# Diffusion Model for Camouflaged Object Detection

Zhennan Chen  Rongrong Gao  Tian-Zhu Xiang  Fan Lin Corresponding Authors.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Informatics,
Xiamen University, Xiamen, China  Department of Computer Science and
Engineering, HKUST, Hong Kong, China  G42, Abu Dhabi, UAE

###### Abstract

Camouflaged object detection is a challenging task that aims to identify
objects that are highly similar to their background. Due to the powerful
noise-to-image denoising capability of denoising diffusion models, in this
paper, we propose a diffusion-based framework for camouflaged object
detection, termed diffCOD, a new framework that considers the camouflaged
object segmentation task as a denoising diffusion process from noisy masks to
object masks. Specifically, the object mask diffuses from the ground-truth
masks to a random distribution, and the designed model learns to reverse this
noising process. To strengthen the denoising learning, the input image prior
is encoded and integrated into the denoising diffusion model to guide the
diffusion process. Furthermore, we design an injection attention module (IAM)
to interact conditional semantic features extracted from the image with the
diffusion noise embedding via the cross-attention mechanism to enhance
denoising learning. Extensive experiments on four widely used COD benchmark
datasets demonstrate that the proposed method achieves favorable performance
compared to the existing 11 state-of-the-art methods, especially in the
detailed texture segmentation of camouflaged objects. Our code will be made
publicly available at: https://github.com/ZNan-Chen/diffCOD.

##  1 Introduction

Camouflage is to use any combination of coloration, illumination, or materials
to hide organisms in their surroundings, or disguise them as something else,
for deception and paralysis purposes. Camouflaged object detection (COD) [13],
that is, segmenting camouflaged objects from the background, is a challenging
vision topic that has emerged in recent years, due to the high similarity of
camouflaged objects to the background. COD has also attracted growing research
interest from the computer vision community, because of its wide range of
real-world applications, such as agricultural pest detection [30], medical
image segmentation [34], and industrial defect detection [51].

With the advent of large-scale camouflaged object detection datasets in recent
years, such as CAMO [31] and COD10K [13] datasets, numerous deep learning-
based methods have been proposed and achieved great progress. Some methods are
inspired by human visual mechanisms and adopt convolutional neural networks to
imitate predation behavior, thus designing a series of models for COD, such as
search identification network [12], positioning and focus network [37], zoom
in and out [41], and PreyNet [61]. Some methods adopt auxiliary cues to
improve network discrimination, or branch tasks to jointly learn camouflage
features. The former typically employ frequency domain [63], edge/texture [24,
65], or motion information [5] to improve feature representation, and the
latter usually introduces boundary detection [50], classification [31],
fixation [36], or saliency detection [32] for multi-task collaborative
learning. More recently, to improve global contextual exploration,
transformer-based approaches have also been proposed, such as HitNet [22] and
FSPNet [23]. Although these methods have greatly improved the performance of
camouflaged object detection, the existing methods still struggle to achieve
accurate location and segmentation in most complex scenarios, due to the
interference of highly similar backgrounds and the complexity of the
appearance of camouflaged objects.

In recent years, diffusion models [20] have demonstrated impressive
performance in the generative modeling of images and videos [10], opening up a
new era of generative models. Diffusion models are a class of generative
models that consist of Markov chains trained using variational inference, to
denoise noisy images blurred by Gaussian noise via learning the reverse
diffusion process. Because of its powerful noise-to-image denoising pipeline,
the computer vision community is curious about its variants for discriminative
tasks [8]. More recently, diffusion models have been found to be highly
effective in other computer vision tasks, such as image editing [19], super-
resolution [33], instance segmentation [17], semantic segmentation [3, 4] and
medical image segmentation [43, 53]. However, despite their great potential,
diffusion models for challenging camouflaged object detection have still not
been well explored.

(a) Mainstream COD paradigm.

(b) Diffusion-based COD paradigm.

Figure 1: (a) The current mainstream COD paradigm inputs images into the
network for prediction in a single direction, generating a deterministic
segmentation mask. (b) Our proposed diffCOD provides a novel paradigm that
decomposes COD into a series of forward-and-reverse diffusion processes.

normal

In this paper, we propose to formulate the camouflaged object detection as a
generative task, through a denoising diffusion process from the noisy mask to
the object mask in the image. Specifically, in the training stage, Gaussian
noise is added to the ground-truth masks to obtain noisy masks, and then the
model learns to reverse this noising process. In the inference stage, the
model progressively refines a set of randomly generated noisy masks from the
image through the learned denoising model, until they perfectly cover the
targeted object without noise. We can see that the denoising diffusion model
is the process of recovering the ground-truth mask from the random noisy
distribution to the learned distribution over object masks. As shown in Figure
1(b), unlike previous deterministic network solutions that produce a single
output for an input image, we decouple the detection of the object into a
novel noise-to-mask paradigm with a series of forward-and-reverse diffusion
steps, which can output masks from single or multi-step denoising, thereby
generating multiple object segmentation masks from a single input image.

To this end, we propose a denoising diffusion-based model, termed diffCOD,
which approaches camouflaged object tasks from the perspective of the noise-
to-mask denoising diffusion process. The proposed model adopts a denoising
network conditioned on the input image prior. The semantic features extracted
from the image by a Transformer encoder are integrated into the denoising
diffusion model to guide the diffusion process at each step. To effectively
bridge the gap between the diffusion noise embedding and the conditional
semantic features, an injection attention module (IAM) is designed to enhance
the denoising diffusion learning by aggregating conditional semantic features
and diffusion model encoder through a cross-attention mechanism. Our
contributions are summarized as follows:

  * •

We extend the denoising diffusion models to the task of camouflaged object
detection, and propose a diffusion-based object segmentation model, called
diffCOD, a novel framework that views camouflaged object detection as a
denoising diffusion process from noisy masks to object masks.

  * •

We design an injection attention module (IAM) to model the interaction between
noise embeddings and image features. The proposed module adopts the cross-
attention mechanism to integrate the conditional semantic feature extracted
from the image into the diffusion model encoder to guide and enhance denoising
learning.

  * •

Extensive quantitative and qualitative experiments demonstrate that the
proposed diffCOD achieves superior performance over the recent 11 state-of-
the-art (SOTA) methods by a large margin, especially in object detail texture
segmentation, indicating the effectiveness of the proposed method.

##  2 Related Work

###  2.1 Camouflaged Object Detection

Existing COD methods [11, 12, 13] are based on a non-generative approach to
segment the objects from the background. The approaches in COD can be broadly
categorized into the following strategies: a) Introducing additional cues to
facilitate the exploration of camouflage features. BGNet [50] uses edge
semantic information to enable the model to extract features that highlight
the structure of the object and thus pinpoint the object boundary. TINet [65]
designs a texture label to find boundaries and texture differences through
progressive interactive guidance. FDCOD [63] incorporates frequency domain
features into CNN models to better detect objects from the background. DGNet
[24] utilizes gradient edge information to facilitate the generation of
contextual and texture features. b) Multi-task learning strategies are used to
improve segmentation capabilities. ANet [31] proposed joint learning of
classification and segmentation tasks to help the model improve recognition
accuracy. UJSC [32] detects both salient and camouflaged objects to improve
the model performance. Rank-Net [36] proposes to use the localization model to
find the obvious discriminative region of the camouflaged object, and the
segmentation model to segment the full range of the camouflaged object. c)
Coarse-to-fine feature learning strategy is utilized to explore and integrate
multi-scale features. SegMaR [27] uses multi-stage detection to focus on the
region where the goal is located. ZoomNet [40] learns multi-scale semantic
information through multi-scale integration and hierarchical hybrid strategies
to promote models that produce predictions with higher confidence. PreyNet
[61] imitates the predation process for stepwise aggregation and calibration
of features. PFNet [37] mimics nature’s predation process by first locating
potential targets from a global perspective and then gradually refining the
fuzzy regions. SINet [13] is designed to improve segmentation performance by
locating the object first and then differentiating the details. $C^{2}$FNet
[49] proposes to use global contextual information to fuse on high-level
features in a cascading manner to obtain better performance. HitNet [22] and
FSPNet [23] propose to explore global context cues by transformers. In this
paper, we introduce generative models, i.e., denoising diffusion models, into
the COD task to gradually refine the object masks from the noisy image, which
achieve excellent performance, especially for objects with fine textures.

###  2.2 Diffusion Model

The diffusion model [20, 47] is a generative model that uses a forward
Gaussian diffusion process to sample a noisy image, and then iteratively
refines it using a backward generative process to obtain a denoised image.
Diffusion models have shown strong potential in several fields, such as image
synthesis [10, 20], image editing [19], and image super-resolution [9].
Moreover, the learning process of diffusion models is able to capture high-
level semantic information that is valuable for segmentation tasks [3], which
has led to a growing interest in diffusion models for image segmentation
including medical image segmentation [53, 54], semantic segmentation [4, 26,
55, 57], and instance segmentation [1, 17]. MedSegDiff [53] proposes the first
DPM-based medical segmentation model, and MedSegDiff-V2 [54] further improves
the performance based on it using transformer. DDeP [4] finds that pre-
training a semantic segmentation model as a denoising self-encoder is
beneficial for performance improvement. DDP [26] designs a dense prediction
framework with stepwise denoising refinement guided by image features. ODISE
[57] combines a trained text image diffusion model with a discriminative model
to achieve open-vocabulary panoptic segmentation. DiffuMask [55] uses a model
for the automatic generation of image and pixel-level semantic annotations,
and it also shows superiority in open vocabulary segmentation. DiffusionInst
[17] proposes the first instance segmentation model based on a diffusion
process to achieve global instance mask reconstruction. Segdiff [1] uses a
diffusion probabilistic approach to design an end-to-end segmentation model
that does not rely on a pre-trained backbone. However, there are no studies
that demonstrate the effectiveness of diffusion models in COD tasks. In this
work, we present the first diffusion model for the COD segmentation task.

Figure 2: Our proposed diffCOD framework for COD, which feeds a given image
into a denoising diffusion model with UNet architecture as the core component
for denoising. An injection attention module (IAM) is designed to implicitly
guide the diffusion process with the conditional semantic features that have
gone through the backbone and feature fusion module (FF), allowing the model
to take full advantage of the correspondence between image features and
diffusion information.

##  3 Methodology

In this section, we first review the diffusion model (Sec. 3.1). Then we
introduce the architecture of diffCOD (Sec. 3.2). Finally, we describe the
specific process of training and inference of diffCOD (Sec. 3.3 & Sec. 3.4).

###  3.1 Diffusion Model

The diffusion probability model has reaped plenty of attention due to its
simple training process and excellent performance. It is mainly divided into
forward process and reverse process. In the forward process, noise is added to
the target image to make it closer to the Gaussian distribution. The reverse
process learns to map the noise to the real image.

The forward process refers to the gradual incorporation of Gaussian noise with
variance $\beta_{t}\in(0,1)$ into the original image $x_{0}\sim
p\left(x_{0}\right)$ at time $t$ until it converges to isotropic Gaussian
distribution. The forward process is described by the formulation:

$q\left(x_{t}\mid
x_{t-1}\right)=\mathcal{N}\left(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}\mathbf{I}\right)$
(1)

where $t\in[1,T]$. We can obtain the latent variable $x_{t}$ directly by using
$x_{0}$ by the following equation:

$q\left(x_{t}\mid
x_{0}\right)=\mathcal{N}\left(x_{t};\sqrt{\bar{\alpha}_{t}}x_{0},\left(1-\bar{\alpha}_{t}\right)\mathbf{I}\right)$
(2)

where $\alpha_{t}:=1-\beta_{t}$, $\bar{\alpha}_{t}:=\prod_{s=0}^{t}\alpha_{s}$
and $\epsilon\sim\mathcal{N}(0,\mathbf{I})$.

The reverse process converts the latent variable distribution $p(x_{T})$ to
$p(x_{0})$ through a Markov chain, and the reverse process can be denoted as
follows:

$p_{\theta}\left(x_{t-1}\mid
x_{t}\right)=\mathcal{N}\left(x_{t-1};\mu_{\theta}\left(x_{t},t\right),\Sigma_{\theta}\left(x_{t},t\right)\right)$
(3)

The combination of $q$ and $p$ is a variational auto-encoder, and the
variational lower bound (VLB) is defined as follows:

$L_{\mathrm{vlb}}:=L_{0}+L_{1}+\ldots+L_{T-1}+L_{T}$ (4) $L_{0}:=-\log
p_{\theta}\left(x_{0}\mid x_{1}\right)$ (5)
$L_{t-1}:=D_{KL}\left(q\left(x_{t-1}\mid x_{t},x_{0}\right)\|\
p_{\theta}\left(x_{t-1}\mid x_{t}\right)\right)$ (6)
$L_{T}:=D_{KL}\left(q\left(x_{T}\mid x_{0}\right)\|\
p\left(x_{T}\right)\right)$ (7)

###  3.2 Architecture

As shown in Figure 2, the proposed diffCOD aims to solve the COD task by the
diffusion model. The denoising network of diffCOD is based on the UNet
architecture [44]. To get effective conditional semantic features, we obtain
multi-scale features by ViT-based backbone and feature fusion (FF) to yield
features containing rich multi-scale details. In addition, to let the texture
patterns and localization information in the conditional semantic features
guide the denoising process, we propose an injection attention module (IAM)
based on cross-attention. This allows the network to reduce the difference
between diffusion features and image features and to combine the advantages of
both.

Feature Fusion (FF). Given an initial input image $x_{o}\in\mathbb{R}^{H\times
W\times 3}$, we adopt the top-three high-level features of the visual backbone
as our multi-scale backbone features, denoted as $\mathcal{X}^{p}_{i}$,
$i\in\\{1,2,3\\}$ whose resolution is $\frac{H}{k}\times\frac{W}{k}$,
$k\in\\{8,16,32\\}$. Here we use PVTv2 [52] as the backbone. Then FF is used
to aggregate these multiscale features. Specifically, FF contains three
branches to process $\mathcal{X}^{p}_{i}$, each branch uses two convolution
operations with 3$\times$3 kernel for feature enhancement, and finally the
three branches are coalesced by a single convolution to obtain
$\mathcal{F}\in\mathbb{R}^{\frac{H}{32}\times\frac{W}{32}\times C}$.

Injection Attention Module (IAM). To introduce texture and location
information of the original features in the noise prediction process, we
employ a cross-attention-based IAM, which is embedded in the middle of the
UNet-based denoising network. Given the multiscale fusion feature
$\mathcal{F}$ from FF and the deepest feature
$\mathcal{D}\in\mathbb{R}^{\frac{H}{32}\times\frac{W}{32}\times C}$ from the
diffusion model as the common input to the IAM. Specifically, $\mathcal{D}$ is
transformed by linear projection to generate the query $\mathbf{Q^{D}}$, the
key $\mathbf{K^{D}}$ and the value $\mathbf{V^{D}}$. $\mathcal{F}$ generates
$\mathbf{P^{F}}$, $\mathbf{V^{F}}$ by linear projection, and it is noteworthy
that $\mathcal{F}$ does not generate the query and the key for similarity
comparison, but uses the generated $\mathbf{P^{F}}$ to act as an intermediary
for similarity comparison with $\mathcal{D}$. This process is defined as
follows:

$\begin{split}\mathbf{Q^{D}}=\mathcal{D}\cdot\mathcal{W_{Q}^{D}}&,\quad\mathbf{K^{D}}=\mathcal{D}\cdot\mathcal{W_{K}^{D}},\quad\mathbf{V^{D}}=\mathcal{D}\cdot\mathcal{W_{V}^{D}}\\\
\mathbf{P^{F}}&=\mathcal{F}\cdot\mathcal{W_{P}^{F}},\quad\mathbf{V^{F}}=\mathcal{F}\cdot\mathcal{W_{V}^{F}}\end{split}$
(8)

where $\mathcal{W_{Q}^{D}}$, $\mathcal{W_{K}^{D}}$, $\mathcal{W_{V}^{D}}$,
$\mathcal{W_{P}^{F}}$, $\mathcal{W_{V}^{F}}$ $\in\mathbb{R}^{d\times d}$. $d$
is the dimensionality.

Thus the IAM operation is defined as follows:

$\mathbf{M}^{att}_{1}=\operatorname{Softmax}\left(\frac{\mathbf{Q^{D}}\cdot(\mathbf{P^{F}})^{T}}{\sqrt{d}}\right)$
(9)
$\mathbf{M}^{att}_{2}=\operatorname{Softmax}\left(\frac{\mathbf{K^{D}}\cdot(\mathbf{P^{F}})^{T}}{\sqrt{d}}\right)$
(10)
$O^{I}=\mathbf{M}^{att}_{1}\cdot\mathbf{M}^{att}_{2}\cdot(\mathbf{V}^{D}+\mathbf{V}^{F})$
(11)

where $\mathbf{M}^{att}_{1}$ and $\mathbf{M}^{att}_{2}$ represent the
attention maps of $\mathbf{Q^{D}}$-$\mathbf{P^{F}}$ and
$\mathbf{K^{D}}$-$\mathbf{P^{F}}$, respectively.
$O^{I}\in\mathbb{R}^{\frac{H}{32}\times\frac{W}{32}\times C}$ denotes the
final generated cross-attention fusion feature.

###  3.3 Training

In the forward process, the Gaussian noise $\epsilon_{t}$ is added to the
ground truth $y_{0}$ to obtain the noise mapping ${y}_{t}\sim
q\left({y}_{t}\mid{y}_{0}\right)$ by $T$-steps. The intensity of the noise is
controlled by $\alpha_{t}$ and conforms to the standard normal distribution.
This process can be defined as follows:

$y_{t}=\sqrt{\alpha_{t}}y_{t-1}+\left(1-\alpha_{t}\right)\epsilon_{t}$ (12)

where $t=[1,\cdots,T]$ and $\epsilon_{t}\sim\mathcal{N}(0,\mathbf{I})$.

By iterative computation, we can directly obtain $y_{t}$. This process can be
further marginalized as:

$y_{t}=\sqrt{\bar{\alpha}_{t}}y_{0}+\left(1-\bar{\alpha}_{t}\right)\epsilon_{t}$
(13)

where $\bar{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}$.

In the reverse process, we map from $y_{t}$ to $y_{t-1}$ until the segmented
image is acquired step by step. The mathematics is defined as follows:

$y_{t-1}=\mu_{\theta}\left(y_{t},t,x_{o}\right)+\Sigma_{\theta}\left(y_{t},t,x_{o}\right)\epsilon_{t}$
(14)

We train a denoising UNet model to predict
${\epsilon}_{\theta}\left({y}_{t},t,{x}_{o}\right)$:

$\mu_{\theta}\left(y_{t},t,x_{o}\right)=\frac{\left(y_{t}-\left(\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\right)\epsilon_{\theta}\left(y_{t},t,x_{o}\right)\right)}{\sqrt{\alpha_{t}}}$
(15)

We follow the improved DDPM [39] to simplify Eq. (4)-(7) to define the hybrid
objective $L_{\mathrm{hybrid}}=L_{\mathrm{simple}}+L_{\mathrm{vlb}}$.
$L_{\mathrm{vlb}}$ learns the term
$\Sigma_{\theta}\left(y_{t},t,x_{o}\right)$. Furthermore, inspired by [54], we
use FF and a convolution layer to provide an initial static mask $y_{m}$ to
reduce the diffusion variance, and its mean square loss is defined as
$L_{\mathrm{static}}$. Total loss function $L_{total}$ is defined as follows:

$\left\\{\begin{array}[]{ll}L_{\mathrm{simple}}&=\mathbb{E}_{t\sim[1,T],y_{0}\sim
q\left(y_{0}\right),\epsilon}\left\|\epsilon-\epsilon_{\theta}\left(y_{t},t,x_{o}\right)\right\|^{2}\\\
L_{\mathrm{static}}&=\mathbb{E}_{y_{0}\sim
q\left(y_{0}\right),y_{m}}\left\|y_{0}-y_{m}\right\|^{2}\\\
L_{\mathrm{total}}&=L_{\mathrm{simple}}+L_{\mathrm{vlb}}+L_{\mathrm{static}}\end{array}\right.$
(16)

Algorithm 1 provides the training procedure for diffCOD.

⬇

def training_loss(images, masks):

"""images: [b, h, w, 3], masks: [b, h, w, 1]"""

# Encode images

X_p = ViT(images)

F = FF(X_p)

# corrupt groundtruth

t = uniform(0, 1)

eps = normal(mean=0, std=1)

mask_crpt = sqrt(gamma(t)) * masks +

sqrt(1 - gamma(t)) * eps

# predict and backward

D = UNet_1(images, mask_crpt, t)

O = IAM(F, D)

preds = UNet_2(O)

# compute loss

loss = loss_function(preds, masks)

return loss

Algorithm 1 diffCOD Training

###  3.4 Inference

In the inference stage, we step-by-step apply Eq. (14) to sample a pure
Gaussian noise $y_{t}\sim\mathcal{N}(0,I)$. In addition, we add conditional
information related to the image features to guide the inference process.
After performing $T$ iterations, we can obtain the segmentation image of the
camouflaged object. Using the setting of [39] for the sampling, the inference
process of diffCOD is shown in Algorithm 2.

⬇

def inference(images, steps):

"""images: [b, h, w, 3], steps: sample steps"""

# Encode images

X_p = ViT(images)

F = FF(X_p)

m_t = normal(mean=0, std=1)

# time intervals

for step in range(steps):

out = p_sample(images, F, m_t, step)

return out

Algorithm 2 diffCOD Inference

Method | COD10K | NC4K | CAMO | CHAMELEON
---|---|---|---|---
$S_{\alpha}\uparrow$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$ | $S_{\alpha}$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$ | $S_{\alpha}$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$ | $S_{\alpha}\uparrow$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$
2019 CPD [56] | 0.736 | 0.547 | 0.607 | 0.801 | 0.053 | 0.769 | 0.652 | 0.713 | 0.822 | 0.072 | 0.688 | 0.552 | 0.623 | 0.728 | 0.114 | 0.876 | 0.809 | 0.821 | 0.914 | 0.036
2019 EGNet [62] | 0.746 | 0.560 | 0.591 | 0.789 | 0.053 | 0.804 | 0.727 | 0.731 | 0.834 | 0.066 | 0.730 | 0.579 | 0.693 | 0.762 | 0.104 | 0.851 | 0.705 | 0.747 | 0.869 | 0.049
2020 SINet [13] | 0.772 | 0.543 | 0.640 | 0.810 | 0.051 | 0.810 | 0.665 | 0.741 | 0.841 | 0.066 | 0.753 | 0.602 | 0.676 | 0.774 | 0.097 | 0.867 | 0.727 | 0.792 | 0.889 | 0.044
2020 MINet [41] | 0.780 | 0.628 | 0.677 | 0.838 | 0.040 | 0.810 | 0.717 | 0.764 | 0.856 | 0.057 | 0.741 | 0.629 | 0.682 | 0.783 | 0.096 | 0.853 | 0.768 | 0.803 | 0.902 | 0.035
2020 PraNet [15] | 0.800 | 0.656 | 0.699 | 0.869 | 0.041 | 0.826 | 0.739 | 0.780 | 0.878 | 0.056 | 0.769 | 0.664 | 0.716 | 0.812 | 0.091 | 0.870 | 0.790 | 0.816 | 0.915 | 0.039
2021 PFNet [37] | 0.797 | 0.656 | 0.698 | 0.875 | 0.039 | 0.826 | 0.743 | 0.783 | 0.884 | 0.054 | 0.774 | 0.683 | 0.737 | 0.832 | 0.087 | 0.889 | 0.823 | 0.840 | 0.946 | 0.030
2021 LSR [36] | 0.805 | 0.660 | 0.703 | 0.876 | 0.039 | 0.832 | 0.743 | 0.785 | 0.888 | 0.053 | 0.793 | 0.703 | 0.753 | 0.850 | 0.083 | 0.890 | 0.824 | 0.834 | 0.932 | 0.034
2022 ERRNet [25] | 0.780 | 0.629 | 0.679 | 0.867 | 0.044 | — | — | — | — | — | 0.761 | 0.660 | 0.719 | 0.817 | 0.088 | 0.877 | 0.805 | 0.821 | 0.927 | 0.036
2022 NCHIT [60] | 0.790 | 0.608 | 0.689 | 0.817 | 0.046 | — | — | — | — | — | 0.780 | 0.671 | 0.733 | 0.803 | 0.088 | 0.874 | 0.793 | 0.812 | 0.891 | 0.041
2022 CubeNet [66] | 0.795 | 0.644 | 0.681 | 0.864 | 0.041 | — | — | — | — | — | 0.788 | 0.682 | 0.743 | 0.838 | 0.085 | 0.873 | 0.787 | 0.823 | 0.928 | 0.037
2023 CRNet [18] | 0.733 | 0.576 | 0.627 | 0.832 | 0.049 | — | — | — | — | — | 0.735 | 0.641 | 0.702 | 0.815 | 0.092 | 0.818 | 0.744 | 0.756 | 0.897 | 0.046
diffCOD | 0.812 | 0.684 | 0.723 | 0.892 | 0.036 | 0.837 | 0.761 | 0.802 | 0.891 | 0.051 | 0.795 | 0.704 | 0.758 | 0.852 | 0.082 | 0.893 | 0.826 | 0.837 | 0.933 | 0.030

Table 1: Quantitative comparisons of our proposed method and other 11 state-
of-the-art methods on four widely used benchmark datasets. The higher the
$S_{\alpha}$, $F_{\beta}^{\omega}$, $F_{m}$, and $E_{m}$, the better the
performance. The smaller the $MAE$, the better. The best results are marked in
$\mathbf{bold}$.

##  4 Experiments

###  4.1 Experimental Setup

Datasets. We conduct experiments on four widely used benchmark datasets of COD
task, i.e., CAMO , CHAMELEON, COD10K and NC4K. The details of each dataset are
as follows:

  * •

CAMO contains 1,250 camouflaged images and 1,250 non-camouflaged images,
covering eight categories.

  * •

CHAMELEON has a total of 76 camouflaged images.

  * •

COD10K consists of 5,066 camouflaged, 1,934 non-camouflaged, and 3,000
background images. It is currently the largest dataset which covers 10
superclasses and 78 subclasses.

  * •

NC4K is a newly published dataset that has a total of 4,121 camouflaged
images.

Following the standard practice of COD tasks, we use 3,040 images from COD10K
and 1,000 images from CAMO as the training set and the remaining data as the
test set.

Evaluation metrics. According to the standard evaluation protocol of COD, we
employ the five common metrics to evaluate our model, i.e., structure-measure
($S_{\alpha}$), weighted F-measure ($F_{\beta}^{\omega}$), mean F-measure
($F_{m}$), mean E-measure ($E_{m}$) and mean absolute error ($MAE$). The
purpose of structure-measure ($S_{\alpha}$) is to evaluate the structural
information of the result and ground truth, including object perception and
region perception. Weighted F-measure $F_{\beta}^{\omega}$ is the weighted
information of the mean F-measure ($F_{m}$) metric, and these two metrics are
a combined assessment of the accuracy and recall of the result. Mean E-measure
($E_{m}$) is able to perform both pixel-level matching and image-level
statistics, and is used to calculate the overall and local accuracy of the
segmentation results. The mean absolute error ($MAE$) metric is often used to
evaluate the average pixel-level relative error between the result and ground
truth.

Implementation details. The proposed method is implemented with the PyTorch
toolbox. We set the time step as $T$ = 1000 with a linear noise schedule for
all the experiments. We use Adam as our model optimizer with a learning rate
of 1e-4. The batch size is set to 64. During the training, the input images
are resized to 256$\times$256 via bilinear interpolation and augmented by
random flipping, cropping, and color jittering.

Baselines. Our diffCOD is compared with 11 recent state-of-the-art methods,
including CPD [56], EGNet [62], SINet [13], MINet [41], PraNet [15], PFNet
[37], LSR [36], ERRNet [25], NCHIT [60], CubeNet [66], CRNet [18]. For a fair
comparison, all results are either provided by the authors or reproduced by an
open-source model re-trained on the same training set with the recommended
setting.

###  4.2 Quantitative Evaluation

The quantitative comparison of our proposed diffCOD with 11 state-of-the-art
methods is shown in Table 1. Our method achieves superior performance over
other competitors, indicating that our model can generate high-quality
camouflaged segmentation masks compared to previous methods. For the largest
COD10K dataset, our method shows a substantial performance jump, with an
average increase of 4.8%, 12.8%, 9.5%, 6.4% and 19.1% for $S_{\alpha}$,
$F_{\beta}^{\omega}$, $F_{m}$, $E_{m}$ and $MAE$, respectively. For another
recent large-scale NC4K dataset, diffCOD also outperforms all methods,
increasing by 3.4%, 7.1%, 6.1%, 4.0% and 14.8% on average for $S_{\alpha}$,
$F_{\beta}^{\omega}$, $F_{m}$, $E_{m}$ and $MAE$, respectively. In addition,
the most significant increases in the CAMO dataset were seen in the
$F_{\beta}^{\omega}$ and $MAE$, with improvements of 10.2% and 11.3%,
respectively. CHAMELEON is the smallest COD dataset, therefore most of the
methods perform inconsistently on this dataset, our method increases 3.0%,
6.2%, 4.0%, 2.6% and 21.2% for $S_{\alpha}$, $F_{\beta}^{\omega}$, $F_{m}$,
$E_{m}$ and $MAE$, respectively.

(a) Image

(b) GT

(c) diffCOD

Figure 3: Visual results of our proposed model in terms of detailed textures.

(a) Image

(b) GT

(c) Ours

(d) CRNet [18]

(e) CubeNet [63]

(f) PFNet [37]

(g) SINet [13]

(h) MINet [41]

Figure 4: Qualitative comparison of our proposed method and other
representative COD methods. Our method provides better performance than all
competitors for camouflaged object segmentation in various complex scenes.

###  4.3 Qualitative Evaluation

Figure 4 shows a comprehensive visual comparison with current state-of-the-art
methods. It can be found that our method achieves competitive visual
performance in different types of challenging scenarios. Our diffCOD is able
to guarantee the integrity and correctness of recognition even under difficult
conditions, such as single object (e.g., row 1-4), multi-objects (e.g., row
5-8), small object (e.g., row 9-11). Nature’s camouflaged organisms often have
strange traits, such as tentacles, tiny spikes, etc. Past models have blurred
the recognition of edge parts even if the location of the target is correctly
targeted. However, we are surprised by the advantages of diffCOD in terms of
detailed textures. As shown in Figure 3, our method is able to accurately
identify every subtlety, and it can depict the textures of the object in
extremely fine detail, solving the blurring problem of segmentation masks in
other methods.

###  4.4 Ablation Studies

Overview. We perform ablation studies on key components to verify their
effectiveness and analyze their impacts on performance, as shown in Table 2.
Experimental results demonstrate that our designed Injection Attention Module
(IAM), Feature Fusion (FF), and ViT can improve detection performance. When
they are combined to build diffCOD, significant improvements in all evaluation
metrics are observed. Note that the Baseline refers to the standard diffusion
model.

No. | Component | COD10K | NC4K | CAMO
---|---|---|---|---
Baseline | IAM | FF | ViT | $S_{\alpha}\uparrow$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$ | $S_{\alpha}\uparrow$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$ | $S_{\alpha}\uparrow$ | $F_{\beta}^{\omega}\uparrow$ | $F_{m}\uparrow$ | $E_{m}\uparrow$ | $MAE\downarrow$
#1 | $\checkmark$ |  |  |  | 0.761 | 0.604 | 0.657 | 0.845 | 0.046 | 0.781 | 0.687 | 0.712 | 0.841 | 0.061 | 0.731 | 0.607 | 0.664 | 0.790 | 0.097
#2 | $\checkmark$ | $\checkmark$ |  |  | 0.788 | 0.638 | 0.687 | 0.861 | 0.041 | 0.805 | 0.711 | 0.747 | 0.863 | 0.056 | 0.749 | 0.631 | 0.694 | 0.805 | 0.093
#3 | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | 0.801 | 0.662 | 0.709 | 0.876 | 0.039 | 0.823 | 0.731 | 0.772 | 0.876 | 0.054 | 0.770 | 0.664 | 0.718 | 0.829 | 0.087
#4 | $\checkmark$ | $\checkmark$ |  | $\checkmark$ | 0.809 | 0.677 | 0.719 | 0.888 | 0.036 | 0.835 | 0.758 | 0.798 | 0.889 | 0.051 | 0.792 | 0.693 | 0.751 | 0.849 | 0.083
#5 | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | 0.799 | 0.657 | 0.708 | 0.868 | 0.039 | 0.820 | 0.727 | 0.770 | 0.872 | 0.054 | 0.772 | 0.663 | 0.722 | 0.831 | 0.086
#OUR | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 0.812 | 0.684 | 0.723 | 0.892 | 0.036 | 0.837 | 0.761 | 0.802 | 0.891 | 0.051 | 0.795 | 0.704 | 0.758 | 0.852 | 0.082

Table 2: Ablation studies of our diffCOD. The best results are marked in
$\mathbf{bold}$.

(a) Image

(b) GT

(c) Sample 1

(d) Sample 2

(e) Sample 3

(f) Sample 4

(g) Sample 5

Figure 5: Visual results of the sampling process. (c)-(g) is the diffCOD
sampling process. The time step is 200, 400, 600, 800, and 1000, respectively.

Effectiveness of IAM. As can be seen in Table 2, the presence or absence of
IAM plays a key role in the performance improvement of the model. Compared to
the experiments without this key component, the average improvement of #2 with
IAM over #1 for $S_{\alpha}$, $F_{\beta}^{\omega}$, $F_{m}$, $E_{m}$ and $MAE$
on the three datasets is 3.0%, 4.3%, 4.7%, 2.1% and 7.7%, respectively.
Furthermore, #Our accuracy improvement over #5 is significant, with an average
increase of 6.0% in $MAE$ metric on the three datasets. This is a good
indication that IAM integrates diffusion features and texture features from
the backbone perfectly.

Effectiveness of FF. The main role of FF is to aggregate the multi-scale
features. As shown in Table 2, compared to No. #2, No. #3 has an average
improvement of 2.2%, 5.0%, 3.8%, 2.5% and 6.0% for $S_{\alpha}$,
$F_{\beta}^{\omega}$, $F_{m}$, $E_{m}$ and $MAE$ on the three datasets,
respectively. The performance of #Ours on $S_{\alpha}$, $F_{\beta}^{\omega}$,
$F_{m}$ and $E_{m}$ is 3.2%, 1.0%, 0.7% and 0.3% higher than that of No. #4.

Effectiveness of ViT. To obtain the location information and texture
information of the objects in the original features, we use a ViT as a
backbone to assist the diffusion process. From Table 2, we can learn that
#Ours containing rich original features has an average improvement of 2.1%,
4.5%, 3.8%, 2.1% and 6.3% over #3 for $S_{\alpha}$, $F_{\beta}^{\omega}$,
$F_{m}$, $E_{m}$ and $MAE$ on the three datasets, respectively. #2, which
contains no original features at all, has an average of 4.0%, 7.5%, 6.6%, 3.9%
and 10.6% lower than #4 for $S_{\alpha}$, $F_{\beta}^{\omega}$, $F_{m}$,
$E_{m}$ and $MAE$ on the three data sets, respectively. In addition, to
further demonstrate the significance of conditional semantic features to guide
the diffusion process, we visualize the sampling process of diffCOD. From
Figure 5, we can see that our model learns part of the location information
and texture patterns of the camouflaged objects at the early stage of
denoising, and the subsequent inference process gradually refines the final
mask by training out the denoising model on this basis. This shows that the
key clues extracted by ViT are perfectly integrated into the diffusion process
with the help of FF and IAM.

##  5 Conclusion

In this paper, we propose a diffusion-based framework for camouflaged object
detection, which changes the previous detection paradigm of the COD community
by using a generative model for the segmentation of camouflaged objects to
achieve significant performance gains. To the best of our knowledge, this is
the first framework that employs a denoising diffusion model for COD tasks.
Our approach decouples the task of segmenting camouflaged objects into a
series of forward and reverse diffusion processes, and integrates key
information from conditional semantic features to guide this process.
Extensive experiments show the superiority over 11 other state-of-the-art
methods on four datasets. As a new paradigm for camouflaged object detection,
we hope that our proposed method will serve as a solid baseline and encourage
future research.

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[1]The University of Texas at Austin, Department of Physics, Austin, Texas
78712-1081, USA. [2]The Helmholtz-Zentrum Dresden-Rossendorf, Institute for
Radiation Physics, 01328 Dresden, Germany<EMAIL_ADDRESS>
# Compact Spectral Characterization of 5-500 MeV X-rays from the Texas
Petawatt Laser-Driven Plasma Accelerator

A. Hannasch  L. Lisi  J. Brooks  X. Cheng  A. Laso Garcia  M. LaBerge  I.
Pagano  B. Bowers  R. Zgadzaj  H. J. Quevedo  M. Spinks  M. E. Donovan  T.
Cowan  M. C. Downercor1

###### Abstract

We reconstruct spectra of secondary x-rays generated from a 500 MeV - 2 GeV
laser plasma electron accelerator. A compact (7.5 $\times$ 7.5 $\times$ 15
cm), modular x-ray calorimeter made of alternating layers of absorbing
materials and imaging plates records the single-shot x-ray depth-energy
distribution. X-rays range from few-MeV inverse Compton scattered x-rays to
$\sim$100 MeV average bremsstrahlung energies and are characterized
individually by the same calorimeter detector. Geant4 simulations of energy
deposition from mono-energetic x-rays in the stack generate an energy-vs-depth
response matrix for the given stack configuration. A fast, iterative
reconstruction algorithm based on analytic models of inverse Compton
scattering and bremsstrahlung photon energy distributions then unfolds x-ray
spectra in $\sim 10$ seconds.

##  1 INTRODUCTION

Accelerator based high energy x-ray sources at photon energies $>1$ MeV find
applications in secondary particle generation Pomerantz  _et al._ (2014),
material science Asoka-Kumar, Lynn, and Welch (1994); Selim  _et al._ (2002),
space science Stassinopoulos and Raymond (1988), nuclear resonance
fluorescence Hayward and Fuller (1957); Cohen, Tobin, and McElhinney (1959)
and homeland security Bertozzi  _et al._ (2007). These x-ray sources require
relativistic electron energies, typically from radio-frequency accelerators,
to generate bright synchrotron or bremsstrahlung radiation. Radio-frequency
accelerator facilities are limited by accelerating gradients of $\sim$100
MeV/m Allen  _et al._ (1989) and require m- to km-scale structures, becoming
expensive to build and operate. Laser plasma accelerators (LPAs) offer a
tabletop complement to RF accelerators with accelerating gradients $\sim 100$
GeV/m from an intense laser pulse interacting with a plasma Tajima and Dawson
(1979); Esarey, Schroeder, and Leemans (2009). LPAs have accelerated electrons
to energies $E_{e}$ approaching 10 GeV Gonsalves  _et al._ (2019) with
bandwidth $\Delta E_{e}/E_{e}\sim 1-15\%$. At lower energies they produce nCs
of total charge, enabling them to generate secondary x-rays with peak
brilliance comparable to that of synchrotron x-rays.

LPAs can generate three types of secondary x-rays Corde  _et al._ (2013):
betatron, inverse Compton scattered (ICS), and bremsstrahlung. Betatron
radiation originates from transverse undulations of accelerating electrons in
the wake’s focusing fields Esarey  _et al._ (2002); Kostyukov, Kiselev, and
Pukhov (2003) and has a synchrotron-like spectrum with critical energy $\sim$
several 10s of keV from a GeV class LPA Rousse  _et al._ (2004). ICS radiation
results from backscatter of counter-propagating laser photons from accelerated
electrons, upshifting the photons to energy $E_{x}\sim 4\gamma_{e}^{2}E_{L}$
Esarey, Ride, and Sprangle (1993) where $E_{L}$ is the laser photon energy and
$\gamma_{e}$ is the Lorentz factor of the electrons. Thus ICS of $E_{L}=1.17$
eV photons from electron bunches with peak energy in the range $500<E_{e}<900$
MeV ($980<\gamma_{e}<1760$) generates x-rays with spectral peaks in the range
$5<E_{x}<15$ MeV. Bremsstrahlung x-rays result from collisions of relativistic
electrons passing through a converter after the accelerator, producing
broadband x-rays with photon energies up to the energy of the electrons Glinec
_et al._ (2005). Secondary x-ray photons from GeV-class LPAs thus span an
energy range from several keV to several hundred MeV, enabling a wide range of
applications Corde  _et al._ (2013); Albert  _et al._ (2014).

Here, we spectrally characterize bremsstrahlung and ICS x-rays generated by
500 MeV - 2 GeV LPA electrons in a single shot, using a single, compact,
inexpensive calorimeter consisting of a stack of absorbers of varying $Z$ and
thickness, interlaced with imaging plates (IPs). We reconstruct spectra that
are bremsstrahlung- or ICS-dominated, and discuss future improvements to
enable unfolding of more complex spectra. The calorimeter used here extends
earlier work by Hannasch et al. (2021) Hannasch  _et al._ (2021) to
significantly higher x-ray photon energies.

Figure 1:  LPA x-ray characterization overview. (a) Schematic set up showing
(left to right) incident laser pulse, gas cell, tilted plasma mirror (PM)
positioned at $z=3.3$ cm from gas cell exit for generating inverse Compton
scattered x-rays, converter at $z=20$ cm for generating bremsstrahlung, 1 T
magnetic electron spectrometer, and stack calorimeter made of alternating
absorbers and image plates. (b) Representative single-shot electron spectrum.
Top left: raw dispersed electron data from image plate. Bottom left: electron
energy distribution averaged over the vertical width of the trace. Right: raw
betatron profile after passing through Ross filter pairs. (c) Representative
energy-depth profiles from the stack calorimeter image plates for a
bremsstrahlung dominated source (top) and an inverse Compton scatter dominated
source (bottom). (d) Integrated energy-depth distributions in the stack for
the bremsstrahlung (blue triangles) and inverse Compton source (black circles)
normalized to the total deposited energy in the stack.

###  1.1 PROCEDURE

Experiments were performed at the Texas Petawatt Laser facility at the
University of Texas at Austin which delivers 100-120 J pulses, 120-140 fs in
duration to a 7 cm long gas cell of pure He at an electron density of $5\times
10^{17}\leavevmode\nobreak\ \mathrm{cm^{-3}}$ [see Fig. 1(a)]. A $\sim 1$ T
magnetic spectrometer placed 20 cm downstream from the gas jet sweeps
accelerated electrons off-axis and measures their angle integrated spectrum at
an image plate (IP) placed 2.7 m after the exit of the gas cell Wang  _et al._
(2013) [see Fig. 1(b)]. The IP records the on-axis betatron radiation after
passing through a Ross-pair filter set for spectral analysis [Fig. 1(b) right]
and wire fiducials placed at two z-positions after the magnet impart shadows
on the dispersed electrons which provides energy and pointing calibration Wang
_et al._ (2013) [Fig. 1(b) left, top and bottom]. The bremsstrahlung and ICS
x-ray spectra are diagnosed with a compact stack calorimeter made of
alternating absorbers of increasing Z and IPs placed 5.4 m after the
accelerator and outside of vacuum. Thin target bremsstrahlung x-rays are
generated by placing a 25 $\mu$m-thick Kapton film 20 cm after the gas cell
and prior to the magnet so that the laser is not intense enough to generate
ICS radiation. Thick target bremsstrahlung x-rays are generated by placing a
860 $\mu$m-thick Sn converter prior to the magnet to compare with the 25
$\mu$m-thick Kapton target. ICS dominated radiation is generated by placing
the 25 $\mu$m Kapton film 3.3 cm after the exit of the gas cell to act as a
plasma mirror and to minimize the bremsstrahlung contribution. We estimate the
scattering parameter $a_{0}<0.5$ when the plasma mirror is placed 3.3 cm after
the exit of the accelerator, generating ICS radiation in a linear regime Corde
_et al._ (2013). Figure 1(c) shows the first 10 IP exposures for the 860
$\mu$m-thick Sn bremsstrahlung case (top) and the ICS dominated case (bottom).
The photostimulated luminescence (PSL) in each IP layer is integrated over the
FWHM of the beams and converted to energy using the value $6.95\pm 1.2\times
10^{-1}$ PSL/MeV from Bonnet et al., (2013) Bonnet  _et al._ (2013). The IP
scans were performed 5-20 minutes after the shot and are corrected for fading
based on Tanaka et al., (2005) Tanaka  _et al._ (2005). These energy-depth
profiles, shown in Fig. 1(d), are inputs for reconstructing the incident
photon spectrum and are normalized to the total deposited energy. The
normalized profiles depend only on the relative energy content of the incident
photons while the absolute number of photons in the incident x-rays then
depends on the unfolded spectra and the energy conversion and fading
correction with an estimated combined uncertainty of $\pm 20\%$.

###  1.2 X-ray spectral reconstruction

Reconstruction of the incident x-ray spectrum at the stack follows the
procedure and error analysis outlined in Hannasch et al. (2021) and is
summarized here. The absorbing layers were tailored to maximize resolution for
photon energies $>1$ MeV. The stack design for this experiment consists of
polycarbonate, aluminum, steel and lead in increasing thickness. We performed
Geant4 simulations of mono-energetic photons interacting with the stack to
generate a response matrix $R_{ij}$, which gives the energy deposited in layer
i by a photon of energy $\hbar\omega_{j}$. We write the integrated energy
deposited in layer $i$ of the calorimeter as a vector with components
$D_{i}\leavevmode\nobreak\ (i=1,2,...,24)$. $D_{i}$ can be calculated from the
response matrix and a discretized photon spectrum $dN_{j}/d(\hbar\omega)$
which describes the number of photons of energy $\hbar\omega_{j}$ in increment
$d(\hbar\omega_{j})$:

$D_{i}^{(calc)}\approx\sum_{j=1}^{N}\frac{dN_{j}}{d(\hbar\omega)}R_{ij}\leavevmode\nobreak\
d(\hbar\omega_{j}).$ (1)

Here, N is the number of energy bins. To unfold the incident photon spectrum,
$dN_{j}/d(\hbar\omega)$ is constrained to take the form of a physics-based
analytic function of $\hbar\omega$ that depends on additional parameters of
the bremsstrahlung and ICS radiation. We wish to reconstruct the incident
photon spectrum $dN_{j}/d(\hbar\omega)$ by finding these additional parameters
that minimize the sum of the squared difference between calculated and
measured deposited energy profiles.

Bremsstrahlung radiation: The model used for unfolding the bremsstrahlung
spectra is based on Kramers’ law Kramers (1923), where the x-ray spectrum
depends on the initial electron energy $E_{0}$ as

$\frac{dN}{d(\hbar\omega)}\propto\frac{1}{E_{0}}(E_{0}-\hbar\omega),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \hbar\omega\leq E_{0}.$ (2)

Kramers’ law has been used extensively to predict the experimental
bremsstrahlung spectra at a given angle Koch (1959) and is a good model for
bremsstrahlung from a broadband electron distribution as in the cases
presented here. The unfolded parameter, $E_{0}$, defines the cutoff energy of
the x-ray spectrum and is approximately the maximum electron energy for
broadband electron spectra.

Inverse Compton scattered radiation: The energy spectrum for linear ICS
radiation, $dI/d(\hbar\omega)$, can be approximated by a Gaussian function
with mean energy $E_{x}$ and standard deviation of $\sigma_{x}$ as

$\frac{dN}{d(\hbar\omega)}=\frac{1}{\hbar\omega}\frac{dI}{d(\hbar\omega)}\propto\frac{1}{\hbar\omega}\exp\left(-\frac{(\hbar\omega-
E_{x})^{2}}{2\sigma_{x}^{2}}\right)$ (3)

This approximation has proven successful at reconstructing the characteristic
features of ICS radiation from peaked electrons with energy spread $>10\%$ and
laser strength parameter $a_{0}\ll 1$, i.e., for linear ICS Hannasch  _et al._
(2021). The unfolded parameters of the energy spectrum are weighted by
$(\hbar\omega)^{-1}$ when converting to photon spectrum, thus the peak energy
appears to be redshifted from the unfolded $E_{pk}$ while the energy spread
appears larger than $\sigma_{x}$.

##  2 RESULTS

###  2.1 Bremsstrahlung radiation

The 860 $\mu$m-thick Sn bremsstrahlung integrated energy profiles
$D^{(meas)}_{i}\leavevmode\nobreak\ (i=1,2,...24)$ for two consecutive shots
(1 and 2) are shown in Fig. 2(a) as squares and circles generated from the
electron spectra shown in Fig. 2(b). Simulations of the anticipated
bremsstrahlung spectra were performed in Geant4 by including the converter
material in the Geant4 geometry and initiating electrons with an energy
distribution given by the measured electron spectra in Fig. 2(b). Figure 2(a)
compares the measured energy deposition profiles with the unfolded profiles
(solid and dashed curves) and simulated profiles ($\times$’s). Figure 2(c)
shows the corresponding unfolded photon spectra (solid and dashed curves) for
both shots along with the simulated spectra (dotted and dash-dotted curves)
that fall within 10% of the unfolded spectral shape for each shot. The
unfolded and simulated bremsstrahlung parameters are compiled in Table 1.

Figure 2: 860 $\mu$m-thick Sn bremsstrahlung x-ray results. (a) Measured energy deposition profile from shot 1 (blue squares) and shot 2 (red circles), corresponding unfolded profiles (solid blue and dashed red curves) based on bremsstrahlung radiation model (Eq. 2) and Geant4-simulated profiles ($\times$’s) based on the measured electron spectra in (b); (c) bremsstrahlung spectra extracted from unfolded and simulated profiles in (a) where shaded regions denote unfolding error in (a). Table 1: Compiled 860 $\mu$m-thick Sn bremsstrahlung x-ray results |  1cbElectron $E_{pk}$ |  |  |  |
---|---|---|---|---|---
(MeV) |  1cbx-ray FWHM |  |  |  |
(mrad) |  1cbUnfolded $E_{avg}$ |  |  |  |
(MeV) |  1cbSimulated $E_{avg}$ |  |  |  |
(MeV) | Unfolded $N_{phot}$ |  |  |  |
Shot 1 | $680\pm 280$ | $4.0\pm 0.1$ | $84\pm 6$ | 86 | $7.9\pm 1.6\times 10^{8}$
Shot 2 | $530\pm 220$ | $4.0\pm 0.1$ | $85\pm 7$ | 83 | $5.1\pm 1.0\times 10^{8}$

The 25 $\mu$m-thick Kapton bremsstrahlung integrated energy profile
$D^{(meas)}_{i}\leavevmode\nobreak\ (i=7,8,...24)$ is shown in Fig. 3(a) as
data points. Betatron radiation that is not blocked by the Kapton deposits
energy in layers 1-6 as evident from the 2D pattern of the Ross pair filters
used to analyze their spectrum observed in those layers [see Fig. 1(c) bottom
panel, left-most layer images]. These layers are removed for unfolding the
bremsstrahlung only signal from 25 $\mu$m-thick Kapton. The dotted red and
solid black curves in Fig. 3(a) show the simulated and unfolded deposited
energy profiles and the corresponding simulated and unfolded spectra are shown
in Fig. 3(b). The simulated average photon energy is 76 MeV and falls within
the uncertainty of the unfolded average energy of $81\pm 8$ MeV. The
bremsstrahlung from the 25 $\mu$m-thick Kapton generates $3.5\pm 0.7\times
10^{6}$ photons within the FWHM, $\sim 200\times$ fewer photons than is
generated by the 860 $\mu$m-thick Sn converter.

Figure 3: 25 $\mu$m-thick Kapton bremsstrahlung x-ray results. (a) Measured
energy deposition profile (black squares), unfolded profile (black solid
curve) based on the bremsstrahlung radiation model (Eq. 2) and
Geant4-simulated profile (red dotted curve) based on the measured electron
spectra in inset of (b); (b) bremsstrahlung spectra extracted from unfolded
and simulated profiles in (a) where shaded regions denote unfolding error in
(a).

###  2.2 Inverse Compton scattered radiation

The ICS integrated energy profiles $D^{(meas)}_{i}\leavevmode\nobreak\
(i=7,8,...24)$ for two consecutive shots (1 and 2) are shown in Fig. 4(a) as
squares and circles generated from the electron spectra shown in the inset
plot of Fig. 4(b). The first 6 layers are discarded to remove the betatron
contribution and Ross pair filter shadows which are evident in Fig. 1(c) in
the bottom panel. The bremsstrahlung contribution from the 25 $\mu$m-thick
Kapton is scaled to the total energy of the electron beam
$N_{e}\left<E_{e}\right>$ for each shot and subtracted from $D_{i}^{(meas)}$
to unfold the ICS only contribution. The unfolded deposited energy profiles
are overlayed in Fig. 4(a) as solid and dashed curves and the corresponding
unfolded photon spectra are shown in Fig. 4(b). The electron peak energy and
unfolded x-ray parameters are compiled in Table 2. The expected
$4\gamma_{e}\hbar\omega_{L}$ peak x-ray energy is $12.5\pm 3$ MeV and $6.3\pm
2.4$ MeV and falls within the uncertainty of the unfolded $E_{pk}$ for shot 1
and 2, respectively. The ICS shots generate $\sim 10\times$ fewer photons than
the 860 $\mu$m-thick Sn bremsstrahlung and $\sim 20\times$ more than the 25
$\mu$m-thick Kapton bremsstrahlung.

Figure 4: ICS x-ray results. (a) Measured energy deposition profile from shot 1 (red squares) and shot 2 (black circles) generated by the electrons spectra in the inset of (b) and unfolded profiles (solid and dashed curves) based on the ICS radiation model (Eq. 3); (b) ICS spectra extracted from unfolded profiles in (a) where shaded regions denote unfolding error in (a). Table 2: Compiled ICS x-ray results |  1cbElectron $E_{pk}$ |  |  |  |
---|---|---|---|---|---
(MeV) |  1cbx-ray FWHM |  |  |  |
(mrad) |  1cbUnfolded $E_{pk}$ |  |  |  |
(MeV) |  1cbUnfolded $\sigma_{x}$ |  |  |  |
(MeV) | Unfolded $N_{phot}$ |  |  |  |
Shot 1 | $840\pm 230$ | $2.35\pm 0.05$ | $12.8\pm 0.8$ | $4.9\pm 0.4$ | $6.4\pm 1.2\times 10^{7}$
Shot 2 | $590\pm 230$ | $2.55\pm 0.05$ | $7.9\pm 0.5$ | $3.6\pm 0.7$ | $9.1\pm 1.8\times 10^{7}$

##  3 DISCUSSION

The stack calorimeter can unfold the average or peak photon energy and
integrated photon number for both the broadband bremsstrahlung and linear
inverse Compton scattered radiation sources based on a parameterized model of
each source. More complex sources such as linear ICS radiation from electron
bunches with multiple energy peaks or nonlinear ICS radiation with multiple
harmonics may require a more elaborate unfolding procedure.

One way to increase the accuracy of the unfolding is to take advantage of
material transitions in the stack design. The dips and subsequent increases in
the integrated energy observed in the ICS profiles in Fig. 4(a) are correlated
with transitions from low to high Z materials. For example, IP layer 11 is
placed between the last Al layer and the first steel layer and absorbs more
energy due to back scattering from the higher Z layer. This occurs again as
the absorbers transition from steel to lead at IP layer 18. These wiggles in
the energy deposition profile for energies between 5 and 10 MeV improve the
accuracy of the unfolded spectrum, but are washed out by energetically broader
bremsstrahlung x-rays [see Fig. 1(d), Fig. 2(a) and Fig. 3(a)].

Another feature of the deposited energy profile that correlates with
transitions from low to high Z materials is the transverse profile size.
Similar to the energy deposition for bremsstrahlung, the beam size of a
broadband source of x-rays $E_{avg}\gg 1$ MeV will experience a relatively
smooth and constant increase through the stack. In contrast, a peaked source
$\sim 5$ MeV such as ICS experiences sharp changes in the deposited energy
profile radius at the material transitions. Figure 5(a) shows the measured and
simulated normalized radius $r/r_{avg}$ as the 860 $\mu$m-thick Sn
bremsstrahlung radiation passes through the stack. The same normalized radius
is shown in Fig. 5(b) for shot 2 of the unfolded ICS source ($E_{pk}=7.9\pm
0.5$) and a simulated mono-energetic beam at 5 MeV. These sharp features in
the radiation profile radius as a function of layer in the stack can be a
powerful tool to aid unfolding, reducing the number of possible solutions that
may match the energy deposition profile.

Figure 5:  (a) The normalized beam radius $r/r_{avg}$ from shot 1 of the 860
$\mu$m-thick Sn bremsstrahlung comparing measured (black squares) and
simulated (red circles) data and (b) the normalized beam radius from shot 2 of
the ICS (black squares) and 5 MeV simulated mono-energetic data (red circles).

##  4 CONCLUSION

We have presented a set of unfolded secondary x-ray spectra from LPA electrons
with energies between 500 MeV and 900 MeV. Bremsstrahlung from an 860 $\mu$m
Sn target is unfolded from two electron spectra with average energies of
$84\pm 6$ MeV and $85\pm 7$ MeV and with $7.9\pm 1.6\times 10^{8}$ and $5.1\pm
1.0\times 10^{8}$ photons within the FWHM. These bremsstrahlung cases are
compared with Geant4 simulations of the anticipated spectrum and agree well
within the uncertainty. Thin-target bremsstrahlung from
$25\leavevmode\nobreak\ \mu$m-thick Kapton is unfolded with an average energy
is $81\pm 8$ MeV and $3.6\pm 0.7\times 10^{6}$ photons in the FWHM. ICS
dominated radiation from electron bunches with different peak energies was
unfolded to observe a shift in peak ICS energy from $7.9\pm 0.5$ MeV to
$12.8\pm 0.8$ MeV and a total of $9.1\pm 1.8\times 10^{7}$ and $6.4\pm
1.2\times 10^{7}$ photons in the FWHM, respectively. The versatility of the
stack calorimeter makes it a valuable diagnostic for LPA based x-ray sources
over a large range of photon energies.

##  5 ACKNOWLEDGEMENTS

U. Texas authors acknowledge support from the U.S. Department of Energy grant
DE-SC0011617 and the DOE Office of Science, Fusion Energy Sciences under
Contract No. DE-SC0019167 through the LaserNetUS initiative at the University
of Texas at Austin. A. H. acknowledges support from the National Science
Foundation Graduate Research Fellowship under Grant No. DGE-1610403.

## AUTHOR CONTRIBUTIONS STATEMENT

A.H., L.L., J.B., X.C., A.L.G., M.L., I.P., and B.B. conducted the
experiments. A.H. analyzed the results. A.H., A.L.G. and L.L. performed the
simulations. H.J.Q., M.S.,and M.E.D. manage and operate the Texas Petawatt
laser facility. R.Z., T.C., and M.C.D. provided overall supervision of the
project. All authors reviewed the manuscript.

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