multiples and translates of $l_{P}\in P^{1,p}(\mathbb{R}^{n})$
correspond to linear elements within the theory of finite elements.

######  Lemma 2.

For $P\in\mathcal{P}_{0}^{n},$

$\langle l_{P}\rangle_{p}=P.$

###### Proof.

Let $P$ have facets $F_{1},...,F_{m}.$ For the facet $F_{i}$, let $u_{i}$ be
its unit outer normal vector and $T_{i}$ the convex hull of $F_{i}$ and the
origin. Since for $x\in T_{i},$

$l_{P}(x)=-\frac{u_{i}}{h(P,u_{i})}\cdot x+1$

and

$\nabla l_{P}(x)=-\frac{u_{i}}{h(P,u_{i})},$

we have that

$\displaystyle n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla l_{P}(x))dx$
$\displaystyle=$ $\displaystyle n\sum_{i=1}^{m}\int_{T_{i}}\Psi^{p}(-\nabla
l_{P}(x))dx$ $\displaystyle=$ $\displaystyle
n\sum_{i=1}^{m}\int_{T_{i}}\Psi^{p}(\frac{u_{i}}{h(P,u_{i})})dx$
$\displaystyle=$ $\displaystyle
n\sum_{i=1}^{m}\Psi^{p}(u_{i})\frac{V_{n}(T_{i})}{h(P,u_{i})^{p}}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\Psi^{p}(u_{i})V_{n-1}(F_{i})h(P,u_{i})^{1-p}$
$\displaystyle=$ $\displaystyle\int_{S^{n-1}}\Psi^{p}(u)dS_{p}(P,u).$

Since $S_{p}(P,\cdot)=S_{p}(\langle 1_{P}\rangle_{p},\cdot),$ we have
$P=\langle 1_{P}\rangle_{p}$ from Theorem 4. ∎

######  Lemma 3.

If $f\in L^{1,0}(\mathbb{R}^{n})$ and $\phi\in SL(n)$, then

$\langle f\circ\phi^{-1}\rangle_{p}=\phi\langle f\rangle_{p}.$

###### Proof.

It is well-known (e.g. see [20]) that if

$V_{p}(\phi
K,L)=V_{p}(J,\phi^{-1}L)~{}for~{}all~{}convex~{}bodies~{}L\in\mathcal{K}_{0}^{n},$

then $K=J.$

Since by the affine invariance of the $L_{p}$ mixed volume, we have

$V_{p}(\phi\langle f\rangle_{p},L)=V_{p}(\langle f\rangle_{p},\phi^{-1}L),$

and we know, by the change of variable $y=\phi(x)$, that

$\displaystyle\int_{\mathbb{R}^{n}}h(L,\nabla(f\circ\phi^{-1})(u))^{p}dy$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}h(L,\phi^{-t}\nabla
f(\phi^{-1}y))^{p}dy$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}h(\phi^{-1}L,\nabla f(x))^{p}dx.$

Therefore, we have that $V_{p}(\langle
f\circ\phi^{-1}\rangle_{p},L)=V_{p}(\langle f\rangle_{p},\phi^{-1}L),$ which
implies that $\langle f\circ\phi^{-1}\rangle_{p}=\phi\langle f\rangle_{p}.$ ∎

As a quick consequence of Lemma 3, we have the following corollary.

######  Corollary 1.

Given $\phi\in SL(n)$ and $f\in L^{1,0}(\mathbb{R}^{n})$, we have

$|\langle f\rangle_{p}|=|\langle f\circ\phi^{-1}\rangle_{p}|.$

The operator
$\langle\cdot\rangle_{p}:L^{1,0}(\mathbb{R}^{n})\rightarrow\mathcal{P}_{0}^{n}$
has the valuation property.

######  Definition 2.

For two polytopes $K,L\in\mathcal{P}_{0}^{n}$, we define the $L_{p}$-Blaschke
sum $K\sharp_{p}L$ of $K,L$ by requiring

(6) $S_{p}(K\sharp_{p}L,\cdot)=S_{p}(K,\cdot)+S_{p}(L,\cdot).$

Remark: By Theorem 4, we know there is a polytope $K\sharp_{p}L$ uniquely
defined through equation (6).

For $f,g\in L^{1,0}(\mathbb{R}^{n})$, the function $f\vee g$ denotes the
pointwise maximum and the function $f\wedge g$ the pointwise minimum of $f$
and $g$.

######  Theorem 7.

The operator
$\langle\cdot\rangle_{p}:(L^{1,0}(\mathbb{R}^{n}),\wedge,\vee)\rightarrow(\mathcal{P}_{0}^{n},\sharp_{p})$
is an $L_{p}$-Blaschke valuation.

###### Proof.

What we need to show is that

$\displaystyle\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle
f\rangle_{p},u)+\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle g\rangle_{p},u)$
$\displaystyle=$ $\displaystyle\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f\wedge
g\rangle_{p},u)+\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f\vee
g\rangle_{p},u),$

for every continuous $\Psi:S^{n-1}\rightarrow\mathbb{R}$ and $f,g\in
L^{1,0}(\mathbb{R}^{n}).$

For this we triangulate the support of $f$ and $g$ so that $f(x)=x\cdot
u_{i}+c_{i}$ on $P_{i}$ for $i=1,...,m$ and $g(x)=x\cdot v_{i}+d_{i}$ on
$Q_{i}$ for $i=1,...,n$, and without loss of generality, we assume

$\\{x:f(x)\geq g(x)~{}and~{}g(x)\neq
0\\}=\bigcup_{k(i)}P_{k(i)}\cup\bigcup_{l(j)}Q_{l(j)},$

where $k(i)\in\\{1,...,m\\}$ and $l(j)\in\\{1,...,n\\}.$ Then we notice

$\displaystyle\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle
f\rangle_{p},u)+\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle g\rangle_{p},u)$
$\displaystyle=$ $\displaystyle n(\int_{\mathbb{R}^{n}}\Psi(-\nabla
f(x))^{p}dx+\int_{\mathbb{R}^{n}}\Psi(-\nabla g(x))^{p}dx)$ $\displaystyle=$
$\displaystyle
n(\sum_{i=1}^{m}\Psi(-u_{i})^{p}|P_{i}|+\sum_{j=1}^{j=n}\Psi(-v_{j})^{p}|Q_{j}|).$

Similarly,

$\displaystyle\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f\wedge
g\rangle_{p},u)+\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f\vee g\rangle_{p},u)$
$\displaystyle=$ $\displaystyle n(\int_{\mathbb{R}^{n}}\Psi(-\nabla(f\wedge
g)(x))^{p}dx+\int_{\mathbb{R}^{n}}\Psi(-\nabla(f\wedge g)(x))^{p}dx)$
$\displaystyle=$ $\displaystyle
n(\sum_{k(i),l(j)}(\Psi(-u_{k(i)})^{p}|P_{k(i)}|+\Psi(-v_{l(j)})^{p}|Q_{l(j)}|)$
$\displaystyle+\sum_{i\neq k(i)}\Psi(-u_{i})^{p}|P_{i}|+\sum_{j\neq
l(j)}\Psi(-v_{j})^{p}|Q_{j}|)$ $\displaystyle=$ $\displaystyle
n(\sum_{i=1}^{m}\Psi(-u)^{p}|P_{i}|+\sum_{j=1}^{n}\Psi(-v_{j})^{p}|Q_{j}|).$

This concludes the proof. ∎

In [13], Monika Ludwig started the search of interesting valuations on
function spaces. She found out that the only reasonable valuation that enjoys
some natural affine invariance property is the LYZ operator defined by
equation (1). Motivated by this, we propose a Ludwig type conjecture: if
$Z:(L^{1,0}(\mathbb{R}^{n}),\wedge,\vee)\rightarrow(\mathcal{P}_{0}^{n},\sharp_{p})$
is a continuous $L_{p}$ Blaschke valuation which satisfies

$Z(f)=Z(f\circ\tau^{-1}),Z(f\circ\phi^{-1})=|\det\phi|^{q}\phi Z(f)$

for all $\tau\in\mathbb{R}^{n}$, $\phi\in GL(n)$ and some $q\in\mathbb{R}$,
then

$Z=c\langle\cdot\rangle_{p}$

for some constant $c\geq 0.$

###  3.2. The homothetic functional $L_{p}$ Minkowski problem.

Given $f\in W^{1,p}(\mathbb{R}^{n})$, its distribution function
$\mu_{f}:[0,\infty)\rightarrow[0,\infty]$ is defined by

$\mu_{f}(t)=|\\{x\in\mathbb{R}^{n}:|f(x)|>t\\}|,$

where $|\cdot|$ denotes Lebesgue measure on $\mathbb{R}^{n}$. The decreasing
rearrangement $f^{\ast}:[0,\infty)\rightarrow[0,\infty]$ of $f$ is defined by

$f^{\ast}(s)=\inf\\{t>0:\mu_{f}(t)\leq s\\}.$

The symmetric rearrangement $f^{\ast}:\mathbb{R}^{n}\rightarrow[0,\infty]$ of
$f$ is the function defined by

$f^{\star}(x)=f^{\ast}(\omega_{n}|x|^{n}),$

where $|\cdot|$ is the standard Euclidean norm.

For a convex body $K\in\mathcal{K}_{0}^{n}$, the convex symmetrization $f^{K}$
of $f$ with respect to $K$ is defined as follows:

$f^{K}(x)=f^{\ast}(\omega_{n}h(\tilde{K}^{\circ},x)^{n}),$

where $h(\tilde{K}^{\circ},x)$ is the support function of $\tilde{K}^{\circ}$,
with $\tilde{K}$ being a dilation of $K$ so that $|\tilde{K}|=\omega_{n}$.

######  Lemma 4.

Given $f^{K}\in W^{1,p}(\mathbb{R}^{n})$ and $K\in\mathcal{K}_{0}^{n}$, there
is a unique convex body $\langle f^{K}\rangle_{p}\in\mathcal{K}_{0}^{n}$ such
that

$n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla
f^{K}(x))dx=\int_{S^{n-1}}\Psi(u)^{p}dS_{p}(\langle f^{K}\rangle_{p},u),$

for every continous $\Psi$ that is homogeneous of degree $1$. Furthermore

$\langle f^{K}\rangle_{p}=\xi_{f}\tilde{K},$

where
$\xi_{f}=(n(n\omega_{n})^{p}\int_{0}^{\infty}t^{np+n-p-1}((-f^{\ast})^{\prime}(\omega_{n}t^{n}))^{p}dt)^{\frac{1}{n-p}}.$

###### Proof.

We remark that the proof here is similar to that in [29].

First, we notice that the uniqueness is a quick consequence of the $L_{p}$
Minkowski inequality (Equation (1)).

Since $h(\tilde{K}^{\circ},\cdot)$ is a Lipschitz function and
$h(\tilde{K}^{\circ},\cdot)=1$ on $\partial\tilde{K},$ for almost all
$x\in\partial\tilde{K}$,

$\sigma_{\tilde{K}}(x)=\frac{\nabla h(\tilde{K}^{\circ},x)}{|\nabla
h(\tilde{K}^{\circ},x)|},$

where $\tilde{\sigma_{K}}(x)$ is the outer unit normal vector of $\tilde{K}$
at the point $x$. And from the definition of the polar body, we have

$h(\tilde{K},\sigma_{K}(x))=\frac{1}{|\nabla h(\tilde{K}^{\circ},x)|}.$

Also, it is well known that $f^{\ast}$ is locally absolutely continuous on
$(0,\infty)$. We have, by the co-area formula applied to
$h(\tilde{K}^{\circ},\cdot)$, that:

$\displaystyle n\int_{\mathbb{R}^{n}}\Psi^{p}(-\nabla f^{K}(x))dx$
$\displaystyle=$ $\displaystyle
n\int_{\mathbb{R}^{n}}\Psi^{p}(-(f^{\ast})^{\prime}(\omega_{n}h(\tilde{K}^{\circ},x)^{n})n\omega_{n}h(\tilde{K}^{\circ},x)^{n-1}\nabla
h(\tilde{K}^{\circ},x))dx$ $\displaystyle=$ $\displaystyle
n\int_{0}^{\infty}\int_{\partial
K}t^{n-1}((-(f^{\ast})^{\prime}(\omega_{n}t^{n}))n\omega_{n}t^{n-1})^{p}\Psi^{p}(\frac{\nabla
h(\tilde{K}^{\circ},x)}{|\nabla h(\tilde{K}^{\circ},x)|})|\nabla
h(\tilde{K}^{\circ},x)|^{p-1}dH^{n-1}(x)dt$ $\displaystyle=$ $\displaystyle
n(n\omega_{n})^{p}\int_{0}^{\infty}t^{np+n-p-1}((-f^{\ast})^{\prime}(\omega_{n}t^{n}))^{p}dt\int_{S^{n-1}}\Psi^{p}(u)h^{1-p}(\tilde{K},u)dS(\tilde{K},u)$
$\displaystyle=$ $\displaystyle
n(n\omega_{n})^{p}\int_{0}^{\infty}t^{np+n-p-1}((-f^{\ast})^{\prime}(\omega_{n}t^{n}))^{p}dt\int_{S^{n-1}}\Psi^{p}(u)dS_{p}(\tilde{K},u)$

for any $\Psi$ that is homogeneous of degree $1$.

So we get

$S_{p}(\langle f^{K}\rangle_{p},u)=\xi_{f}S_{p}(\tilde{K},u),$

where
$\xi_{f}=n(n\omega_{n})^{p}\int_{0}^{\infty}t^{np+n-p-1}((-f^{\ast})^{\prime}(\omega_{n}t^{n}))^{p}dt.$
This concludes the proof. ∎

######  Corollary 2.

If $f\in L^{1,0}(\mathbb{R}^{n})$ and $K,L\in\mathcal{P}_{0}^{n},$ then
$\langle f^{K}\rangle_{p}$ is a dilate of $K$ and

$|\langle f^{K}\rangle_{p}|=|\langle f^{L}\rangle_{p}|.$

###### Proof.

It is an immediate corollary of Lemma 4. Because from Lemma 4, we have

$\langle f^{K}\rangle_{p}=\xi_{f}\tilde{K},$

and

$\langle f^{L}\rangle_{p}=\xi_{f}\tilde{L}.$

We know from the definition that $|\tilde{K}|=|\tilde{L}|$, therefore
Corollary 2 follows. ∎

##  4\. Applications

###  4.1. General $L_{p}$ Affine Sobolev Inequalities.

The general $L_{p}$ Sobolev inequality established in [1] could be stated as
follows:

######  Theorem 8 ([1]).

Let $K\in\mathcal{K}_{0}^{n}$ be normalized such that $|K|=n\omega_{n}$. Then

$\alpha_{n,p}(\int_{\mathbb{R}^{n}}h(K,\nabla
f(x))^{p}dx)^{1/p}\geq(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}},$

for $1<p<n$ and $f\in W^{1,p}(\mathbb{R}^{n})$, where
$p^{\star}=\frac{np}{n-p}$. The optimal constants $\alpha_{n,p}$ are given by

(7)
$\alpha_{n,p}=n^{-1/p}(\frac{p-1}{n-p})^{1-1/p}(\frac{\Gamma(n)}{\omega_{n}\Gamma(\frac{n}{p})\Gamma(n+1-\frac{n}{p})})^{1/n}.$

By optimizing this inequality among all norms that have unit balls in
$\mathcal{P}_{0}^{n}$, we get the following result:

######  Theorem 9.

For $f\in L^{1,0}(\mathbb{R}^{n})$ with $1<p<n$, we have

$(\omega_{n})^{1/n}\alpha_{n,p}|\langle
f\rangle_{p}|^{\frac{n-p}{np}}\geq(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}},$

where the best constant $\alpha_{n,p}$ is defined in Theorem 8 and
$p^{\star}=\frac{np}{n-p}$.

###### Proof.

Since we have

$\alpha_{n,p}(\int_{\mathbb{R}^{n}}h(K,\nabla
f(x))^{p}dx)^{1/p}\geq(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}},$

for arbitrary $K\in\mathcal{K}_{0}^{n}$ with fixed volume $n\omega_{n}$, we
have, in particular,

$\alpha_{n,p}(\int_{\mathbb{R}^{n}}h(\tilde{\langle f\rangle_{p}},\nabla
f(x))^{p}dx)^{1/p}\geq(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}}.$

Therefore, we have

$\alpha_{n,p}|\langle
f\rangle_{p}|^{\frac{n-p}{np}}\geq(\omega_{n})^{-1/n}(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}}.$

∎

######  Theorem 10.

Given $0\leq\lambda\leq 1$ and a function $f\in W^{1,p}(\mathbb{R}^{n})$ with
$1<p<n$, we have

$2^{1/p}\alpha_{n,p}(\int_{S^{n-1}}(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)_{-}^{p})dx)^{-n/p}dv)^{-1/n}\geq(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}},$

where the best constant $\alpha_{n,p}$ is defined in Theorem 8 and
$p^{\star}=\frac{np}{n-p}$.

###### Proof.

For $f\in L^{1,0}(\mathbb{R}^{n})$, since we have

$|\langle
f\rangle_{p}|^{\frac{n-p}{np}}\geq(\omega_{n})^{-1/n}\alpha_{n,p}^{-1}(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}},$

by using Theorem 3 we have

$|\Phi_{\lambda,p}\langle f\rangle_{p}|^{-1}\geq\alpha|\langle
f\rangle_{p}|^{n/p-1},$

where $\Phi_{\lambda,p}(\langle f\rangle_{p})$ is defined through equation (4)
as:

$\Phi_{\lambda,p}(\langle f\rangle_{p})=(1-\lambda)\cdot\Pi_{p}^{+}(\langle
f\rangle_{p})+_{p}\lambda\cdot\Pi_{p}^{-}(\langle f\rangle_{p}),$

with $0\leq\lambda\leq 1$ and $\alpha$ is the best possible constant depending
on $p,n$.

Combining the above inequalities we get

$|\Phi_{\lambda,p}\langle
f\rangle_{p}|^{-\frac{1}{n}}\geq\alpha(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}}.$

Using the polar formula, we get

$(\int_{S^{n-1}}(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx)^{-{n/p}}dv)^{-1/n}\geq\alpha(\int_{\mathbb{R}^{n}}|f|^{p^{\star}}dx)^{1/{p^{\star}}},$

where $\alpha$ is the best possible constant depending on $n,p.$

For general $f\in W^{1,p}(\mathbb{R}^{n})$, we choose a sequence $f_{k}\in
L^{1,0}(\mathbb{R}^{n}),$ such that $f_{k}\to f$ in $W^{1,p}(\mathbb{R}^{n})$.
We know that

$\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})_{+}^{p}+\lambda(D_{v}f_{k})^{p})dx\to\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p})dx$

pointwise for all $v\in S^{n-1}$ as $k\to\infty$.

Since both
$(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})_{+}^{p}+\lambda(D_{v}f_{k})^{p}_{-})dx)^{1/p}$
and
$(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)_{-}^{p})dx)^{1/p}$
are support functions of convex bodies, we know from Lemma 1 that

$\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})_{+}^{p}+\lambda(D_{v}f_{k})^{p})dx\to\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p})dx$

uniformly for all $v\in S^{n-1}$ as $k\to\infty.$

Therefore, using Fatou’s lemma we arrive at

$(\int_{S^{n-1}}(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx)^{-n/p}dv)^{-1/n}\geq\alpha(\int_{\mathbb{R}^{n}}|f|^{p^{\star}})^{1/{p^{\star}}}$

for all $f\in W^{1,p}(\mathbb{R}^{n})$, where $\alpha$ is the best possible
constant depending on $n,p.$

By calculating the optimal constant $\alpha$, we conclude the proof. ∎

###  4.2. General Affine Pólya-Szegö Principles.

While the usual Pólya-Szegö principles states that the Dirichlet integral does
not increase by symmetric rearrangement. The affine Pólya-Szegö principles
states the affine energy, which is affine invariant, defined in [22] does not
increase by symmetric rearrangement. Yet the affine Pólya-Szegö principles is
stronger than the usual Pólya-Szegö principles in the sense that the affine
Pólya-Szegö principles implies the usual Pólya-Szegö principles by the Hölder
inequality.

####  4.2.1. The case $p\neq n$.

We assume in this subsection that $1<p<\infty$ and $p\neq n$. We will deal
with the case $p=n$ in later parts because we need a different normalization
factor.

In [1] the authors studied convex symmetrization for general norms. In terms
of support functions of convex bodies, the result can be stated as follows:

######  Theorem 11 ([1]).

For $K\in\mathcal{K}_{0}^{n},$

$\int_{\mathbb{R}^{n}}h(K,\nabla
f(x))^{p}dx\geq\int_{\mathbb{R}^{n}}h(K,\nabla f^{K}(x))^{p}dx$

for all $f\in L^{1,0}(\mathbb{R}^{n}).$

As an application of the above theorem, we have the following lemma:

######  Lemma 5.

Given $f\in L^{1,0}(\mathbb{R}^{n}),$ we have

$\langle f\rangle_{p}=(1+\alpha)\langle f^{\langle f\rangle_{p}}\rangle_{p}$

for some $\alpha\geq 0.$

###### Proof.

Since we know

$\displaystyle n\int_{\mathbb{R}^{n}}h(K,\nabla f(x))^{p}dx$ $\displaystyle=$
$\displaystyle\int_{S^{n-1}}h(K,u)^{p}dS(\langle f\rangle_{p},u)$
$\displaystyle=$ $\displaystyle nV_{p}(\langle f\rangle_{p},K).$

Similarly, we have

$\displaystyle n\int_{\mathbb{R}^{n}}h(K,\nabla f^{K}(x))^{p}dx$
$\displaystyle=$ $\displaystyle\int_{S^{n-1}}h(K,u)^{p}dS(\langle
f^{K}\rangle_{p},u)$ $\displaystyle=$ $\displaystyle nV_{p}(\langle
f^{K}\rangle_{p},K).$

Since from Lemma 4 we know that $\langle f^{K}\rangle_{p}$ is a dilation of
$K$, we have, by choosing $K=\langle f\rangle_{p}$ in the above equation and
using Theorem 11 and Theorem 6, that

$V_{p}(\langle f\rangle_{p},\langle f^{\langle f\rangle_{p}}\rangle_{p})\leq
V_{p}(\langle f\rangle_{p},\langle f\rangle_{p}).$

This implies that

$\langle f\rangle_{p}=(1+\alpha)\langle f^{\langle f\rangle_{p}}\rangle_{p}$

for some $\alpha\geq 0.$ ∎

We are ready to prove the following general affine Pólya-Szegö principle. We
remark that the case $\lambda=\frac{1}{2}$ is due to Cianchi, Lutwak, Yang and
Zhang [5] and the case $\lambda=0$ is due to Haberl, Schuster and Xiao [9].

######  Theorem 12.

Given $f\in W^{1,p}(\mathbb{R}^{n})$, if we denote

$\Omega_{\lambda,p}(f)=\alpha_{n,p}(\int_{S^{n-1}}(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx)^{-n/p}du)^{-1/n},$

where $0\leq\lambda\leq 1$,
$\alpha_{n,p}=(n\omega_{n})^{1/n}(\frac{n\omega_{n}\omega_{p-1}}{\omega_{n+p_{2}}})^{1/p}$
and $\omega_{n}=\frac{\pi^{n/2}}{\Gamma(1+\frac{n}{2})},$ we have

$\Omega_{\lambda,p}(f)\geq\Omega_{\lambda,p}(f^{\star}).$

###### Proof.

If $f\in L^{1,0}(\mathbb{R}^{n})$, by Lemma 5, we have

$\langle f\rangle_{p}=(1+\alpha)\langle f^{\langle f\rangle_{p}}\rangle_{p}$

for some $\alpha\geq 0.$ Therefore $\Phi_{\lambda,p}(\langle
f\rangle_{p})\supseteq\Phi_{\lambda,p}(\langle f^{\langle
f\rangle_{p}}\rangle_{p}),$ where $\Phi_{\lambda,p}(\langle f\rangle_{p})$ is
defined through Equation (4). It is not difficult to see that

$|\Phi_{\lambda,p}(\langle f\rangle_{p})|\leq|\Phi_{\lambda,p}(\langle
f^{\langle f\rangle_{p}}\rangle_{p})|,$

therefore we have

$\Omega_{\lambda,p}(f)\geq\Omega_{\lambda,p}(f^{\langle f\rangle_{p}}).$

On the other hand, by Lemma 4 and Theorem 3, we have

$|\Phi_{\lambda,p}(\langle f^{\langle
f\rangle_{p}}\rangle_{p})|\leq|\Phi_{\lambda,p}(\langle
f^{\star}\rangle_{p})|,$

which implies

$\Omega_{\lambda,p}(f^{\langle
f\rangle_{p}})\geq\Omega_{\lambda,p}(f^{\star}).$

Therefore, we have

$\Omega_{\lambda,p}(f)\geq\Omega_{\lambda,p}(f^{\star}),$

for $f\in L^{1,0}(\mathbb{R}^{n}).$

For general $f\in W^{1,p}(\mathbb{R}^{n})$, consider an approximation sequence
$f_{k}\in L^{1,0}(\mathbb{R}^{n})$ in the $W^{1,p}(\mathbb{R}^{n})$ norm. We
know that

$\Omega_{\lambda,p}(f_{k}^{\star})\leq\Omega_{\lambda,p}(f_{k})~{}~{}~{}~{}for~{}k\in
N.$

We know that

$\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})_{+}^{p}+\lambda(D_{v}f_{k})^{p}_{-})dx\to\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx$

pointwise for all $v\in S^{n-1}$ as $k\to\infty$.

Since both
$(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})^{p}_{+}\lambda(D_{v}f_{k})^{p}_{-})dx)^{1/p}$
and
$(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx)^{1/p}$
are support functions of convex bodies, we know from Lemma 1 that

$\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})_{+}^{p}+\lambda(D_{v}f_{k})^{p}_{-})dx\to\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx$

uniformly for all $v\in S^{n-1}$ as $k\to\infty.$

Moreover, the function

$v\to(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx)^{1/p}$

is strictly positive and continuous on $S^{n-1}$, hence attains a positive
minimum on $S^{n-1}$. Consequently,

$(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f_{k})^{p}_{+}+\lambda(D_{v}f_{k})^{p})dx)^{-\frac{n}{p}}\to(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{p}_{+}+\lambda(D_{v}f)^{p}_{-})dx)^{-\frac{n}{p}}$

uniformly for $v\in S^{n-1}$ as $k\to\infty$. Therefore

$\lim_{k\to\infty}\Omega_{\lambda,p}(f_{k})=\Omega_{\lambda,p}(f).$

On the other hand, $f^{\star}\to f^{\star}$ in $L^{p}(\mathbb{R}^{n})$,
because of the contractivity of the spherically symmetric rearrangement in
$L^{p}(\mathbb{R}^{n}).$ Hence we get that $f_{k}^{\star}\to f^{\star}$ weakly
in $W^{1,p}(\mathbb{R}^{n}).$ Since
$\Omega_{\lambda,p}(f_{k}^{\star})=\|\nabla f_{k}^{\star}\|_{p}$ and
$\Omega_{\lambda,p}(f^{\star})=\|\nabla f^{\star}\|_{p}$, and since the
$L^{p}(\mathbb{R}^{n})$ norm of the gradient is lower semi-continuous with
respect to weak convergence in $W^{1,p}(\mathbb{R}^{n}),$

$\liminf_{k\to\infty}\Omega_{\lambda,p}(f_{k}^{\star})\geq\Omega_{\lambda,p}(f^{\star}).$

Therefore, we have

$\Omega_{\lambda,p}(f^{\star})\leq\Omega_{\lambda,p}(f),$

for all $f\in W^{1,p}(\mathbb{R}^{n}).$ ∎

####  4.2.2. The case $p=n$.

For dealing with the case $p=n$, we use the following normalized version of
the discrete functional $L_{p}$ Minkowski problem.

######  Theorem 13.

Given a function $f\in L^{1,n}(\mathbb{R}^{n})$, there exists a unique convex
polytope $\langle f\rangle_{n}\in\mathcal{P}_{0}^{n}$ such that

$\int_{\mathbb{R}^{n}}\Psi^{n}(-\nabla f(x))dx=\frac{1}{|\langle
f\rangle_{n}|}\int_{S^{n-1}}\Psi(u)^{n}dS_{n}(\langle f\rangle_{n},u),$

for every continuous function $\Psi:\mathbb{R}^{n}\to[0,\infty)$ that is
homogeneous of degree $1$.

We remark that we omit the proof of Theorem 13, because it is essentially the
same as the proof of Theorem 6. We remark that Lemma 6 still holds. We will
not give a proof of it. The proof is essentially the same as the proof of
Lemma 5, if as we define the normalized $L_{n}$ optimal Sobolev body $\langle
f^{K}\rangle_{n}$ as the convex body satisfying

$\int_{\mathbb{R}^{n}}\Psi^{n}(-\nabla f^{K}(x))dx=\frac{1}{|\langle
f^{K}\rangle_{n}|}\int_{S^{n-1}}\Psi(u)^{n}dS_{n}(\langle
f^{K}\rangle_{n},u),$

for every continuous function $\Psi:\mathbb{R}^{n}\to[0,\infty)$ that is
homogeneous of degree $1$.

######  Lemma 6.

Given $f\in L^{1,0}(\mathbb{R}^{n})$, we have

$\langle f^{\langle f\rangle_{n}}\rangle_{n}=(1+\alpha)\langle f\rangle_{n}$

for some $\alpha\geq 0.$

Given $0\leq\lambda\leq 1$, we define the normalized general $L_{n}$
projection body $\tilde{\Phi}_{n}K$ of $K$ by:

$h^{n}(\tilde{\Phi}_{\lambda,n}(K),v)=\frac{1}{|K|}\int_{S^{n-1}}((1-\lambda)(u\cdot
v)^{n}_{+}+\lambda(u\cdot v)^{n}_{-})dS_{n}(K,u).$

We need the following normalized version of the general $L_{n}$ Petty
projection inequality.

######  Lemma 7 ([20]).

If $K$ is a convex body, then

$\frac{|\tilde{\Phi}_{\lambda,n}K|}{|K|}\leq\frac{|\tilde{\Phi}_{\lambda,n}B_{2}|}{|B_{2}|},$

and equality holds if and only if $K$ is an ellipsoid.

With all these tools at hand, we establish the following theorem. We remark
again that the case $\lambda=\frac{1}{2}$ is due to Cianchi, Lutwak, Yang and
Zhang [5] and the case $\lambda=0$ is due to Haberl, Schuster and Xiao [9].

######  Theorem 14.

Given $f\in W^{1,n}(\mathbb{R}^{n})$, we have

$\Omega_{\lambda,n}(f)\geq\Omega_{\lambda,n}(f^{\star}),$

if we denote

$\Omega_{\lambda,n}(f)=2^{1/n}\alpha_{n}(\int_{S^{n-1}}(\int_{\mathbb{R}^{n}}((1-\lambda)(D_{v}f)^{n}_{+}+\lambda(D_{v}f)^{n}_{-})dx)^{-1}dv)^{-1/n},$

where
$\alpha_{n}=(n\omega_{n})^{1/n}(\frac{n\omega_{n}\omega_{n-1}}{2\omega_{2n-2}})^{1/n}.$

###### Proof.

By Lemma 6, we have $\langle f^{\langle
f\rangle_{n}}\rangle_{n}\supseteq\langle f\rangle_{n}.$ Therefore, we have

$|\tilde{\Phi}^{\circ}_{\lambda,n}\langle f^{\langle
f\rangle_{n}}\rangle_{n}|^{-\frac{1}{n}}\leq|\tilde{\Phi}^{\circ}_{\lambda,n}\langle
f\rangle_{n}|^{-\frac{1}{n}},$

for $f\in L^{1,n}(\mathbb{R}^{n})$. By Lemma 7 we have

$|\tilde{\Phi}^{\circ}_{\lambda,n}\langle
f^{\star}\rangle_{n}|^{-\frac{1}{n}}\leq|\tilde{\Phi}^{\circ}_{\lambda,n}\langle
f^{\langle
f\rangle_{n}}\rangle_{n}|^{-\frac{1}{n}}\leq|\tilde{\Phi}^{\circ}_{\lambda,n}\langle
f\rangle_{n}|^{-\frac{1}{n}},$

which implies

$\Omega_{\lambda,n}(f)\geq\Omega_{\lambda,n}(f^{\star})~{}~{}for~{}all~{}f\in
L^{1,n}(\mathbb{R}^{n}).$

For general $f\in W^{1,n}(\mathbb{R}^{n})$, the approximation argument is the
same as in Theorem 12. ∎

###  4.3. General Affine Sobolev Type Inequalities.

For the symmetric case, the affine energy coincides with the Dirichlet
integral:

######  Lemma 8.

Given $f\in W^{1,p}(\mathbb{R}^{n})$ with $1<p<\infty$ and $0\leq\lambda\leq
1$, we have

$\Omega_{\lambda,p}(f^{\star})=\|\nabla f^{\star}\|_{p}.$

We omit the proof of Lemma 8 because it is essentially the same as that in
[9].

With the general affne Pólya-Szegö principle, Theorem 12 and Theorem 14 at
hand, it is straightforward to get some affine Sobolev type inequalities which
generalize the results in [9]. We will list some (and not all) of them here.
Because the proofs are essentially the same as those in [9], we will omit
them.

######  Corollary 3 (General affine Moser-Trudinger inequalities).

If $f\in W^{1,n}(\mathbb{R}^{n})$ with $0<|sprtf|<\infty$ and
$0\leq\lambda\leq 1$, then

(8)
$\frac{1}{|sprtf|}\int_{sprtf}\exp(\frac{n\omega_{n}^{1/n}|f(x)|}{\Omega_{\lambda,n}(f)})^{\frac{n}{n-1}}dx\leq
m_{n}.$

Here $sprtf$ is the support of $f$ and the constant $n\omega_{n}^{1/n}$ is
optimal, in that equation (7) would fail for any real number $m_{n}$ if
$n\omega_{n}^{1/n}$ were to be replaced by a larger number. And the best
constant $m_{n}$ is characterized as

$m_{n}=\sup_{g}\int_{0}^{\infty}\exp(g(t)^{\frac{n}{n-1}}-t)dt,$

where the supremum ranges over all nondecreasing and locally absolutely
continuous functions $g$ on $[0,\infty)$ such that $g(0)=0$ and
$\int_{0}^{\infty}g^{\prime}(t)^{n}dt\leq 1.$

######  Corollary 4 (General affine Morrey-Sobolev inequalities).

If $f\in W^{1,p}(\mathbb{R}^{n})$, $p>n,$ such that $|sprtf|<\infty$, then

$\|f\|_{\infty}\leq\alpha_{n,p}|sprtf|^{\frac{p-n}{np}}\Omega_{\lambda,p}(f),$

where
$\alpha_{n,p}=n^{-1/p}\omega_{n}^{-1/n}(\frac{p-1}{p-n})^{\frac{p-1}{p}}$ is
the best constant.

Remark: Everything that is done in this paper has a counterpart for the case
$p=1$. The extension is similar to what we’ve done in this paper, as long as
one observes

$\int_{\mathbb{R}^{n}}\nabla f(x)dx=0$

for every $f\in W^{1,1}(\mathbb{R}^{n}).$

##  5\. Acknowledgements

The work of the author was supported by Austrian Science Fund (FWF) Project
P25515-N25.

## References

  * [1] A. Alvino; V. Ferone; P.L. Lions; G. Trombetti, _Convex symmetrization and applications._ Ann. Inst. H. Poincare, Anal. Non Lineaire 14, 275-293 (1997).
  * [2] K.J. Boroczky, _Stronger versions of the Orlicz-Petty projection inequality._ Submitted.
  * [3] S. Campi; P. Gronchi, _The $L_{p}$-Busemann-Petty centroid inequality,_ Adv. Math. 167 (2002), 128-141.
  * [4] K.-S. Chou; X.-J. Wang, _The $L_{p}$-Minkowski problem and the Minkowski problem in centroaffine geometry_, Adv. Math. 205 (2006), 33-83.
  * [5] A. Cianchi; E. Lutwak; D. Yang; G. Zhang, _Affine Moser-Trudinger and Morrey-Sobolev inequalities_ , Calc. Var. Partial Differ. Equ., 36 (2009), 419-436.
  * [6] R.J. Gardner, _Geometric tomography._ Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 1995.
  * [7] C. Haberl; F.E. Schuster, _General $L_{p}$ affine isoperimetric inequalities_, J. Differential Geom., 83 (2009), 1-26.
  * [8] C. Haberl; F.E. Schuster, _Asymmetric affine $L_{p}$ Sobolev inequalities_, J. Functional Anal. 257 (2009), 641-658.
  * [9] C. Haberl; F.E. Schuster; J. Xiao, _An Asymmetric affine Polya-Szegoe Principle,_ Math. Ann., 352 (2012), 517-542.
  * [10] D. Hug; E. Lutwak; D. Yang; G. Zhang, _On the $L_{p}$ Minkowski problem for polytopes_, Discrete Comput. Geom. 33 (2005), 699-715.
  * [11] M. Ludwig, _Ellipsoids and matrix-valued valuations_ , Duke Math. J. 119 (2003), 159-188.
  * [12] M. Ludwig, _Minkowski valuations,_ Trans. Amer. Math. Soc. 357 (2005), 4191-4213.
  * [13] M. Ludwig, _Valuations on Sobolev spaces_ , Amer. J. Math. 134 (2012), 824-842.
  * [14] M. Ludwig; M. Reitzner, _A classification of $SL(n)$ invariant valuations_, Ann. Math., 172, 1219 - 1271 (2010).
  * [15] E. Lutwak, _On the affine isoperimetric inequality._ Differential geometry and its applications (Brno, 1989), 109-118, World Sci. Publ., Teaneck, NJ, 1990.
  * [16] E. Lutwak, _On some affine isoperimetric inequalities_ , J. Differential Geom. 23 (1986), 1-13.
  * [17] E. Lutwak, _The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem_ , J. Differential Geom. 38 (1993), 131-150.
  * [18] E. Lutwak, _The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas_ , Adv. Math. 118 (1996), 244-294.
  * [19] E. Lutwak; V. Oliker, _On the regularity of solutions to a generalization of the Minkowski problem_ , J. Differential Geom. 41 (1995), 227-246.
  * [20] E. Lutwak; D. Yang; G. Zhang, _Optimal Sobolev norms and the $L_{p}$ Minkowski problem._ Int. Math. Res. Not., 2006, Art. ID 62987, 21 pp.
  * [21] E. Lutwak; D. Yang; G. Zhang, _$L_{p}$ affine isoperimetric inequalities_, J. Differential Geom. 56 (2000), 111-132.
  * [22] E. Lutwak; D. Yang; G. Zhang, _Sharp affine $L_{p}$ Sobolev inequalities_, J. Differential Geom. 62 (2002), 17-38.
  * [23] E. Lutwak; D. Yang; G. Zhang, _A new ellipsoid associated with convex bodies_ , Duke Math. J. 104 (2000), 375-390.
  * [24] E. Lutwak; D. Yang; G. Zhang, _On the Lp Minkowski problem_ , Trans. Amer. Math. Soc. 356 (2004), 4359-4370.
  * [25] C.M. Petty, _Isoperimetric problems._ Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971), pp. 26-41.
  * [26] R. Schneider, _Convex bodies: the Brunn-Minkowski theory._ Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.
  * [27] T. Wang, _The affine Sobolev-Zhang inequality on $BV(\mathbb{R}^{n})$_, Adv. Math. 230 (2012), 2457-2473.
  * [28] T. Wang, _Semi-Valuations on $BV(\mathbb{R}^{n})$_, submitted.
  * [29] T. Wang _The affine Polya-Szego principle: equality cases and stability_ , J. Functional Anal. 265 (2013), 1728-1748.
  * [30] G. Zhang, _The affine Sobolev inequality_ , J. Differential Geom. 53 (1999), 183-202.


# A Study on Trees’s Knots Prediction from their Bark Outer-Shape

Mohamed Mejri1,Antoine Richard 2
School of Electrical and Computer Engineering, Georgia Institute of
Technology, Atlanta, USA1,2
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Cédric Pradalier3
UMI2958 GT-CNRS, Metz, France3
<EMAIL_ADDRESS>
###### Abstract

In the industry, the value of wood-logs strongly depends on their internal
structure and more specifically on the knots’ distribution inside the trees.
As of today, CT-scanners are the prevalent tool to acquire accurate images of
the trees internal structure. However, CT-scanners are expensive, and slow,
making their use impractical for most industrial applications. Knowing where
the knots are within a tree could improve the efficiency of the overall tree
industry by reducing waste and improving the quality of wood-logs by-products.
In this paper we evaluate different deep-learning based architectures to
predict the internal knots distribution of a tree from its outer-shape,
something that has never been done before. Three types of techniques based on
Convolutional Neural Networks (CNN) will be studied. The architectures are
tested on both real and synthetic CT-scanned trees. With these experiments, we
demonstrate that CNNs can be used to predict internal knots distribution based
on the external surface of the trees. The goal being to show that these
inexpensive and fast methods could be used to replace the CT-scanners.
Additionally, we look into the performance of several off-the-shelf object-
detectors to detect knots inside CT-scanned images. This method is used to
autonomously label part of our real CT-scanned trees alleviating the need to
manually segment the whole of the images.

##  1 Introduction

Knots are one of the most important factors in the wood-processing chain-
value. A knot is the portion of a branch embedded in the trunk that usually
originates at the trunk’s pith [26]. The knots’ size, shape and distribution
depend on the tree specie and are influenced by environmental factors [9]. For
both forestry experts and saw-millers, knowing the exact knot distribution
inside wood-logs is crucial as it directly impacts the value of the logs. When
processing trees, the most valuable parts are long straight continuous portion
of wood without knots because these chunks are used to make wood planks. The
rest of the trees are usually processed to make small wood pieces used in
furniture or shavings that are later used to make either paper or chipboard
wood. Their values are significantly inferior to wood planks.

Nowadays, CT-scanners are the way to go to acquire accurate representation of
the trees internal density. Yet, these machines are prohibitively expensive
and take a long time to image a whole tree. Furthermore, extracting the knots
out of the CT-scanned images is still an active field of research: modern
methods struggle to extract knots out of all tree species. This motivates our
work to predict the knot distribution inside a tree from its outer-shape which
can be acquired using fast and cheap sensors such as 2D-laser scanners. A
representation of the tree’s outer surface is fed to the network that outputs
a representation of the knots. The choice for data representation depends on
the network’s architecture. Such learning is possible thanks to the strong
correlation between surface defects on the tree’s bark and the tree’s internal
knot distribution as observed in [28]. Based on this observation, we first
extract the knots from the CT-scanned images using an object detection
algorithm. Once we have extracted the knots from the CT-scanned images, we
train various neural network architectures to predict the knots distribution
using the outer-shape of trees. More precisely, three different data
representation will be studied. First, we study the 2D architectures: they
take as an input a patch of the log’s surface map and output the density of
the wood-log on the plane normal to the surface. Then we use 3D architectures:
they take as an input a voxel that represents the whole log’s surface and
output a voxel that contains the knots. Finally, we try Convolutional-LSTM
(CLSTM) networks, the input is similar to the 3D networks but the data is read
layer by layer starting from the top of the log to its bottom. To the best of
our knowledge, it is the very first time that a model is used to predict the
internal knot distribution from the outer-shape of a tree. In contrast,
current methods are applied solely on CT-scanned images to detect the knots.

The challenge of predicting the knots from the outer-shape arises from the
high variability of the surface defects that indicates the presence of knots
underneath them. Hence, to predict the knots distribution of a wood-log from
its external shape, deep learning algorithms require a large dataset with many
real CT-scanned logs and their corresponding outer-shape, and labeled knots.
However, this data is expensive and difficult to acquire. Thus, we propose to
pre-train our models on a synthetic dataset and then fine-tune them on a small
dataset acquired from real-trees. To acquire the real dataset we use a hybrid
method. We first label some knots out of the real CT-scanned images, then we
train an object detector on it and use the said object detector to detect
knots in the remaining CT-scanned images. Furthermore, we compare how
different state-of-the-art detectors fare on this sample limited task.

In the end, this paper’s contributions are the following:

  * •

Using Convolutional Neural Networks (CNN), we demonstrate that we can predict
the internal knot density of a tree from its external shape. Something that
has never been done before.

  * •

We evaluate different data representations to find the ones that are best
suited to solve the aforementioned task.

  * •

We evaluate various object detectors in a sample limited scenario to detect
knots inside CT-scanned images.

##  2 Related work

###  2.1 Detecting Knots from CT-Scanned Images

Detecting knots inside CT-scanned cross-sections of logs is an active field of
research, where the availability of the data and the high variability in the
knots appearance are key challenges. [18] and [27] propose a method to
automatically detect knots inside the heartwood of Scandinavian Scots pine
(Pinus sylvestrisL.), Norwayspruce (Picea abies(L.) Karst.) and Douglas-fir
(Pseudotsuga menziesii (Mirb.) Franco) logs. In [18] they suggest to model the
knots by non-concentric ellipses inside the log:First, they try to fit all the
knots (i.e, a region with low-density value) inside CT-scanned Roundwood to
ellipses. Then they tracked those ellipses across the cross-section of the
tree until reaching the tree bark.[21, 19] exploited the geometrical
properties of the wood combining discrete connected component
extractions.However, these methods heavily rely on user expertise to adjust
the parameters of the algorithms correctly. Additionally, they are only valid
on two tree species, the Scandinavian Scots Pine and the Douglas-fir. [29]
showed that deep learning techniques and specifically convolutional neural
networks outperformed a commercial detector based on feature descriptors and
SVMs when it comes to detecting knots in oak trees planks. Following the
observation made by [29], we decided to use neural networks architectures
instead of relying on the traditional computer vision methods usually used in
this field to locate the knots inside the CT-scanned wood-logs.

###  2.2 Object detection algorithms

As of today, a large variety of deep-learning based object detectors can be
used to find knots inside CT-scanned cross sections.Among those, we find two-
stages object detectors such as Fast R-CNN [11] and Faster-RCNN [17]; who
first propose a set of regions of interest in which deep feature maps are
cropped and classified. Although those algorithms achieve good performance,
they are time-consuming due to their cascaded pipeline. Unlike two-stage
detectors, a single-stage detector is faster and hence more adapted to real-
time object detection purposes. The Single Shot Detector (SSD) algorithm
introduced by [23] is one of them. It consists of a pre-trained backbone
network to extract a compressed feature map and an SSD head for predicting the
offsets to default boxes of different scales and aspect ratios and their
associated confidences. For all these approaches, a wide range of neural
network architecture can be used as a backbone structure. These include
Resnet50 [12] and mobilenet [15], among others. Another well known single
stage object-detector is yolo family [30, 31, 32, 4]. This family of models
offers a good trade-off between accuracy and inference time thanks to its
relatively simple decoder. Something that could be interesting in our case as
often simpler architectures are less prone to over-fitting and exhibit better
generalization capacities. Finally, the current state of the art network:
RetinaNet-500 [22] uses a FCN [24] followed by a ResNet to generate rich,
multi-scale feature maps.
Extracting the knots from CT-scanned wood-logs is important: it helps us build
our dataset, yet, predicting the knot distribution of the log from its
external shape stands as our main goal.

###  2.3 Pixel-wise regression methods

Predicting the knots distibution based on three outer-shape can be seen as a
pixel wise regression task. In this paper, we will focus on 2-D, 3-D, and
convolutional long short term memory (CLSTM) based encoder-decoders. However,
in our case, the encoder will find defects on the outer-shape of the tree, and
the decoder will reconstruct the inside of the tree.

####  2.3.1 2D encoder-decoder structures

2D Encoder-Decoder architectures such as SegNet [3], and U-Net [33] take an
image as an input, and, when used for pixel-wise regression, output a matrix
of floats of same height and width as the input. These architectures are
comprised of a contractive path, (the encoder) and an expansive path (the
decoder).

Fully Convolutional Networks such as DeepLab [8], FCN [25], and PSPNet [37]
rely on a very similar concept, however, the decoding process is slightly
different. The decoding is done using a spatial pooling pyramid, instead of
the expansive path. This results in higher accuracy in pixel-wise
classification but not in pixel-wise regression as we will show later. Indeed,
due to the upsampling mechanism used in the Spatial Pyramidal Pooling, the
regression results are not as good as SegNet [3]’s or U-Net [33]’s results.

####  2.3.2 3D encoder-decoder structures

With the advent of 3D data acquisition techniques, especially in medical
imaging, several 3D neural network architectures have been designed to perform
volumetric segmentation/regression. 3D-U-Net [38] is designed to perform
kidney [38] segmentation. It extends the U-Net architecture [33] by replacing
all 2D operations with their 3D counterparts. [6] introduced a voxel-wise
residual neural network based on residual neural network [13]. It consists of
three stacked residual modules followed by four 3D-deconvolutional layers.
According to [6], VoxResNet achieves better results than 3D-U-Net [38] after
being tested on MICCAI MRBrainS challenge data [6].

While those technics have their own advantages they each suffer of a major
drawback. The 2D encoder-decoder architectures cannot capture the correlation
between the successive cross sections of a log. And the volumetric
segmentation/regression networks are heavily bottlenecked by their huge memory
requirements which imposes low resolution input and output voxels.

####  2.3.3 Convolutional-LSTM based encoder-decoder structures

An alternative that alleviates both of these issues is to leverage the
advantage of the 2D encoders: their high resolution, and add a mean for them
to process sequences of input. A natural solution emmerges: Recurrent Neural
Network and, more specifically, Long Short Term Memory (LSTM) structures. This
family of network is widely used in natural language processing to achieve
sequence-to-sequence processing. Those neural networks aim to retrieve
correlations between words in sentences. [34] introduces Convolutional-LSTMs
(CLSTM), and [35]) based encoder-decoder structure.
The encoder structure generates feature-maps from the input images that are
fed to the CLSTM module. It consists of a Long Short Term Memory module where
the weights are replaced with a filter bank of a convolutional layer. The
decoder is composed of several deconvolutional [10] layers and combines the
outputs of different CLSTM modules and generates the segmentation map for the
next time-step. Bidirectional CLSTM based segmentation structures have also
been introduced and aim to capture the temporal information in both
directions.

In the following sections, we are going to test and evaluate all the
aforementioned architectures to predict the internal knot distribution of the
log from its outer-shape.

##  3 Method

Properly predicting the knots distribution inside trees is a challenging task
due to the limited amount of CT-scanned wood-log publicly available, and the
complexity of the logs’ internal structures. Hence, correctly extracting the
knots from the CT-scanned images is essential.

###  3.1 Feature identification and extraction

####  3.1.1 Internal Knot labeling from CT-scanned cross sections

According to forestry experts, a wide range of knots exists with different
spatial behaviours and shape [20]. In this paper, only the knots that start
from the pith center and extends to bark will be considered because of their
direct relationship with the tree bark’s shape. To assess which detector is
best suited to label CT-scanned wood-logs, various object detection algorithms
will be trained on a labeled subset(30$\%$) of real CT-scanned wood-logs of
four different tree species: fir, elm, ash, and aspen trees. The knots have
been modeled as ellipses within the labeled bounding boxes. A Faster R-CNN
[17] with an Inception-ResNet-v2 and a RetinaNet-500 [22] have been selected
as the best detectors based on their performance on the MS-COCO detection
challenge. They were trained on a previously annotated CT-scanned
subset(30$\%$ of the total dataset) of wood-logs of different tree species.
This model is then used to perform coarse automatic-labeling on the 70$\%$
remaining CT-scanned cross-section images. Although Object detectors can
localize the knots inside the wood-log by identifying their main features,
they do not consider the correlation of the position of the knots between
successive cross-section images. Object detectors can localize accurately the
knots inside wood-log. However, round-woods usually contain more than one
knot, and one of them is often overshadowed by other defects due to its
relatively small size or low density. From a biological point of view, knots
are a continuous defect across the headwood and hence their position might be
tracked along the round-woods. Therefore, we proposed to use tracking
methods(e.g,CSR algorithms) to detected missed knots. Furthermore, to
guarantee consistency in knots size along the log’s height, a median filter is
applied, so that very small or huge ellipses are filtered.

####  3.1.2 External surface extraction from CT-scanned cross sections

To extract the tree’s external shape, we used a modified version of Canny
algorithm[5]. Common edge detection algorithms such as Canny combined with
filtering and other processing are often used to detect edges in Computer
Vision. However, applying that algorithm failed to extract only the tree bark
shape since it considers both the tree bark and the growth rings of the log as
a potential edge. A solution to this is a thresholding along different axes
crossing the center of the tree to extracts the shape of the tree bark
accurately.
The extraction of the surfaces is challenging due to the high noise level that
is formed by cracking and splitting of the tree bark, small branches,etc.
Hence, we decided to use an object detection neural network to extract
relevant extrusion on the bark surface and that might be an extension of a
knot inside the log. Unlike the knots extraction that requires the tracking
process, the position of the external defects remains relatively unchanged
when moving throughout the cross-sections.

###  3.2 wood-log density prediction from tree bark shape

As our current database of real CT-scanned logs is limited, we proposed to
design a look-like CT-scanned dataset based on mathematical knot modelling
[1]. All the deep learning based model used to predict the internal density of
the logs based on the their external shape will be first be applied on our
synthetic dataset to assess the performance of the models and to pretrain them
on a large scale dataset. To predict the internal density we propose 3
different strategies that we present in the following subsubsection.

####  3.2.1 2D neural network architectures

Figure 1: Different steps for 2D prediction and 3D reconstruction

The first strategy relies on predicting a slice of the density
$d=f(r,\theta=\theta_{0},z)$ within the radial longitudinal plane using its
correspondent surface
$s=S(\theta\in[\theta_{0}-\frac{\pi}{12},\theta_{0}+\frac{\pi}{12}],z)$ using
auto-encoders like architectures. The figure 1 illustrates the prediction
pipeline used in image-to-image predictions. As stated in the previous
section, three variety of 2D-Encoder decoder will be used: SegNet [3], U-Net
[33], and DeepLabV3+ [8]. To use SegNet architecture as a pixel-wise
regression encoder-decoder, we use an $L2$ loss(mean-squared error).
Furthermore, we change the depth of the encoder and the decoder of the
original SegNet due to the low-resolution input to prevent the model from
overfitting. We remove the last and the first convolutional block respectively
from the encoder and the decoder architectures. Additionally, we replace the
rectified linear function (ReLU) with parametric-rectified linear function
$P-ReLU$[14] which is a learned-slope value version of leaky-ReLU. It improves
model convergence while avoiding over-fitting issues. SegNet is known to be a
very Deep-Network, often in Deep Networks, part of the information is lost due
to a large amount of operation the data has to go through. This can results in
degraded performance. Hence, we teste other encoder-decoder architectures that
feature skip-connections which alleviate the very-deep networks issue.
U-Net[33] is one of them. Here, due to low-resolution constraints, the
original contracting and expanding path of the U-Net[33] architecture are
reduced (i.e., the 1024 filters convolution layer is removed). The DeepLabV3+
[8] is a revisited version of DeepLab [7] architecture containing the
following structures: Atrous Spatial Pooling Pyramids, Xception blocks,Depth-
wise separable convolutions. In the original implementation of DeepLabV3+ [8],
the L2 regularization strategy aims to avoid overfitting with a weight decay
of $4e-5$. Due to the instability issue while using DeepLabV3+ [8] for
regression, we add dropout at the end of all the convolution layers instead of
performing L2 regularization. This modification help stabilize the network’s
training.

####  3.2.2 3D neural network architectures

In this section, we present various 3D architectures, they take as an input
the outer-shape of the tree as a voxel, and they output the knot distribution
as a voxel. The three dimensions of the voxels represent the Cartesian
coordinates x, y, and z. For the input the value of a cell is the distance
between that cell and the center of the tree. For the output the cells values
are the internal density.

For this data reprentation we test three different deep learning models: 3D-U-
Net [38], 3D-SegNet and VoxResNet [6]. 3D-SegNet is inspired by 2D-SegNet [3]:
the 2D layers have been replaced with their 3D counterpart and the
encoder/decoder have been shorten to reduce the memory, time expenses, and
prevent overfitting. More precisely, in comparison to the original SegNet who
is based on VGG-16 [36] the last three convolution layers whose depth are 512
have been removed in both the encoder and the decoder. As in the original
SegNet architecture, a Batch-normalization layer, and a Parametric-ReLU
activation follow each convolution layer. To prevent overfitting, a dropout of
0.1 is added at the end of each convolution layer.

Regarding 3D-U-Net [38] we kept the original structure but made some changes
in the loss function and overfitting strategy. The original 3D-U-Net [38] is
trained on partially annotated voxels. Hence, the loss function has to be
weighted with zero weight when the data is not annotated and one otherwise. In
our case, we are performing regression with a fully-annotated dataset; we use
unweighted mean-squared as a loss function. Furthermore, to prevent
overfitting, a dropout of 0.1 is added after each 3D-convolution layer.

One advantage of VoxResNet [6] over 3D-U-Net [38] is the ability to build
deeper encoder and decoder networks while avoiding overfitting. We do not make
a significant change in the original architecture. We remove the final
n-classes projection layer and keep the multi-level contextual information
consisting of 4 3D-deconvolution layers with respectively 1,2,4 and 8 degrees
of strides. Dropout of 0.1 is added after each convolutional layer to prevent
overfitting.

###  3.3 CLSTM based network architectures

The CLSTMs take as an input a sequence of cross section containing the outer-
shape of the tree and output a cross section with the predicted knot inside.
We have modified the original architecture due to computational limitation.
The SegNet [3] structure is used to build the encoder and the decoder of this
model. To capture the correlation between the different cross-sections of the
internal density of the log while reducing the computation costs, we only keep
the CLSTM[35] layer at the end of the encoder (bottleneck). To avoid
overfitting, a dropout of 0.1 is added after each convolutional layer and
inside the CLSTM Layer.

Intuitively the correlation between a successive cross-section of log exists
in forward and backward directions. To capture the correlation in both
directions, the bidirectional CLSTM version of SegNet [3] could be useful. It
has almost the same architecture as the original CLSTM based SegNet. The main
difference consists of a Bidirectional layer that replaces the original CLSTM
layer. Finally, to prevent overfitting, we decided to increase the dropout
from 0.1 to 0.2 inside the CLSTM layer.

##  4 Experiments

###  4.1 Dataset

####  4.1.1 Synthetic look-like CT-scanned dataset

The dataset is composed of 1800 synthetic logs. For the 3D network and the
CLSTMs the inputs have a size of 256x256x64. For the 2D networks the input
size is 64x64. While it may seems that the resolution of the 2D network data
is lower it is not: the input of the 2D network is an angular slice of 15
degrees of the trees’ surface.

The dataset is balanced in terms of the number of knots per logs. It is
composed of 300 k-knots logs where k varies between 2 and 7. Based on the
observations made in [2], we chose to represent the tree surface as a function
$s(r,\theta,z)$ where $\theta$ and $r$ refer to the polar coordinate of the
tree and $z$ is the height.To add texture onto our synthetic tree we add high
and low-frequencies cosines. This creates a pattern somewhat similar to an oak
bark. The longitudinal shape of the log is modeled as a decreasing function
starting from its basis. Furthermore, The branches, from which the knots
originates are modeled as a 2D-Gaussian function. We model the internal
density of the log as a function $d(r,\theta,z)$. The branch, a region with
high density, is modeled as a combination of square root functions and linear
functions. The figure 2 illustrates two different sections of a log, as well
as two external surfaces projection.

Figure 2: From left to right: surface with a branch, surface without branch,
longitudinal section of a log, cross section of a log

We provide codes and scripts for reproducibility.111
https://github.com/WACV2021SyntheticTree/synthetic-dataset.git

####  4.1.2 Real CT-scanned wood-log dataset

The dataset consist of four log of different species: fir,elm,ash and aspen
tree. Each log is comprised of a sequence of 831 to 862 512x512 high
resolution CT-scanned cross-section images. Each CT-scanned image is
translated and cropped so that the pith center and the image center coincide.
Then each wood-log image is smoothed through a 3x3 kernel size median filter
to remove the noise generated by the acquisition process.

###  4.2 Network training and validation

Table 1: Mean Average Prediction results of the different object detection algorithms on four tree species. Higher is better. | Ash tree | Fir tree | Elm tree | Aspen tree
---|---|---|---|---
Object detector | $mAP_{50}$ | $mAP_{75}$ | $mAP_{50}$ | $mAP_{75}$ | $mAP_{50}$ | $mAP_{75}$ | $mAP_{50}$ | $mAP_{75}$
Faster R-CNN [17] | 35.2 | 34.7 | 34.2 | 32.7 | 36.0 | 33.4 | 30.3 | 28.7
SSD-513 [23] | 25.9 | 24.7 | 26.9 | 23.7 | 23.6 | 22.7 | 21.4 | 20.1
RetinaNet-500 [22] | 43.7 | 40.9 | 42.7 | 41.9 | 44.3 | 41.2 | 36.9 | 34.7
YoloV3-608 [32] | 37.4 | 32.6 | 36.4 | 33.6 | 38.9 | 36.3 | 31.0 | 29.8

####  4.2.1 Object Detection

To extract the knots and the external defect from the CT-scanned images we
chose to use object detection algorithms provided by the Tensorflow object
detection API [16]. It allows to easily reproduce our experiments and provides
a diverse set of contemporary architectures 222
https://github.com/tensorflow/models/tree/master/research/object_detection.
Each detector (i.e. SSD [23], Faster RCNN and RetinaNet-500 [22]) are pre-
trained on MS COCO dataset and fine-tuned on 32000 annotated synthetic CT-
scanned images for 60 epochs.

####  4.2.2 Internal Knots Prediction

To predict the knot distribution of the log from its outer-shape, we split the
synthetic dataset into three subsets: training set, validation set, and
testing set with the respective proportion $80\%$, $4\%$, and $16\%$.
The hyperparameter of the 2D and 3D predictors are tuned as follows: we chose
an Adam optimizer with an initial learning rate of $10^{-3}$, a learning rate
scheduler is also used to avoid undesirable divergent behavior while
increasing the number of epochs(the learning rate drops to $10^{-4}$ after 20
epochs). For computational constraints, the batch size is fixed to two voxels,
100 images, and a sequence of 64 images for respectively the 3D models, 2D
models, and CLSTM based models. Each model is trained for 50 epochs. At the
end of each epoch, a the models are evaluated on the validation set and we
save only the weights of the model with a minimum validation error.

####  4.2.3 Evaluation Metrics

To evaluate the performances of the object detecion algorithms we use the most
common metrics: the mean Average Precision (mAP) with two thresholds: $50\%$
and $75\%$.
Quantitative and qualitative metrics were used to assess the quality of the
internal density prediction. The Root means square error (RMSE) of the
difference voxel to voxel between the ground truth and the predicted
volumetric structure is applied to evaluate the performance of both the 3D and
the CLSTM based encoder-decoder models. It was also used to assess the
prediction quality of the 2D models after 3D reconstruction of the ground
truth and the predicted logs.

##  5 Results

###  5.1 Internal and external knots extraction

In this section, we assess and compare the results of the different object
detectors applied on our four real CT-scanned trees. Table 1 shows the mAP of
the different detectors used to detect the knots as well as the branches(i.e.,
external defects) with and without the tracking process.

Table 1 shows the average precision of the four different object detection
algorithms on high resolution CT-scanned wood-log (i.e. 512x512). Compared to
Faster detectors like SSD-513 [23] or YoloV3-608 [32], RetinaNet-500 [22]
consitently achieves more accurate results in detecting the knots of all the
tree species. Even though, RetinaNet-500 [22] is more time consuming and
computationally expensive its superior performances make it the most relevant
choice to extract the knots and branches.

Table 2: RMSE of the different type of architectures : On the synthetic dataset. Lower is better. Architecture | RMSE $(10^{-2})$ | Parameters
---|---|---
2 $Branches$ | 5 $Branches$ | 7 $Branches$
SegNet [33] | 1.27 | 1.66 | 2.46 | 34 M
U-Net | 1.33 | 1.68 | 2.42 | 36 M
DeepLabV3+ [8] | 3.17 | 3.23 | 3.48 | 42 M
3D-SegNet | 3.42 | 3.68 | 4.22 | 144 M
3D-U-Net [38] | 3.53 | 3.72 | 4.30 | 114 M
VoxResNet [6] | 3.6 | 3.92 | 4.2 | 35 M
CLSTM-SegNet | 3.61 | 3.89 | 4.26 | 31 M
Bidir-CLSTM-SegNet | 3.1 | 3.29 | 3.9 | 52 M

When looking at the results, the difference betzeen the mAP50 and mAP75 is
minimal meaning that the objects that the object detected by the detectors are
well detected but it misses quite a lot of them. While it’s far from being an
autonomous solution it can be used as a prelabeling tool where the user then
adds or remove the incorrect detection. Moreover, all of the object detection
algorithms have a relatively high false negatives rate when it comes to
distinguish between knots that extends to a branch and small surface defects
generated by a cracking inside the log. Hence, we decided to keep only
trackable defects that extends to the tree bark and could be predicted from
the outer-shape of the log. As explained earlier, our current database of real
CT-scanned trees is extremely limited. As of today only four trees are
available. This explains why the detectors are struggling, yet RetinaNet
clearly shows that it is at least the most sample efficient architecture if
not simply the most accurate architecture. This is extremely important as the
database that we are currently building will not exceed hundred of trees.

(a) (Upper Left) Elm , (Upper Right) ash, (Lower Left) aspen (Lower Right) fir

(b) (Upper Left) Elm , (Upper Right) ash , (Lower Left) fir (Lower Right)
aspen

Figure 3: (Left) Retina-Net detected bounding boxes and their corresponding
ellipses representing knots.(Right) Retina-Net detected bounding boxes of
external branches and epicormic shoots

###  5.2 Knots Distribution Prediction

####  5.2.1 Synthetic Dataset

In this section, we will assess the knots prediction based on 288 synthetic
log external surfaces. Table 2 shows the mean performance of the different 2D,
3D, and CLSTM based neural network architectures when tested on 40 k-knots
logs ($k\in\\{2,5,7\\}$). Each density prediction model has been trained five
times, and we compute the average RMSE to get accurate results.

From the results shown in table 2, it can be seen that the 2D networks largely
outperform their 3D counterparts. Within the 2D networks, SegNet achieves
better prediction results than DeepLabV3+ [8] and U-Net for two branches logs.
In contrast, U-Net achieves better results for predicting the internal density
of logs with a high number of branches. Those results are expected because,
unlike the SegNet model, U-Net architecture focuses more on capturing the
information lost through the encoder structure, which may be relevant when it
comes to more complex log structure (i.e., logs with a high number of
branches). DeepLabV3+ fails compared to the other architectures to achieve
good results: The DeepLabV3+ architecture features four times upsampling at
the end of the network that may explain why its performances are lower than
the other networks.
When looking at the voxel-based networks, 3D-SegNet and 3D-U-Net [38] achieve
better performance than VoxResNet [6]. Unlike 3D-U-Net and VoxResNet,
3D-SegNet doesn’t contain skip-connections; hence it may be affected by
information loss within the encoding process, as seems to suggest the decrease
in performance of the network as the number of branches increases. Finally,
the sequence-based models offered the best performance of all the 3D models on
the synthetic dataset; this remains true as the number of branches increases.
Moreover, the LSTMs have fewer parameters than the other 3D models making them
particularly interesting with real data as they will have less chances of
overfitting.

All in all, on the synthetic data, the 2D approaches performed better than
their 3D counterparts. This can be explained by the different types of inputs
the 2D networks and the voxel-based/sequence-based networks have. Indeed to
predict the internal density, the 2D networks rely on a slice of the tree
surface, which in our configuration yields an angular resolution of about 4.2
pixels per degree. On the other hand, the 256x256 voxels offer, at most, the
equivalent of an angular resolution of 2.2 pixels per degree. This drastic
difference in resolution explains why the 2D networks performed better. As of
today the voxel-based networks would have to be able to handle much larger
voxels to match the resolution of the 2D nets, which is not possible yet on
our 16Gb GPUs.

Unfortunately, the 2D networks cannot be applied to the real trees, as they
need a high-resolution surface map that we cannot generate from the CT-scanned
trees. The CT-scanned images at our disposal do not have the resolution and
finesse required to create proper surface maps of the trees. However, they
should be applicable to datasets that features both CT-scans and lidar
acquired external tree shapes. A dataset that is currently under construction.

Table 3: RMSE of the 3D architectures applied on real trees. Lower is better.  Architecture | RMSE $(10^{-2})$
---|---
Fir tree | elm tree | ash tree | aspen tree
3D-SegNet | 5.98 | 6.13 | 7.92 | 6.51
Bidir-CLSTM-SegNet | 4.56 | 4.4 | 5.23 | 4.1

####  5.2.2 Real Dataset

The prediction of the knots distribution in the real wood-logs is more
challenging due to the few samples available and the high variability of the
surface patterns indicating the presence of knots beneath them.This
observations might explain the results shown in table 3. Like in the previous
experiment, the CLSTMs show superior performances when compared to 3D SegNet,
a voxel-based network. While these results are not as good as on the synthetic
dataset, it shows that the CLSTMs are not only more accurate but also remains
more accurate with a limited dataset.

The CLSTMs results are encouraging as they were acquired using a single real-
log in the fine-tuning process. As shown in 5, the results on the fir and elm
trees are particularly promising.

###  5.3 3D visual results

We used $Paraview^{\copyright}$, an open-source multi-platform application for
3D visualization, to assess the 3D rendering of the predicted iso-surface of
the internal density of the tree. Figure 4 shows the predicted iso-surface
with different models for the 6-branches synthetic log. Based on the RMSE
evaluation, we chose to assess only the 2D, 3D, and CLSTM based models with
the highest performance. We conclude that the 2-D models achieve better
results than 3-D models in terms of iso-surface correspondence between the
ground truth (Green volume) and the predicted log (Red volume).

Figure 4: From left to right: 3D-U-Net [38], U-Net, Bidirectional CLSTM based
SegNet

As stated in the previous section, only predicted iso-surface derived from
real CT-scanned wood-log using pre-trained Bidir-CLSTM [35] SegNet and 3D-Seg-
Net will be visually assessed. The figure 5 shows a volumetric representation
of the predicted internal density of the trees using a CLSTM and their
corresponding ground truth.

Figure 5: (Upper Left) fir tree, (Upper Right) ash tree, (Lower Left) aspen
tree (Lower Right) elm tree

It is easier for the neural network architectures to predict knots closer to
the bark tree (ex: elm tree) than those closer to the pith (ex: ash tree).

##  6 Conclusion

In this paper, and for the first time, we show that the internal knots
distribution can be inferred by CNNs from the outer-shape of the trees. In
doing so we study the performances of a wide diversity of neural networks.
First to automatically label the CT-scanned cross-section images using a
object detectors. To do so, we leveraged manually annotated images and
extended our current dataset by applying object-detectors on the remaining
non-labeled images. Amongst the evaluated object detectors RetinaNet performed
best even though the dataset was limited. Then we evaluated 3 different data
representations on both synthetic and real data. Amidst the different data
representation applied on the synthetic dataset the 2D one performed best. The
3D CNNs and the CLSTM falling far behind. However, the 2D representation could
not be tested on the data we currently have at our disposal, hence the real
data was tested only on the CLSTMs and 3D CNNs. On the real dataset like on
the synthetic one, the best performing model were the bi-directionnal CLSTMs.
Overall, the performances of the CLSTMs model on the real trees, are not that
different from the synthetic trees indicating that those results should be
applicable to the larger and more sophisticated dataset that we are currently
acquiring. This first proof of concept comfort us our belief that CT-scanners
are not the only solution to infer the internal structure of trees. In future
work we will apply CLTMs and 2D CNNs to a larger and more diverse dataset. And
finally, we will improve the realism of our synthetic dataset by leveraging
GANs.

## References

  * [1] Jean-Philippe Andreu and Alfred Rinnhofer.  Modeling knot geometry in norway spruce from industrial ct images.  In Scandinavian Conference on Image Analysis, pages 786–791. Springer, 2003.
  * [2] Jean-Philippe Andreu and Alfred Rinnhofer.  Modeling of internal defects in logs for value optimization based on industrial ct scanning.  pages 23–26, 2003.
  * [3] Vijay Badrinarayanan, Alex Kendall, and Roberto Cipolla.  Segnet: A deep convolutional encoder-decoder architecture for image segmentation.  2015\.
  * [4] Alexey Bochkovskiy, Chien-Yao Wang, and Hong-Yuan Mark Liao.  Yolov4: Optimal speed and accuracy of object detection.  2020\.
  * [5] J. Canny.  A computational approach to edge detection.  IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8(6):679–698, 1986.
  * [6] Hao Chen, Qi Dou, Lequan Yu, and Pheng-Ann Heng.  Voxresnet: Deep voxelwise residual networks for volumetric brain segmentation.  2016\.
  * [7] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L. Yuille.  Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs.  2016\.
  * [8] Liang-Chieh Chen, Yukun Zhu, George Papandreou, Florian Schroff, and Hartwig Adam.  Encoder-decoder with atrous separable convolution for semantic image segmentation.  2018\.
  * [9] Emmanuel Duchateau, David Auty, Frederic Mothe, Longuetaud Fleur, C. Ung, and Alexis Achim.  Models of knot and stem development in black spruce trees indicate a shift in allocation priority to branches when growth is limited.  PeerJ, 3, 04 2015.
  * [10] Vincent Dumoulin and Francesco Visin.  A guide to convolution arithmetic for deep learning.  2016\.
  * [11] Ross B. Girshick.  Fast R-CNN.  CoRR, abs/1504.08083, 2015.
  * [12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  CoRR, abs/1512.03385, 2015.
  * [13] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  2015\.
  * [14] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Delving deep into rectifiers: Surpassing human-level performance on imagenet classification.  2015\.
  * [15] Andrew G. Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam.  Mobilenets: Efficient convolutional neural networks for mobile vision applications.  CoRR, abs/1704.04861, 2017.
  * [16] Jonathan Huang, Vivek Rathod, Chen Sun, Menglong Zhu, Anoop Korattikara, Alireza Fathi, Ian Fischer, Zbigniew Wojna, Yang Song, Sergio Guadarrama, and Kevin Murphy.  Speed/accuracy trade-offs for modern convolutional object detectors.  CoRR, abs/1611.10012, 2016.
  * [17] L. Jiao, F. Zhang, F. Liu, S. Yang, L. Li, Z. Feng, and R. Qu.  A survey of deep learning-based object detection.  IEEE Access, 7:128837–128868, 2019.
  * [18] Erik Johansson, Dennis Johansson, Johan Skog, and Magnus Fredriksson.  Automated knot detection for high speed computed tomography on pinus sylvestris l. and picea abies (l.) karst. using ellipse fitting in concentric surfaces.  Computers and Electronics in Agriculture, 96:238 – 245, 2013.
  * [19] Adrien Krähenbühl, Bertrand Kerautret, and Isabelle Debled-Rennesson.  TKDetection: a software to detect and segment wood knots.  imagen-a, 3(5), Mar. 2013.
  * [20] A. Krähenbühl, B. Kerautret, I. Debled-Rennesson, F. Longuetaud, and F. Mothe.  Knot detection in x-ray ct images of wood.  In George Bebis, Richard Boyle, Bahram Parvin, Darko Koracin, Charless Fowlkes, Sen Wang, Min-Hyung Choi, Stephan Mantler, Jürgen Schulze, Daniel Acevedo, Klaus Mueller, and Michael Papka, editors, Advances in Visual Computing, pages 209–218, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg.
  * [21] Adrien Krähenbühl, Bertrand Kerautret, and Isabelle Debled-Rennesson.  Knot segmentation in noisy 3d images of wood.  volume 7749, pages 383–394, 03 2013.
  * [22] Tsung-Yi Lin, Priya Goyal, Ross B. Girshick, Kaiming He, and Piotr Dollár.  Focal loss for dense object detection.  CoRR, abs/1708.02002, 2017.
  * [23] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott E. Reed, Cheng-Yang Fu, and Alexander C. Berg.  SSD: single shot multibox detector.  CoRR, abs/1512.02325, 2015.
  * [24] Jonathan Long, Evan Shelhamer, and Trevor Darrell.  Fully convolutional networks for semantic segmentation.  CoRR, abs/1411.4038, 2014.
  * [25] Jonathan Long, Evan Shelhamer, and Trevor Darrell.  Fully convolutional networks for semantic segmentation.  2014\.
  * [26] Bruna Longo, Franka Brüchert, Gero Becker, and Udo Sauter.  Validation of a ct knot detection algorithm on fresh douglas-fir (pseudotsuga menziesii (mirb.) franco) logs.  Annals of Forest Science, 76, 06 2019.
  * [27] Bruna L. Longo, Franka Brüchert, Gero Becker, and Udo H. Sauter.  Validation of a ct knot detection algorithm on fresh douglas-fir (pseudotsuga menziesii (mirb.) franco) logs.  Annals of Forest Science, 76:1–16, 2019.
  * [28] Jean-Baptiste Morisset, Frédéric Mothe, and Francis Colin.  Observation of quercus petraea epicormics with x-ray ct reveals strong pith-to-bark correlations: Silvicultural and ecological implications.  Forest Ecology and Management, 278:127 – 137, 2012.
  * [29] Rickard Norlander, Josef Grahn, and Atsuto Maki.  Wooden knot detection using convnet transfer learning.  pages 263–274, 06 2015.
  * [30] Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi.  You only look once: Unified, real-time object detection.  2015\.
  * [31] Joseph Redmon and Ali Farhadi.  YOLO9000: better, faster, stronger.  CoRR, abs/1612.08242, 2016.
  * [32] Joseph Redmon and Ali Farhadi.  Yolov3: An incremental improvement.  CoRR, abs/1804.02767, 2018.
  * [33] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.  U-net: Convolutional networks for biomedical image segmentation.  2015\.
  * [34] Seyed shahabeddin Nabavi, Mrigank Rochan, Yang, and Wang.  Future semantic segmentation with convolutional lstm.  2018\.
  * [35] Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai kin Wong, and Wang chun Woo.  Convolutional lstm network: A machine learning approach for precipitation nowcasting.  2015\.
  * [36] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  2014\.
  * [37] Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia.  Pyramid scene parsing network.  2016\.
  * [38] Özgün Çiçek, Ahmed Abdulkadir, Soeren S. Lienkamp, Thomas Brox, and Olaf Ronneberger.  3d u-net: Learning dense volumetric segmentation from sparse annotation.  2016\.