# Iterative Reconstruction Based on Latent Diffusion Model for Sparse Data
Reconstruction

Linchao He 1,2, Hongyu Yan11footnotemark: 1 1,2, Mengting Luo1,2, Kunming
Luo3,
Wang Wang4, Wenchao Du2, Hu Chen 2 Hongyu Yang2 Yi Zhang5
1Department of National Key Laboratory of Fundamental Science on
Synthetic Vision, Sichuan University, Chengdu, China
2College of Computer Science, Sichuan University, Chengdu, China
3The Hong Kong University of Science and Technology, Hong Kong, China
4State Key Laboratory of Information Engineering in Surveying,
Mapping and Remote Sensing (LIESMARS), Wuhan University, Wuhan, China
5School of Cyber Science and Engineering, Sichuan University, Chengdu, China
{hlc, hongyuyan<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>Equal contribution.Corresponding
Authors.

###### Abstract

Reconstructing Computed tomography (CT) images from sparse measurement is a
well-known ill-posed inverse problem. The Iterative Reconstruction (IR)
algorithm is a solution to inverse problems. However, recent IR methods
require paired data and the approximation of the inverse projection matrix. To
address those problems, we present Latent Diffusion Iterative Reconstruction
(LDIR), a pioneering zero-shot method that extends IR with a pre-trained
Latent Diffusion Model (LDM) as a accurate and efficient data prior. By
approximating the prior distribution with an unconditional latent diffusion
model, LDIR is the first method to successfully integrate iterative
reconstruction and LDM in an unsupervised manner. LDIR makes the
reconstruction of high-resolution images more efficient. Moreover, LDIR
utilizes the gradient from the data-fidelity term to guide the sampling
process of the LDM, therefore, LDIR does not need the approximation of the
inverse projection matrix and can solve various CT reconstruction tasks with a
single model. Additionally, for enhancing the sample consistency of the
reconstruction, we introduce a novel approach that uses historical gradient
information to guide the gradient. Our experiments on extremely sparse CT data
reconstruction tasks show that LDIR outperforms other state-of-the-art
unsupervised and even exceeds supervised methods, establishing it as a leading
technique in terms of both quantity and quality. Furthermore, LDIR also
achieves competitive performance on nature image tasks. It is worth noting
that LDIR also exhibits significantly faster execution times and lower memory
consumption compared to methods with similar network settings. Our code will
be publicly available.

Figure 1: LDIR is capable of reconstructing both nature images and medical
data in a zero-shot manner. In this paper, we demonstrate the reconstruction
ability of our method on various datasets and measurements. Different from
previous works, LDIR can reconstruct high-resolution images (512 x 512) from
sparse measurements using unconditional latent diffusion models. FBP is
filtered backprojection. The display window of CT images is set to [-150, 256]
HU.

##  1 Introduction

Computed tomography (CT) is a crucial medical imaging technique in
contemporary medicine to aid physicians in diagnosing relevant conditions.
Measurements in CT are obtained by X-rays projections of an object from
different views. However, the use of X-rays in CT exposes the human body to
potentially harmful doses of radiation, raising concerns among the public
regarding radiation-induced diseases. Therefore, reducing the exposure dose,
such as sparse-view and limited-angle imaging, while maintaining the quality
of imaging has beneficial implications for both public health and medical
diagnosis, specifically in intraoperative CT. Due to the sparse information,
the CT reconstruction processes are well-known ill-posed inverse problems. In
the past decade, numerous works have focused on Iterative Reconstruction (IR),
which is considered a solution to CT reconstruction Donoho (2006); Candès et
al. (2006). Iterative reconstruction aims to recover signals $x$ from noisy
measurements $y=\mathbf{A}x+n$, where $n$ represents the noise in the
measuring process and $\mathbf{A}$ is the linear projection matrix that
typically maps $x$ to a lower dimension. As a result, a typical IR process can
be formulated as follows:

$\displaystyle\hat{x}=\operatorname*{arg\,min}_{x}\left|\left|\mathbf{A}x-y\right|\right|_{2}^{2}+\lambda
R\left(x\right),$ (1)

here, $\left|\left|\mathbf{A}x-y\right|\right|_{2}^{2}$ is the data-fidelity
term that ensures the reconstructed results are consistent with the
measurements, while $\lambda R\left(x\right)$ is a prior term that ensures the
reconstructed results are realistic and follow the distribution
$p\left(x\right)$ of the ground truth images.

The key challenge of IR is to find appropriate data prior or sparse
transformation to generate the prior term. Traditional IR methods Beck,
Teboulle (2009); Kim et al. (2016); Zhao et al. (2000) leverage the total
variation, nonlocal means, or wavelets to gain the reconstructed results by a
handcrafted prior and the approximation of the inverse projection matrix.
Inspired by the success of deep learning and neural network, recent data-
driven methods Bora et al. (2017); Chen et al. (2018) achieve impressive
performance by learning the prior and data-fidelity terms. However, most of
these methods require large-scale paired data to train their networks.
Besides, they directly project the measurement to the results and they also
need to retrain while the detector geometry changes. Although method Song et
al. (2022) solves these problems mentioned above by introducing a score-based
generative model, it still needs to know the approximation of the inverse
projection matrix which is difficult to be obtained in the real-world
application. Additionally, these methods use direct projection to replace the
data-fidelity term which makes they make them need to design different
sampling procedures for different detector geometries.

In this paper, we introduce a novel Latent Diffusion Iterative Reconstruction
(LDIR) for sparse CT data reconstruction in a zero-shot manner. The LDIR is
the first to extend traditional IR techniques by incorporating a latent
diffusion model. Specifically, we train a latent unconditional diffusion model
to learn the data distribution. In the reverse process, the trained diffusion
model is utilized to replace the prior term in the regular IR. In particular,
to generate specified prior from the unconditional diffusion model, the
gradient from the data-fidelity term is applied to guide the sampling process
of the diffusion model. Therefore, we do not need any paired data or inverse
of the measurement matrix to train our network. Moreover, LDIR can solve
various CT reconstruction tasks with a single model. In addition, by guiding
the reverse diffusion sampling process in the latent space, our zero-shot
method can generate high-resolution images with impressive performance. Since
LDIR does not make any assumption on the data-fidelity term, we can use any
differentiable measurement function to keep the consistency of data. Further
on, we propose a novel guidance strategy to adaptively adjust the sample-level
gradient by fusing the history gradient, thereby improving the performance of
our zero-shot diffusion model.

Extensive experiments demonstrate our method outperforms state-of-the-art
supervised and unsupervised methods for extremely sparse CT data
reconstruction. Additionally, due to the commonality of iterative
reconstruction on CT images and natural images, we extend LDIR to the natural
image restoration task, and our approach achieves competitive performance
compared to other state-of-the-art zero-shot methods. Our approach provides a
valuable zero-shot tool for solving inverse problems with latent diffusion
models, allowing us to leverage the vast amount of available latent diffusion
models. Fig. 1 shows some visual results of the proposed method.

##  2 Background

###  2.1 Diffusion models

Consider a $T$-step Gaussian diffusion process, where
$x_{t}\in\mathbb{R}^{n},t\in[0,\dots,T-1]$ and initial $x_{0}$ is sampled from
the original data distribution $P_{data}$. We define the forward diffusion
process using stochastic differential equation (SDE) Song et al. (2021):

$\displaystyle dx=f\left(x,t\right)\,dt+g\left(t\right)\,dw,$ (2)

where $f\left(\cdot,t\right):\,\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a drift
coefficient function, $g\left(t\right)\in\mathbb{R}$ is defined as a diffusion
coefficient function, and $w\in\mathbb{R}^{n}$ is a standard $n$-dimensional
Brownian motion. Thus, the reverse SDE of Eq. (2) can also defined as:

$\displaystyle dx=\left[f\left(x,t\right)-g\left(t\right)^{2}\nabla_{x}\log
p_{t}\left(x\right)\right]\,dt+g\left(t\right)\,dw,$ (3)

where $dt$ is a negative infinitesimal time step. The reverse SDE defines a
generative process that transforms standard Gaussian noise into meaningful
content. To accomplish this transformation, the score function $\nabla_{x}\log
p_{t}\left(x\right)$ needs to be matching, which is typically replaced with
$\nabla_{x}\log p_{0|t}\left(x_{t}\middle|x_{0}\right)$ in practice.
Therefore, we can train a score model $s_{\theta}\left(x,t\right)$, so that
$s_{\theta}\left(x,t\right)\approx\nabla_{x}\log
p_{t}\left(x\right)\approx\nabla_{x}\log
p_{0|t}\left(x_{t}\middle|x_{0}\right)$ using the following score-matching
objective:

$\displaystyle\min_{\theta}\mathbb{E}_{t\in\left[0,\dots,T-1\right],x_{0}\sim
P_{data},x_{t}\sim
p_{0|t}\left(x_{t}\middle|x_{0}\right)}\left[\left|\left|s_{\theta}\left(x,t\right)-\nabla_{x}\log
p_{0|t}\left(x_{t}\middle|x_{0}\right)\right|\right|_{2}^{2}\right]\,.$ (4)

Therefore, the reverse SDE can yield meaningful contents $x_{0}\sim P_{data}$
from random noises $x_{T-1}\sim\mathcal{N}\left(0,\mathbf{I}\right)$ by
iteratively using $s_{\theta}\left(x,t\right)$ to estimate the scores
$\nabla_{x}\log p_{t}\left(x\right)$. In our experiments, we adopt the
standard Denoising Diffusion Probabilistic Models (DDPM) Ho et al. (2020)
which is equivalent to the above variance preserving SDE (VP-SDE Song et al.
(2021)).

###  2.2 Diffusion model for inverse problem solving

To solve the inverse problems using the diffusion model, various workarounds
are proposed Rombach et al. (2022); Saharia et al. (2022); Gao et al. (2023);
Luo et al. (2023). These methods use conditional diffusion models to
iteratively denoise Gaussian noise and obtain reconstructions. However, these
approaches have limitations, as they rely on conditional diffusion models that
require paired data for training and can only handle specific tasks without
retraining. To address these issues, several zero-shot diffusion-based inverse
solvers Lugmayr et al. (2022); Song et al. (2022); Kawar et al. (2022); Chung
et al. (2022); Wang et al. (2023) have been proposed. Typically, assuming
$n\equiv 0$, for each denoising step, they Lugmayr et al. (2022); Song et al.
(2022); Wang et al. (2023) unconditionally estimate new denoised samples based
on the previous step, followed by replacing the corresponding items in the
denoised samples using the measurement $\mathbf{A}^{-1}y$, which is also known
as range-null space decomposition Wang et al. (2023). This approach ensures
data consistency, but it fails in the case of noisy measurements, since
$\mathbf{A}^{-1}y$ is not a correct corresponding item for the denoised
samples. To address this limitation, alternative methods Chung et al. (2023b,
a) have been proposed to solve the inverse problems with noised measurements.
Rather than directly replace items, these approaches use the gradient of
$\left|\left|y-\mathbf{A}x\right|\right|_{2}^{2}$ to conditionally guide the
generative process. These methods are robust to noise and can process
nonlinear projection operators. However, these methods try to solve inverse
problems on the pixel space and make strong assumptions on the data-fidelity
term, which significantly underestimates the complexity of real-world problems
and can not reconstruct high-resolution results.

##  3 Method

###  3.1 Diffusion iterative reconstruction

Generally, it is possible to transform the IR methods presented in Eq. (1)
into a more generic form:

$\displaystyle\hat{x}$
$\displaystyle=\operatorname*{arg\,min}_{x}E\left(x\right)$ (5)
$\displaystyle=\operatorname*{arg\,min}_{x}\mathcal{U}\left(\mathbf{A}x,y\right)+\lambda
R\left(x\right),$ (6)

where $\mathcal{U}$ is a measurement function that ensures data consistency.
Assuming that both the measurement function and prior term are differentiable,
it is possible to apply gradient descent to Eq. (6). This yields an ordinary
differential equation (ODE) for iterative reconstruction:

$\displaystyle x_{t-1}$ $\displaystyle=x_{t}-\frac{\partial
E\left(x\right)}{\partial x_{t}}$ (7)
$\displaystyle=x_{t}-\left(\epsilon\nabla_{x_{t}}\,\mathcal{U}\left(\mathbf{A}x_{t},y\right)+\lambda_{t}\,\nabla_{x_{t}}\,R\left(x_{t}\right)\right),$
(8)

where guidance rate $\epsilon$ and $\lambda_{t}$ are used to balance the
consistency and realness. In practice, we can replace
$\lambda\,\nabla_{x_{t}}\,R\left(x_{t}\right)$ with a step-dependent prior
function $\lambda_{t}\,\nabla_{x_{t}}\,R\left(x_{t},t\right)$ to balance
realness and data consistency, as demonstrated by Chen et al. (2018):

$\displaystyle
x_{t-1}=\left(x_{t}-\lambda_{t}\,\nabla_{x_{t}}\,R\left(x_{t},t\right)\right)-\epsilon\nabla_{x_{t}}\,\mathcal{U}\left(\mathbf{A}x_{t},y\right).$
(9)

As demonstrated by Song et al. (2021), the score function is a powerful tool
for representing probability distributions. The score function does not
require the computation of a tractable normalizing constant or its
approximation. This makes it possible to estimate $\log p\left(x\right)$ using
the score function. By replacing the prior term
$\nabla_{x_{t}}\,R\left(x_{t},t\right)$ with the score function
$\nabla_{x_{t}}\,\log p_{t}\left(x\right)$, it is possible to ensure that the
new prior term accurately represents the probability distribution without
requiring heavy computation:

$\displaystyle x_{t-1}=\left(x_{t}-\lambda_{t}\,\nabla_{x}\,\log
p_{t}\left(x\right)\right)-\epsilon\nabla_{x_{t}}\,\mathcal{U}\left(\mathbf{A}x_{t},y\right),$
(10)

In Eq. (4), we can use a score model $s_{\theta}\left(x_{t},t\right)$ to
approximate $\nabla_{x_{t}}\,\log p_{t}\left(x\right)$ using the score
matching technique:

$\displaystyle
x_{t-1}\simeq\left(x_{t}-\lambda_{t}\,s_{\theta}\left(x_{t},t\right)\right)-\epsilon\nabla_{x_{t}}\,\mathcal{U}\left(\mathbf{A}x_{t},y\right).$
(11)

In fact, the first term $x_{t}-\lambda_{t}\,s_{\theta}\left(x,t\right)$ is Eq.
(3) with a noise-free constraint 111$f\left(x_{t},t\right)=g\left(t\right)z=0$
and $g\left(t\right)^{2}=\lambda_{t}$. Thus, we can relax the noise-free
constraint and use Eq. (3) to replace this term:

$\displaystyle x_{t-1}\simeq
x_{t}-\underbrace{\left(\lambda_{t}\,s_{\theta}\left(x_{t},t\right)-g\left(t\right)z\right)}_{\text{Prior
term}}-\underbrace{\epsilon\nabla_{x_{t}}\,\mathcal{U}\left(\mathbf{A}x_{t},y\right)}_{\text{Data-
fidelity term}},z\sim\mathcal{N}\left(0,\mathbf{I}\right).$ (12)

We propose a new iterative reconstruction method called Diffusion Iterative
Reconstruction (DIR). The model uses a score model to represent the original
data distribution and employs it as a learned prior. Algo. 1 demonstrates the
pixel guidance process of DIR.

It is worth noting that our DIR has a mathematical form that is similar to the
approach proposed by Chung et al. (2023b). However, our approach is more
general and can be applied to a wider range of scenarios. Diffusion Posterior
Sampling (DPS) Chung et al. (2023b) models the data consistency term
$\mathcal{U}\left(\cdot,\cdot\right)$ as a noise measurement problem. The
evaluation function for the data-fidelity term is derived based on the prior
assumption of the noise distribution (such as the $L2$ function for Gaussian
noise). This approach works well when the noise type is known in the
measurement process. However, it is challenging to estimate the noise
distribution and derive the correct evaluation function in real-world
applications. In contrast, our DIR models the data-fidelity term as a quality
evaluation function without any assumptions about the noise distribution.
Thus, the evaluation function $\mathcal{U}\left(\cdot,\cdot\right)$ can be any
differentiable evaluation function, allowing for more flexibility.

Algorithm 1 Pixel-space data reconstructing

1:: $N,y$

2:$x_{T}\sim\mathcal{N}$

3:for $\ \textbf{do}\left\\{i=T-1,\dots,0\right\\}$

4: $\hat{s}\leftarrow s_{\theta}\left(x_{i},i\right)$

5:
$x_{0|i}\leftarrow\frac{1}{\sqrt{\bar{\alpha}_{i}}}\left(x_{i}+\left(1-\bar{\alpha}_{i}\right)\hat{s}\right)$

6: $z\sim\mathcal{N}\left(0,\mathbf{I}\right)$

7:
$x^{\prime}_{i-1}\leftarrow\frac{\sqrt{\bar{\alpha}_{i-1}}\beta_{i}}{1-\bar{\alpha}_{i}}x^{\prime}_{0|i}+\frac{\sqrt{\alpha_{i}}\left(1-\bar{\alpha}_{i-1}\right)}{1-\bar{\alpha}_{i}}x_{i}+g\left(i\right)z$

8: $x_{i-1}\leftarrow
x^{\prime}_{i-1}-\epsilon\nabla_{x_{i}}\,\mathcal{U}\left(\mathbf{A}x^{\prime}_{i-1},y\right)$

9:end for

10:return $x_{0|0}$

Algorithm 2 Latent-space data reconstructing

1:: $N,y,\mathcal{D}$

2:$\ell_{T}\sim\mathcal{N}$

3:for $\ \textbf{do}\left\\{i=T-1,\dots,0\right\\}$

4: $\hat{s}\leftarrow s_{\theta_{l}}\left(\ell_{i},i\right)$

5:
$\ell_{0|i}\leftarrow\frac{1}{\sqrt{\bar{\alpha}_{i}}}\left(\ell_{i}+\left(1-\bar{\alpha}_{i}\right)\hat{s}\right)$

6: $z\sim\mathcal{N}\left(0,\mathbf{I}\right)$

7:
$\ell^{\prime}_{i-1}\leftarrow\frac{\sqrt{\bar{\alpha}_{i-1}}\beta_{i}}{1-\bar{\alpha}_{i}}\ell^{\prime}_{0|i}+\frac{\sqrt{\alpha_{i}}\left(1-\bar{\alpha}_{i-1}\right)}{1-\bar{\alpha}_{i}}\ell_{i}+g\left(i\right)z$

8:
$\ell_{i-1}\leftarrow\ell^{\prime}_{i-1}-\epsilon\nabla_{\ell_{i}}\,\mathcal{U}(\mathbf{A}\mathcal{D}\left(\ell^{\prime}_{i-1}),y\right)$

9:end for

10:$x_{0|0}\leftarrow\mathcal{D}\left(\ell_{0|0}\right)$

11:return $x_{0|0}$

###  3.2 History gradient update

In what follows, we show that introducing history gradient update policies can
provide better reconstruction results. The basic formulation of gradient
guidance in Eq. (12) corresponds to a simple gradient descent scheme. The
guidance rate $\epsilon$ can be thought of as the learning rate value in the
Stochastic Gradient Descent (SGD). Thus, in order to further improve the data
consistency, we adopt gradient information from the previous steps. Because
the history gradient information can provide sample-level information to
decide the optimization direction of the guidance process. This is also known
as the first-order gradient-based optimization. Here, we demonstrate two
variants of gradient update policies based on two typical optimizers.

Momentum-like gradient update policy.  Similar to the Momentum optimizer
Sutskever et al. (2013), we consider to use the moving averages $m^{t}$ of the
gradients $\nabla_{x_{t}}\mathcal{U}\left(\mathbf{A}x_{t},y\right)$ to perform
gradient descent:

$\displaystyle m^{t}$ $\displaystyle=\eta
m^{t-1}+\left(1-\eta\right)\nabla_{x_{t}}\mathcal{U}\left(\mathbf{A}x_{t},y\right),$
(13) $\displaystyle x_{t-1}$ $\displaystyle=x_{t}-\epsilon m^{t},$ (14)

where $\eta$ is a hyper-parameter to adjust the factor of momentum.

Adam-like gradient update policy.  Adam optimizer Kingma, Ba (2014) uses
momentum and adaptive learning rate to perform gradient descent:

$\displaystyle m^{t}$
$\displaystyle=\eta_{1}m^{t-1}+\left(1-\eta_{1}\right)\nabla_{x_{t}}\mathcal{U}\left(\mathbf{A}x_{t},y\right),$
(15) $\displaystyle v^{t}$
$\displaystyle=\eta_{2}v^{t-1}+\left(1-\eta_{2}\right)\nabla_{x_{t}}\mathcal{U}\left(\mathbf{A}x_{t},y\right)^{2},$
(16) $\displaystyle x_{t-1}$
$\displaystyle=x_{t}-\epsilon\frac{\hat{m}^{t}}{\sqrt{\hat{v}^{t}}+\varepsilon},$
(17)

where $\left(\eta_{1},\eta_{2}\right)$ are the coefficients used to calculate
the exponentially weighted moving averages of gradient and its square, while
$\varepsilon$ helps improve the numerical stability. With the help of the
moving averages $m^{t}$ and $v^{t}$, we can efficiently locate the flat
minima.

Figure 2: Our LDIR can handle both CT reconstruction and nature image
restoration in the same pipeline. We guide latent diffusion models by using
the gradients
$d_{i}=\nabla_{\ell_{i}}\mathcal{U}\left(\mathbf{A}\mathcal{D}\left(\ell_{i}\right),y\right)$
along with history gradients. The guidance is obtained through gradients,
rather than direct projection.

It is worth noting that the choice of $\epsilon$ and gradient policy is
dependent on the evaluation function $\mathcal{U}$ (Details and ablation
studies can be found in Appendix. C). Our gradient update policies are
compatible with DIR, LDIR, and DPS whether on pixel space or latent space. The
details are demonstrated in the ablation studies. The detail of LDIR
algorithms with the above gradient update policies are presented in Appendix
A.

###  3.3 Latent diffusion iterative reconstruction

Our review of previous works on diffusion-based data reconstruction, including
Chung et al. (2023b, 2022, a, c); Song et al. (2022); Wang et al. (2023),
reveals that they all perform reconstruction in the pixel space, which
requires significant computational resources. To address this limitation, we
draw inspiration from the Latent Diffusion Models (LDMs) proposed by Rombach
et al. (2022) and our DIR. We introduce a novel data reconstruction diffusion
model called Latent Diffusion Iterative Reconstruction (LDIR). LDIR offers
several advantages over previous methods: (i) Instead of processing images in
the pixel space, we encode images into a low-dimensional latent space,
enabling us to process them more efficiently with fewer computational demands.
(ii) The latent space contains significantly higher information density
compared to the pixel space, allowing us to incorporate natural priors such as
sparsity and improve the quality of the restored data.

From pixels to latents.  To encode pixels to latents, we construct an
autoencoder comprising an encoder $\mathcal{E}$ and a decoder $\mathcal{D}$.
Specifically, given an input image $x$ in the pixel space, $\mathcal{E}$ maps
$x$ to a low-dimensional latent vector $\ell=\mathcal{E}\left(x\right)$.
$\mathcal{D}$ reconstructs the image
$\bar{x}=\mathcal{D}\left(\mathcal{E}\left(x\right)\right)$ from $\ell$. To
incorporate the sparsity prior of $x$, we use the Vector Quantized Variational
Autoencoder (VQ-VAE) proposed by Kingma, Welling (2013); Rezende et al.
(2014); Esser et al. (2021) with the quantization layer Van Den Oord et al.
(2017).

Score matching for latents.  With our semantic compression model $\mathcal{E}$
and $\mathcal{D}$, we can now establish the score of latents
$\nabla_{\ell}\,\log p_{t}\left(\ell\right)$ and use score model
$s_{\theta_{\ell}}$ to approx it with the following objective:

$\displaystyle\min_{\theta}\mathbb{E}_{t\in\left[0,\dots,T-1\right],\ell_{0}\sim
p_{\ell},\ell_{t}\sim
p_{0|t}\left(\ell_{t}\middle|\ell_{0}\right)}\left[\left|\left|s_{\theta_{\ell}}\left(\ell,t\right)-\nabla_{\ell}\log
p_{0|t}\left(\ell_{t}\middle|\ell_{0}\right)\right|\right|_{2}^{2}\right],$
(18)

where $p\left(\ell\right)=p\left(\mathcal{E}\left(x\right)\right)$ and $x\sim
p\left(x\right)$.

Conditional guidance process on the latent space.  Similar to the conditional
guidance process in the pixel space, we begin by using the score model
$s_{\theta_{\ell}}$ to generate latents from standard Gaussian noise:

$\displaystyle\ell^{\prime}_{t-1}=\ell_{t}-f\left(\ell,t\right)-g\left(t\right)^{2}s_{\theta_{\ell}}\left(\ell,t\right)+g\left(t\right)z,z\sim\mathcal{N}\left(0,\mathbf{I}\right),$
(19)

We need to use $\mathcal{D}$ to decode $\ell$ to pixel space, and compute the
data consistency term
$\mathcal{U}\left(\mathbf{A}\mathcal{D}\left(\ell\right),y\right)$, which can
be derived to:

$\displaystyle\ell_{t-1}=\ell^{\prime}_{t-1}-\epsilon\nabla_{\ell_{t}}\,\mathcal{U}\left(\mathbf{A}\mathcal{D}\left(\ell_{t}\right),y\right).$
(20)

Thus, we can get the conditional guidance algorithm of LDIR as:

$\displaystyle\ell_{t-1}\simeq\ell_{t}-\underbrace{\left(\lambda_{t}\,s_{\theta_{\ell}}\left(\ell_{t},t\right)-g\left(t\right)z\right)}_{\text{Prior
term}}-\underbrace{\epsilon\nabla_{\ell_{t}}\,\mathcal{U}\left(\mathbf{A}\mathcal{D}\left(\ell_{t}\right),y\right)}_{\text{Data-
fidelity term}},z\sim\mathcal{N}\left(0,\mathbf{I}\right).$ (21)

The final results can be obtained by decoding the final latent
$x_{0}=\mathcal{D}\left(\ell_{0}\right)$ using the decoder $\mathcal{D}$.

##  4 Experiments

###  4.1 Experimental setup

Models and datasets. For medical image reconstruction, we train our DDPM and
LDM model on the 2016 American Association of Physicists in Medicine (AAPM)
grand challenge dataset. The dataset has normal-dose data from 10 patients. 9
patients’ data are used for training, and 1 for validation which contains 526
images. To simulate low-dose imaging, a parallel-beam imaging geometry with
180 degrees was employed. Regarding inpainting and super-resolution tasks, we
test our method on CelebAHQ 1k $256\times 256$ dataset Liu et al. (2015) and
LSUN-bedroom $256\times 256$ dataset Yu et al. (2015). We utilize pretrained
DDPM and LDM models from the open-source model repository from Ho et al.
(2020); Rombach et al. (2022). All the images are normalized to range
$\left[0,1\right]$. More details including the hyper-parameters are listed in
Appendix. B.

Measurement operators. For sparse-view CT reconstruction, we uniformly sample
18 and 32 views. For limited-angle CT reconstruction, we restrict the imaging
degree range to $45$ and $90$ degrees with 128 views using parallel beam
geometry. For random inpainting, following Chung et al. (2022, 2023b), we mask
out $99\%$ of the total pixels (including all the channels). For super-
resolution, we use $8\times$ bilinear downsampling. Gaussian noise is added in
the nature image evaluation after a forward operation performed with
$\sigma=0.05$. The medical data are evaluated without noise.

Figure 3: Qualitative results of medical sparse data reconstruction. The
resolution of CT images is $256\times 256$ The display window of CT images is
set to $\left[-150,256\right]$ HU. The standard measurement of CT is 512 views
around 180 degrees. Table 1: Quantitative evaluation (PSNR, SSIM) of medical
image reconstruction on AAPM test $256\times 256$ dataset. We mark bold for
the best and underline for the second best. CLEAR Zhang et al. (2021) is a
supervised method.

Method | Sparse view | Limited angle
---|---|---
18 | 32 | 45 | 90
PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$
FBP | 24.76 | 0.5296 | 28.03 | 0.6779 | 16.65 | 0.5422 | 20.35 | 0.5113
FISTA-TV | 24.86 | 0.5408 | 28.14 | 0.6888 | 16.66 | 0.5463 | 20.40 | 0.5241
CLEAR | 32.28 | 0.8798 | 36.24 | 0.9257 | 25.71 | 0.8559 | 31.60 | 0.9223
MCG | 28.54 | 0.8135 | 28.98 | 0.8242 | 26.08 | 0.7418 | 28.44 | 0.8079
DPS | 28.55 | 0.8140 | 28.97 | 0.8242 | 28.25 | 0.8204 | 28.25 | 0.8088
LDIR | 39.01 | 0.9552 | 39.77 | 0.9612 | 30.05 | 0.8747 | 32.68 | 0.9032

Table 2: Quantitative evaluation (PSNR, SSIM) of medical image reconstruction
on AAPM test $512\times 512$ dataset for zero-shot methods. We mark bold for
the best and underline for the second best.

Method | Sparse view | Limited angle
---|---|---
18 | 32 | 45 | 90
PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$
FBP | 23.48 | 0.5096 | 26.70 | 0.6423 | 16.53 | 0.5480 | 19.88 | 0.4932
FISTA-TV | 23.93 | 0.5566 | 27.11 | 0.6768 | 16.59 | 0.5695 | 20.08 | 0.5348
LDIR | 27.30 | 0.8443 | 27.33 | 0.8441 | 26.18 | 0.8355 | 26.70 | 0.8381

###  4.2 Evaluation on medical data

To assess the performance of LDIR in reconstructing medical sparse data, we
compare it with several recent state-of-the-art methods: manifold constrained
gradients (MCG) Chung et al. (2022), diffusion posterior sampling (DPS) Chung
et al. (2023b), comprehensive learning enabled adversarial reconstruction
(CLEAR) Zhang et al. (2021), fast iterative shrinkage-thresholding algorithm
with total-variation (FISTA-TV), and the analytical reconstruction method,
filtered backprojection (FBP). Peak-signal-to-noise-ratio (PSNR) and
structural similarity index measure (SSIM) are used for quantitative
evaluation.

The quantitative results of medical sparse data reconstruction are
demonstrated in Tab. 1 and Tab. 2. Our method outperforms all other state-of-
the-art methods by a significant margin across all experiment settings. We
also compare our method in the high-resolution CT image reconstruction task
with zero-shot methods. However, due to the large memory consumption of DDPM,
it is challenging to train DDPM models for high-resolution reconstruction.
Thus, we exclude MCG and DPS which rely on DDPM from Tab. 2. The results show
that LDIR provides noise-free reconstruction results, although there is still
a significant gap between the reconstructed images and the ground truth. In
contrast, other zero-shot methods fail to reconstruct meaningful results.

The qualitative results of medical sparse image reconstruction are
demonstrated in Fig. 3 which are consistent with the quantitative results
reported in Tab. 1. In Fig. 3, we compare our method with the state-of-the-art
zero-shot unsupervised and supervised methods. We observe that our method can
provide high-quality reconstructions, especially for the sparse view
reconstruction task. Specifically, LDIR can provide better overall structure
and nearly artifact-free reconstruction. Additionally, our method also
provides better reconstructions than other methods in limited angle
reconstruction tasks. (More qualitative results of medical sparse data
reconstruction can be found in Appendix. D).

Figure 4: Qualitative results of nature sparse image reconstruction with
Gaussian noise ($\sigma=0.05$). Table 3: Quantitative evaluation (PSNR, SSIM,
LPIPS) of nature image reconstruction on CelebAHQ and LSUN-bedroom dataset. We
mark bold for the best and underline for the second best.

Method | Type | CelebAHQ | LSUN-bedroom
---|---|---|---
Inpaint | SR (8$\times$) | Inpaint | SR (8$\times$)
PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$
PnP-ADMM | Traditional IR | 3.97 | 0.3017 | 0.8916 | 22.94 | 0.6303 | 0.6820 | 5.059 | 0.3236 | 0.8940 | 20.14 | 0.5458 | 0.7944
MCG | Pixel Diffusion | 18.91 | 0.5600 | 0.2544 | 12.47 | 0.1655 | 0.6713 | 16.89 | 0.4555 | 0.5486 | 9.39 | 0.0606 | 0.8698
DPS | Pixel Diffusion | 18.95 | 0.5614 | 0.2543 | 24.36 | 0.7116 | 0.1089 | 17.03 | 0.4587 | 0.5414 | 19.15 | 0.5614 | 0.3074
LDIR | Latent Diffusion | 22.14 | 0.6647 | 0.2280 | 25.27 | 0.7530 | 0.0878 | 20.33 | 0.5845 | 0.4858 | 19.83 | 0.5762 | 0.3253

###  4.3 Evaluation on nature images

In order to further test the performance of our method, we compare our method
against state-of-the-art methods, namely, MCG, DPS, and plug-and-play
alternating direction method of multipliers (PnP-ADMM) Chan et al. (2016). For
quantitative analysis, we utilize three widely-used perceptual evaluation
metrics: LPIPS distance, PSNR, and SSIM.

The quantitative results of nature image reconstruction are illustrated in
Tab. 3. Our method achieves competitive results compared to the previous
state-of-the-art. Specifically, we observe that our method is able to
accurately reconstruct the original data and preserve the most data
consistency, even when dealing with highly sparse measurements such as $99\%$
random inpainting. Additionally, we note that LDIR gains some advantages over
the previous best method on the super-resolution task.

The qualitative results of nature sparse image reconstruction are demonstrated
in Fig. 4. Notably, the traditional iterative method PnP-ADMM failed to
produce satisfactory results for both the inpainting and super-resolution
tasks due to its limited prior terms. In contrast, our method outperforms the
comparison methods, particularly in terms of color and structure in the
inpainting task. In the super-resolution task, the results obtained by MCG
exhibit many artifacts, which are likely due to the projection stepChung et
al. (2023b). Our method, on the other hand, achieves competitive results with
DPS, the most advanced method, with small gaps. (More qualitative results of
nature sparse image reconstruction can be found in Appendix. E).

Table 4: Ablation evaluation (PSNR, SSIM) on the effect of latent-based
iterative reconstruction. We mark bold for the best and underline for the
second best. DPS Chung et al. (2023b) is a special case of DIR where
$\mathcal{U}\left(\cdot\right)\triangleq\left|\left|\cdot\right|\right|_{2}$
and use naive gradient update policy.

Method | Type | Sparse view | Limited angle | Speed(iter/s) $\uparrow$ | Memory(MB) $\downarrow$
---|---|---|---|---|---
18 | 32 | 45 | 90
PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | SSIM $\uparrow$
DPS | Pixel Diffusion | 28.55 | 0.8140 | 28.97 | 0.8242 | 28.25 | 0.8204 | 28.25 | 0.8088 | 20.88 | 6338
DIR (Ours) | Pixel Diffusion | 31.45 | 0.8654 | 32.82 | 0.8898 | 27.31 | 0.8133 | 28.98 | 0.8280 | 20.75 | 6338
LDIR (Ours) | Latent Diffusion | 39.01 | 0.9552 | 39.77 | 0.9612 | 29.60 | 0.8779 | 32.89 | 0.9116 | 36.67 | 4268

###  4.4 Ablation studies

We conducted ablation studies to validate the effectiveness of our approach.
we compared the performance of our latent-based iterative reconstruction
approach against a pixel-based iterative reconstruction approach. To ensure a
fair comparison, we conducted these ablation studies on the medical image
reconstruction task, as both the DDPM and LDM models were trained using the
same protocol.

In Table 4, we can observe that our LDIR outperforms both pixel-based
iterative reconstruction methods, DPS and DIR, by a large margin. This result
confirms that the latent-based approach is superior to the pixel-based
approach in terms of both speed and accuracy. Additionally, we can see that
changing the evaluation function to $L1$ and using a Momentum-like gradient
update policy in the pixel space can improve the performance of DIR, allowing
it to surpass DPS. Compared to pixel-space models, LDIR achieves significant
speed-up with less memory consumption. Although LDIR decodes latent into
pictures at each step, it still has a greater advantage than processing
directly in pixel space.

##  5 Conclusion

In this paper, we propose Latent Diffusion Iterative Reconstruction (LDIR) as
a novel approach for reconstructing CT sparse data in a zero-shot manner. We
show theoretically that utilizing the latent diffusion models as the prior
term is able to ignore the depends for the paired data. In addition, we
generate the prior term in the latent space instead of the pixel space, which
encourages us to form a high-resolution image with lower computational
complexity and sampling time. By using sample-level historical gradient
information from the data-fidelity term, we can guide the reconstruction
process in the latent space. Our experimental results demonstrate that LDIR
outperforms state-of-the-art methods including the supervised method on sparse
CT data reconstruction and achieves competitive results on nature image
restoration. We believe that our work offers the community a promising tool
for leveraging the rapidly growing field of latent diffusion models to restore
high-quality and high-resolution data from degraded measurements.

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# Spectral edge-to-edge topological state transfer in diamond photonic
lattices

Gabriel Cáceres-Aravena† Departamento de Física, Facultad de Ciencias Físicas
y Matemáticas, Universidad de Chile, Chile  Millenium Institute for Research
in Optics - MIRO, Chile  Bastián Real† Departamento de Física, Facultad de
Ciencias Físicas y Matemáticas, Universidad de Chile, Chile  Millenium
Institute for Research in Optics - MIRO, Chile  Diego Guzmán-Silva
Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile, Chile  Millenium Institute for Research in Optics -
MIRO, Chile  Paloma Vildoso  Departamento de Física, Facultad de Ciencias
Físicas y Matemáticas, Universidad de Chile, Chile  Millenium Institute for
Research in Optics - MIRO, Chile  Ignacio Salinas  Departamento de Física,
Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Chile
Millenium Institute for Research in Optics - MIRO, Chile  Alberto Amo  Univ.
Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Molécules, F-59000
Lille, France  Tomoki Ozawa  Advanced Institute for Materials Research (WPI-
AIMR), Tohoku University, Sendai 980-8577, Japan  Rodrigo A. Vicencio
<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias Físicas y
Matemáticas, Universidad de Chile, Chile  Millenium Institute for Research in
Optics - MIRO, Chile

(August 29, 2024)

###### Abstract

Transfer of information between topological edge states is a robust way of
spatially manipulating quantum states while preserving their coherence in
lattice environments. This method is particularly efficient when the edge
modes are kept within the topological gap of the lattice during the transfer.
In this work we show experimentally the transfer of photonic modes between
topological edge states located at opposite ends of a dimerized one-
dimensional photonic lattice. We use a diamond lattice of coupled waveguides
and show that the transfer is insensitive both to the presence of a high
density of states in the form of a flat band at an energy close to that of the
edge states, and to the presence of disorder in the hoppings. We explore
dynamics in the waveguide lattice using wavelength-scan method, where
different input wavelength translates into different effective waveguide
length. These results open the way to the implementation of more efficient
protocols based on the active driving of the hoppings.

Topological edge states are a remarkable resource to engineer photonic systems
with isolated modes protected from the presence of disorder. In two-
dimensional lattices, they can be used to fabricate topological edge mode
lasers with distributed gain and quantized orbital momentum [1, 2], to
transfer single photons around corners in elaborated photonic circuits [3, 4],
and to design topological frequency combs with enhanced efficiency [5, 6]. One
dimensional systems such as the Su-Schrieffer-Heeger (SSH) lattice are
particularly interesting because topological edge and interface modes are
hosted deep into the topological gap of the lattice. This gap protection has
been shown to be beneficial to preserve the quantum state of photons in
boundary modes [7, 8]. Interestingly, the presence of topological edge modes
on both sides of one-dimensional lattices can be used to transfer a state from
one edge of the lattice to the other with high fidelity with the advantage of
being protected from certain types of disorder due to the topological nature
of the system. Such edge state transfer is a promising route to store and
manipulate photonic quantum states in on-chip lattice environments.

Most topological edge transfer protocols rely on the adiabatic evolution of
the lattice such that an edge mode is driven into quasi-bulk modes and again
into an edge mode at the other side [9, 10, 11, 12, 13, 14, 15, 16, 17]. While
these protocols present an optimized transfer rate and fidelity, they are
limited by the adiabaticity condition that requires the adiabatic passage to
be slow enough to avoid the Zenner coupling of the edge state information into
the bulk modes [18, 19]. Furthermore, the presence of disorder in the lattice
would enhance this coupling. A variation of these protocols include counter-
adiabatic driving methods [20]. Recently, a different route has been proposed
based on the coherent coupling of edge modes within the gap [21, 19, 22]. The
great advantage of this approach is that edge modes are kept well into the
topological gap throughout the protocol, ensuring a high fidelity in reduced
times. The simplest version of the coherent state transfer of topological edge
states is via passive evanescent coupling of the exponential tails of edge
modes at opposite sides of the finite size lattice. In this case, coherent
transfer between edge modes takes place at a frequency determined by the tail
overlap, which can be controlled via the size of the gap. Observation of such
coherent oscillations was reported in a short SSH lattice for Rydberg atoms
[23].

In this work, we demonstrate coherent edge-to-edge transfer of light in a
dimerized diamond lattice of coupled waveguides employing a spectral
tomographic technique. More importantly, we show experimentally that the
fidelity of the transfer is robust to a number of perturbations in the system.
First, we show that orthogonality of eigenmodes in our undriven protocol
preserves the transfer even in the presence of a high density of states in the
form of a flat band at energies close to that of the edge states. Second, we
demonstrate that the transfer mechanism is robust to the presence of lattice
defects thanks to the underlying chiral symmetry of the system. The
experimental proof of principle we report in this work can be significantly
sped up by applying a number of driven techniques based on the modulating of
the hoppings in time and the use of concatenated topological lattices [19, 22,
24]. Such techniques are readily implementable in lattices of coupled
waveguides.

Figure 1: (a) Sketch of a dimerized diamond lattice, with A, B and C the sites
of the unit cell. Thick (thin) line denotes a strong (weak) hopping, and
$t_{1}$ ($t_{2}$) indicates the intra(inter)-cell coupling constant. Top
(bottom) panel schematizes the trivial (topological) case $t_{1}>t_{2}$
($t_{1}<t_{2}$). (b) Spectrum as a function of $\delta$ for a finite (lines)
and an infinite (shaded area) lattice. The vertical line denotes $\delta=1$.
The color indicates the IPR for all the states. Inset: amplitude profiles of
edge states at $\delta=0.4$. (c) Sketch of the fs laser writing technique. (d)
Microscope image of a diamond lattice with $\\{d_{1},d_{2}\\}=\\{35,25\\}\
\mu$m ($\delta=0.37$) and $d_{v}=32\,\mu$m. Output images for the lattice in
(d) and for the excitation at (e1) a $B$ left edge site and (e2) an in-phase
$A$-$C$ right edge sites. Yellow ellipses indicate the excited sites.

To demonstrate the topological edge transfer we use a diamond lattice of
coupled waveguides with different intracell ($t_{1}$) and intercell ($t_{2}$)
hoppings, as sketched in Fig. 1(a). The lattice has three sites per unit cell,
denoted as A, B and C sites. Considering a tight-binding coupled-mode
approach, the evolution of the optical field at every site of the n-th unit
cell is written as:

$\displaystyle-i\partial_{z}A_{n}$ $\displaystyle=t_{1}B_{n}+t_{2}B_{n+1}\,,$
$\displaystyle-i\partial_{z}B_{n}$
$\displaystyle=t_{1}(A_{n}+C_{n})+t_{2}(A_{n-1}+C_{n-1})\,,$ (1)
$\displaystyle-i\partial_{z}C_{n}$ $\displaystyle=t_{1}B_{n}+t_{2}B_{n+1}\,.$

Here $A_{n}$, $B_{n}$ and $C_{n}$ are the amplitudes of the optical field at
the n-th unit cell. $z$ describes the coordinate along the waveguides and the
dynamical variable. Moreover, the hopping strengths among nearest-neighbor
(NN) sites can be varied experimentally by adjusting the lattice distances
[25]. We then define the control parameter $\delta\equiv t_{1}/t_{2}$ to
characterize the different regimes. We assume an infinite system and impose a
Bloch-like ansatz in Eq. (Spectral edge-to-edge topological state transfer in
diamond photonic lattices), obtaining the following bands:

$\beta_{z}(k_{x})=0,\pm
t_{2}\sqrt{2\left[\delta^{2}+2\delta\cos(k_{x}a)+1\right]}\ ,$ (2)

where $\beta_{z}$ is the propagation constant (energy), $a$ is the lattice
constant and $k_{x}$ the quasimomentum. The spectrum is composed of two
dispersive and one flat band (FB) [see shaded areas and horizontal light-blue
line at $\beta_{z}=0$ in Fig. 1(b), respectively]. The gap in between both
dispersive bands has a size equal to $2\sqrt{2}t_{2}|\delta-1|$. For
$\delta=1$, this gap closes and the three bands touch each other at the edges
of the Brillouin zone [26]. The diamond lattice possesses the smallest
experimentally reported FB states [27, 26], with an inverse participation
ratio (IPR) [28] of $1/2$ [represented as light-blue color in Fig. 1(b)].
Specifically, in the bases of Wannier functions in the A, B and C sites, the
FB eigenvector is given by: $\ket{v^{FB}}=\\{1,0,-1\\}/\sqrt{2}$, and the ones
corresponding to the dispersive bands are
$\ket{v^{\pm}}=\\{e^{i\phi(k_{x})},\pm\sqrt{2},e^{i\phi(k_{x})}\\}/\sqrt{2}$,
where $\phi(k_{x})=\arctan(-\sin(k_{x}a)/[\delta+\cos(k_{x}a)])$.

Even though this lattice has three sites per unit cell, it exhibits similar
topological features to the SSH model [29], when varying the parameter
$\delta$ [30]. Indeed, a quantized Zak phase of a value $0$ or $\pi$ can be
found when $\delta>1$ ($t_{1}>t_{2}$) or $\delta<1$ ($t_{1}<t_{2}$),
respectively. In this case, the nontrivial phase is protected by inversion
symmetry between $A_{n}$ and $C_{n}$ and by chiral symmetry [31, 30]. Thus, we
expect the appearance of two edge states at zero propagation constant on a
lattice with open boundaries when $\delta<1$. To corroborate this, we compute
the spectrum as a function of $\delta$ for dimerized diamond lattices of $9$
unit cells [see full lines in Fig. 1(b)]. It can be clearly seen that two
states at zero frequency (lighter blue) transform into two dispersive states
(darker blue) around $\delta=1$. When increasing $\delta$, the degeneracy
between them is removed at around $\delta=0.7$ (splitting
$\Delta\beta_{z}^{e}\sim 0.06$) [32] due to the finite size of the lattice.
The flat band remains unchanged at $\beta_{z}=0$, for any value of $\delta$.
The IPR (denoted by color) shows very clearly the transition from localized
edge states (IPR $=1$ or $1/2$, light blue) into extended propagating modes
(IPR $\sim 1/N$, black).

Figs. 1(b)-insets show the two edge states for $\delta=0.4$. They exhibit
exponentially localized amplitudes at both edges. On the left edge, these
states present a null amplitude at A and C sites, whereas the states have a
null amplitude at B sites at the right edge. Moreover, one edge state is
antisymmetric (bottom inset) and the other one is symmetric (top inset), with
respect to the opposite edge. They decay exponentially into the bulk as
$(-\delta)^{|n-n_{e}|}$ for a semi-infinite system, exhibiting a phase shift
of $\pi$ at consecutive B or A,C sites, depending on the specific edge
($n_{e}$). Notice in Fig. 1(b) that the edge states are degenerate for
$\delta\lesssim 0.7$; consequently, the sum of these states gives a state
fully localized at the left edge with amplitude on B sites only and,
conversely, the subtraction of them gives a state fully localized at the right
edge with amplitude on A and C sublattices. For $\delta\gtrsim 0.7$, the
degeneracy of the edge states is lifted, and their frequency deviates from $0$
to $\pm\beta_{z}^{e}$. Therefore, the excitation of sites at the edges is
expected to induce an oscillatory pattern in between both surfaces with a
frequency $\beta_{z}^{e}$ [33], with a long-distance state transfer occurring
on a dynamical scale $z_{transfer}=\pi/\beta_{z}^{e}$ [32].

We fabricate several dimerized diamond photonic lattices, of $9$ unit cells
each, by using a femtosecond (fs) laser writing technique [34, 25, 32], as it
is sketched in Fig. 1(c). For a first set of experiments, the diamond geometry
is defined by distances $d_{1}$, $d_{2}$ and $d_{v}=32\ \mu$m, as described in
Fig. 1(d). For these values, the diagonal (NN) distance was swept in the
interval $\\{25.6,43.1\\}\ \mu$m, as $d_{1}$ and $d_{2}$ were varied in the
interval $\\{20,40\\}\ \mu$m in steps of $1\,\mu$m. The hopping coefficients
(which decay exponentially on waveguide separation [25]) range in the interval
$\sim\\{0.03,0.21\\}$ cm-1 at a wavelength of $640$ nm [32]. Fig. 1(d) shows
an output facet of a lattice with $d_{1}=35$ and $d_{2}=25\ \mu$m, with
$t_{1}=0.05$ and $t_{2}=0.14$ cm-1 ($\delta=0.37$). We first test the quality
of the lattices by exciting them at different input positions using a $640$ nm
laser beam (see Ref. [32] for a complete characterization). For example,
topological edge states can be efficiently excited by injecting light directly
at the lattice boundaries [35, 28, 36, 32]. Fig. 1(e1) shows the output
profile after a B-edge site excitation, with a clear exponential decaying
profile from the edge into the bulk. The excitation at the right boundary
requires a more complicated input condition with two in-phase beams. The
result of this is shown in Fig. 1(e2), with an output profile formed by A and
C sites mostly.

Figure 2: (a) Output intensity profiles for a B-site central excitation, for a
lattice with $d_{1}=d_{2}=30\ \mu$m, at the indicated wavelength. (b) $m2$
versus $\lambda$. (c) $m_{d}$ versus $d_{1}$ (bottom) and $\delta$ (top).
Insets in (c) show the profile at $700$ nm for the indicated case. The bar
shows the standard deviation. Yellow ellipses indicate the excited sites.

We propose a novel technique to characterize the lattice dynamics. Instead of
measuring the output profiles at different $z$ values (which also implies the
fabrication of a larger number of lattices), we implement a wavelength-scan
method: The dynamics of a wavepacket injected in the input facet of a lattice
is revealed when varying the input wavelength coming from a Supercontinuum
(SC) laser source. In general, the lattice dynamics depends on the excitation
wavelength $\lambda$: the longer the wavelength the wider the mode profile and
the larger the coupling constants [25, 32]. In this way, by tuning the input
wavelength, we can study the same lattice at different effective lengths.

We first consider a diamond lattice with $d_{1}=d_{2}=30\ \mu$m. We excite a B
site at the central $5$-th unit cell and scan the input wavelength in the
interval $600-760$ nm, with a step of $10$ nm. Fig. 2(a) shows the output
intensity for three selected $\lambda$’s [32]. Fig. 2(b) shows the second
moment (width), defined as $m2\equiv\sum_{n}(n-n_{c})^{2}P_{n}$, versus
wavelength, with $P_{n}\equiv|A_{n}|^{2}+|B_{n}|^{2}+|C_{n}|^{2}$ the unit
cell power and $n_{c}\equiv\sum_{n}nP_{n}$ the center of mass. We observe a
growing diffraction pattern [27], with a width that increases almost linearly
with the input wavelength [a linear fit is included in Fig. 2(b)]. $m2\sim z$
corresponds to a diffusive regime [37], as expected for discrete diffraction;
therefore, a $\lambda$ increment produces an effectively larger propagation
distance $z$ or a larger coupling constant $t_{1,2}$.

A dimerized diamond lattice has two hoppings which simultaneously change while
$\lambda$ is modified. Since we observe a linear dependence of coupling
constants on wavelength, we can assume $\delta$ as a constant, as a first
approximation. We use the wavelength-scan method to experimentally determine
$n_{c}$ for all the output profiles, after exciting a B site at the central
($5$-th) unit cell of $17$ dimerized lattices, having different values of
$\delta$. For each lattice, we average $n_{c}$ over $\lambda$ and obtain the
averaged beam displacement $m_{d}$, from which the topological invariant can
be inferred [38, 39]. A topologically trivial lattice has a $m_{d}=0$, as an
indication of a zero Zak phase. A topologically non-trivial system will shift
this value to $m_{d}\sim 0.5$, corresponding to a $\pi$ Zak phase [38]. Our
collected results are shown in Fig. 2(c). We observe that for $d_{1}<30\ \mu$m
($\delta>1$) the lattice is topologically trivial and the propagation shows a
$m_{d}$ around zero. For $30\leqslant d_{1}\leqslant 34\ \mu$m
($0.4<\delta\leqslant 1$), a transition region without a well defined
topological phase is observed. For $d_{1}\geqslant 35\ \mu$m ($\delta\leqslant
0.4$), the lattices express a clear averaged beam displacement around $0.5$,
implying a nontrivial Zak phase. Therefore, the wavelength-scan method gives
us valuable information about the dynamics on a specific lattice, and it
becomes a key method to determine its topological phase on a finite size
configuration.

Figure 3: (a) Output profiles of a non-trivial diamond photonic lattice at
different $\lambda$’s, after a B-edge excitation (see yellow ellipse). (b)
Fidelity versus wavelength for topological (black), trivial (gray),
topological + defect (red), and trivial + defect (orange) lattices. (c)
Microscope image of a topological lattice plus two coupling defects (see
dashed rectangle). (d) Same than (a) for a topological lattice with a coupling
defect.

The number of unit cells in the lattice affects the edge state properties: the
fewer unit cells, the shorter the range of $\delta$ in which the edge states
keep degenerate in $\beta_{z}$ [32]. When the degeneracy is lifted, the two
edge modes hybridize. Therefore, an input on one edge will excite coherently
both modes and result in periodic oscillations of the amplitude at the two
edges. Then, a transfer of light from one edge to the other becomes possible
[33, 23]. To experimentally demonstrate this we fabricate a topological
lattice with $9$ unit cells and distances $d_{1}=18$, $d_{2}=14$, and
$d_{v}=14\ \mu$m ($t_{1}=0.30$ and $t_{2}=0.42$ cm-1 at $640$ nm, and
$\delta=0.71$). The trivial lattice ($\delta=1.40$) is obtained by inverting
these distances to $d_{1}=14$ and $d_{2}=18\ \mu$m. We decreased the distances
to increase the coupling coefficients and favor a faster transport in between
the edges, while staying at the non-degenerate situation. Again, we use a SC
laser source in the range $610-740$ nm and sweep the input wavelength in steps
of $10$ nm. We excite the system by injecting light at the B left edge
waveguide, as shown in Fig. 3(a). For $\lambda\lesssim 670$ nm, the intensity
profiles are well localized at the left edge, with most of the light intensity
at the B sublattices, with a profile resembling the edge state [Fig. 1(e1)].
The edge states splitting manifests for $\lambda\approx 680$ nm, where we
start observing a smooth population of the opposite edge, with a weak
excitation of the lattice center (a weak background radiation is always
observed because of the excitation of dispersive modes [32]). The connection
in between both edge patterns [see Figs. 1(b)-insets and (e)], with a B-site
exponential decaying profile at the left surface and an A,C exponential
profile at the right edge, becomes evident for $\sim 710$ nm. The spectral
state transfer phenomenon starts occurring at $\lambda\gtrsim 720$ nm: the
light injected at one edge is mostly transferred into the opposite edge [32].
This shows a very interesting transport mechanism which does not require that
the light explores the whole lattice; in this case, the light is suddenly
transferred from one edge into the other without interacting with the lattice
bulk.

We define the fidelity $F$ for an edge-to-edge light transfer by measuring the
normalized transferred intensity at the opposite lattice edge:
$F\equiv(|A_{edge}|^{2}+|C_{edge}|^{2})/\sum_{n}P_{n}$. If all the light
reaches the two rightmost sites $F=1$, and $F=0$ in the fully opposite case.
We show our results in Fig. 3(b), where we plot the fidelity $F$ versus
$\lambda$, for topological and trivial lattices. We observe how the
topological (black) and the trivial (gray) cases have a similar dynamical
scale; i.e., both processes occur approximately at the same speed. However,
the fidelity at the $A,C$ surface is larger for the topological lattice ($\sim
61\%$). The trivial lattice presents a standard discrete diffraction pattern
[27], with the energy exploring the whole lattice while it moves from one edge
into the other [32], as the wavelength increases [similar to Fig. 2(a)].
Therefore, once the light arrives at the $A,C$ right edge it is reflected back
due to the absence of the edge states. The fidelity in this case decreases to
a $\sim 40\%$.

A remarkable feature of the state transfer between topological edge states is
the resilience to certain types of disorder. Although the fabrication process
can produce random on-site or inter-site defects, we fabricate a couple of
lattices with a symmetric coupling defect, as the one shown in Fig. 3(c). We
design a different distance in between the fourth and the fifth cells and
inside the fifth cell [see dashed rectangle in Fig. 3(c)]. Specifically, we
set this distance to $23\ \mu$m, implying a coupling defect of $0.18$ cm-1.
Fig. 3(d) shows a set of output images at the indicated values of $\lambda$,
for the topological lattice with a defect. We notice that this defect produces
some reflection and trapping of energy at short wavelengths, consequently, not
all the energy is edge-to-edge transferred. Despite this, a significant amount
of the light excites the topological right-edge state composed of $A$ and $C$
sites. The fidelity is $\sim 26\%$ for the topological case, whereas it drops
to $\sim 15\%$ for the trivial lattice.

These numbers show that a trivial lattice undergoes a stronger back reflection
caused by the defect, because the light explores the whole lattice and
interacts strongly with it. On the other hand, in the topological case, the
light does not travel across the lattice and excite efficiently the edge state
without the need of arriving at the boundary by standard transportation. The
fidelity is not perfect in none of the topological cases because a single
$B$-site input always excites part of the dispersive spectrum, in which the
modes extend over the entire lattice. Nonetheless, the strong difference
between the topological and trivial cases is the key of success for a
topological state transfer process, which occurs due to the excitation of
exponentially localized edge states which live at both edges simultaneously
and deep in the gap of the lattice spectrum. This could be a proof of concept
for a long distance sensor, which detects away from the interaction region.

The wavelength-scan method proposed in this work offers a tool for
investigating the dynamics in lattices of coupled waveguides. Using this
method, we evidenced the nontrivial topology of dimerized diamond lattices by
measuring the averaged beam displacement and, additionally, we demonstrated an
edge-to-edge transfer of light via the excitation of the topological edge
states. This transfer is partially robust to defects across the lattice bulk
and, hence, it has the potential for a precise wavelength filter, as well as
an efficient information transport mechanism between two distant ports or by
concatenating several topological lattices in the quantum domain [12, 14].

Acknowledgments. This work was supported in part by Millennium Science
Initiative Program ICN17-012, and FONDECYT Grant 1191205. A.A. acknowledges
the support of European Research Council grant EmergenTopo (865151), the
French government through the Programme Investissement d’Avenir (I-SITE ULNE /
ANR-16-IDEX-0004 ULNE) managed by the Agence Nationale de la Recherche, the
Labex CEMPI (ANR-11-LABX-0007) and the CPER Wavetech. T.O. acknowledges the
support from JSPS KAKENHI Grant No. JP20H01845, JST PRESTO Grant No.
JPMJPR19L2, and JST CREST Grant No.JPMJCR19T1.

†Both authors contributed equally.

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