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<title>Physics Based Image Deshadowing Using Local Linear Model</title>
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<h1 class="intro-heading">Physics Based Image Deshadowing Using Local Linear Model<br><br><br></h1>
<div class="intro-text">Tamir Einy<sup>1*</sup>, Efrat Immer<sup>2*</sup>, Gilad Vered<sup>1</sup>, Shai Avidan<sup>2</sup>
<br><br>Applied Materials<sup>1</sup>,    Tel-Aviv University<sup>2</sup>
<br><p style="font-size:18px">{tamireiny, efratimmer, vgilad}@gmail.com, avidan@eng.tau.ac.il</p>
<p style="font-size:14px">*Denotes equal contribution</p></div>
<div class="intro-buttons">
<a class="btn btn-primary" href="pdf/Physics Based Image Deshadowing Using Local Linear Model.pdf" target="_blank">
<span>Paper</span></a>
<a class="btn btn-primary" href="pdf/Physics Based Image Deshadowing Using Local Linear Model-supp.pdf" target="_blank">
<span>Supplemental</span></a>
<a class="btn btn-primary" href="https://github.com/tamireiny/local_linear_deshadow" target="_blank">
<span>Code</span> </a>
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<h2 class="section-title"> Abstract</h2>
<div class="section-desc">Image deshadowing algorithms remove shadows from images.
This requires both detecting where the shadow is and, once detected, removing it from the image.
This work focuses on the shadow removal part. We follow a common physical shadow formation model
and learn its parameters using a deep neural network. The problem is challenging because the model
assumes six parameters per pixel, which makes the problem ill-posed. Common deshadowing methods use
neural networks to estimate a simplified model that requires only three parameters per pixel.
This leads to good results, but close inspection on real images reveals its limitations. In contrast,
we estimate the full model (i.e., six parameters per pixel) directly. Remarkably, our deshadowing network is
considerably smaller, compared to alternative methods, while producing results that are better, on standard datasets.</div>
</div>
</div>
</section>
<!--
<section class="section-header-w" id="sectionCoeffRed">
<div class="os-container">
<div class="section-header">
<h2 class="section-title"> Estimated Coefficient Maps</h2>
<div class="section-desc">
Coefficients maps <i> w </i> and <i> b </i> for the red color channel, estimated with our network.
</div>
</div>
</div>
</section>
<div>
<section class="demos-w">
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<div class="col-sm-10 col-xl-10 demo-item sreveal is-visible" style="-webkit-transition: none !important">
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</a>
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-->
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<div class="os-container">
<div class="section-header">
<h2 class="section-title">Estimated shadow coefficients maps</h2>
<div class="section-desc">We show an example for our network estimated shadow coefficients maps <i> w </i> and <i> b </i>
for a single color channel (red) in the figure below. As can be seen, the coefficients maps are piecewise
smooth and the values of <i> w </i> and <i> b </i> depend on the pixel's color and spatial location.
For example, higher values of <i> w </i> can be found at the upper shadow part close to the occluding figure,
where the shadow intensity is greater.</div>
</div>
</div>
</section>
<section class="demos-w" ng-controller="demosController as demos">
<div class="row">
<div class="col-sm-6 col-xl-4 demo-item sreveal is-visible {{demo.class}}" ng-repeat="demo in demos.demo" style="-webkit-transition: none !important">
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<div class="demo-header" style="text-transform: none; font-size: 1.5rem;" ng-bind-html='demo.header'></div>
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<div class="demo-shot" style="background-image: url('{{demo.image}}'); background-repeat: no-repeat; background-size: 70% 100%;"></div>
</div>
</a>
</div>
</div>
</section>
<!-- End new -->
<section class="section-header-w" id="image1">
<div class="os-container">
<div class="section-header">
<h2 class="section-title">Shadow Masks Fine-tuning</h2>
<div class="section-desc">We estimate the shadow masks using the shadow detection network BDRAR developed by Zhu et al [1].
We fine-tuned the original model on the ISTD+ train set and evaluated the results with IoU and BER (balance error rate).
Using fine-tuning, we managed to increase the mean IoU, on the ISTD+ test set, from <i>0.794</i> to <i>0.91</i>.
At the same time, the BER dropped from <i>5.61</i> to <i>1.94</i>.
We present below an example for the shadow mask estimation before and after fine-tuning. For this image, after the fine-tuning, the IoU increased from <i>0.06</i> to <i>0.95</i> and the BER dropped from <i>11.4</i> to <i>0.75</i>. <br><br>
<a style="color:#868686" href="https://openaccess.thecvf.com/content_ECCV_2018/papers/Lei_Zhu_Bi-directional_Feature_Pyramid_ECCV_2018_paper.pdf" target="_blank">
[1] Lei Zhu, Zijun Deng, Xiaowei Hu, Chi-Wing Fu, Xuemiao Xu, Jing Qin and Pheng-Ann Heng. Bidirectional Feature Pyramid Network with Recurrent Attention
Residual Modules for Shadow Detection</a><br>
</div>
</div>
</div>
</section>
<section class="demos-w" ng-controller="industriesController as industries">
<div class="row">
<div class="col-sm-6 col-xl-4 demo-item sreveal is-visible {{demo.class}}" ng-repeat="industry in industries.industry" style="-webkit-transition: none !important">
<a class="demo-w">
<div class="demo-header" style="text-transform: none; font-size: 1.5rem;" ng-bind-html='industry.header'></div>
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<div class="demo-shot" style="background-image: url('{{industry.image}}'); background-repeat: no-repeat; background-size: 70% 100%;"></div>
</div>
</a>
</div>
</div>
</section>
<!--
<section class="section-header-w" id="table1">
<div class="os-container">
<div class="section-header">
<h2 class="section-title"> Analysis of the optimal constant and local linear models</h2>
<div class="section-desc">
We conducted an ablation study to investigate the gap between our solution and the optimal solution obtained by computing <i> w, b </i>
maps using the shadow and shadow-free images. We created <i> w, b </i> maps in the following manner. For the constant linear model, we
calculated a single set of shadow parameters to the entire umbra area by performing least-squares regression between the corresponding
pixels in the shadow and shadow-free images. For the local linear model, we calculated a set of shadow parameters per pixel similar to
our network. First, we calculated three coefficients maps using least-squares on a disk-shaped neighborhood sized <i> 9 </i>, <i> 19 </i> and <i> 33 </i> pixels.
Then, the maps were combined together according to weights determined by the absolute normalized cross correlation values. Values closer to
one represent an area with a good linear fit and thus is weighted higher. To further improve the results around the shadow border, we
follow with an aggregation step. Since each pixel belongs to several overlapping disk shaped neighborhoods we can combine the estimations
of all the relevant neighborhoods by weighted average. The weight is again determined by the absolute normalized cross correlation value.
<br> <br> The table below reports the MAE calculated using the optimal constant and local linear models.
</div>
</div>
</div>
</div>
</section>
<div>
<section>
<div class="container">
<div class="table-responsive-sm center">
<table class="table" style="width: 100%">
<thead>
<tr>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Model</th>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Shadow (MAE)</th>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Non-Shadow (MAE)</th>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">All (MAE)</th>
</tr>
</thead>
<tbody>
<tr>
<th class="text-center" scope="row">Constant Linear Model</th>
<td class="text-center">5.56</td>
<td class="text-center">2.53</td>
<td class="text-center">3.02</td>
</tr>
<tr>
<th class="text-center" scope="row">Local Linear Model</th>
<td class="text-center">2.46</td>
<td class="text-center">1.45</td>
<td class="text-center">1.61</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
</div>
<section class="section-header-w" id="table1">
<div class="os-container">
<div class="section-header">
<div class="section-desc">
We find that there is a gap in performance between the optimal solution (<i>2.46</i> MAE) and our method (<i>6.56</i> MAE),
as expected. Given that in a previous ablation experiment we found little difference in the results between using the
predicted shadow mask and the ground truth mask, we conclude that improving the <i> w, b </i> estimation is key to future progress.
</div>
</div>
</div>
</div>
</section> -->
<section class="section-header-w" id="table2">
<div class="os-container">
<div class="section-header">
<h2 class="section-title"> Network Architecture</h2>
<div class="section-desc">
Our network is based on the multi-scale context aggregation network (CAN32 configuration)
developed by Chen et al [1] in the context of semantic image analysis. The network is a fully-convolutional
network with 10 layers. Except for the last layer, each convolutional layer follows with an adaptive normalization
and a leaky ReLU layer. The parameters of the convolutions are summarized in the table below. As can be seen,
the network contains convolutions with a growing dilation rate. The usage of dilation allows the network to process
information from a large receptive field (513 x 513), while using a small number of parameters.
The input to the network is a 4-channel tensor containing the shadow image and the shadow mask. The output is a 6-channel
tensor containing the shadow coefficients maps with the same resolution as the input. <br><br>
<a style="color:#868686" href="https://openaccess.thecvf.com/content_ICCV_2017/papers/Chen_Fast_Image_Processing_ICCV_2017_paper.pdf" target="_blank">
[1] Qifeng Chen, Jia Xu, and Vladlen Koltun. Fast image processing with fully-convolutional networks </a><br>
<!--
We use the multi-scale context aggregation network (CAN32), developed in the context of semantic
image analysis. The network is Fully convolutional, the input image and output have the same resolution.
The network was suggested by Chen etal [1], it includes 10 layers. The first 8 layers contain convolutions
with increasing dilation step size. The 9<sup>th</sup> layer is convolution layer without dilation.
For these 9 layers we use the leaky ReLU activation function presented by Xu etal [2], for the last layer
we use a linear transformation (1 x 1 convolution) without non-linearity, and changed its output to be 6
channels, instead of 3. The usage of dilation allows the network to process information from large receptive
field (513 x 513), while using small number of parameters.<br><br>
<a style="color:#868686" href="https://openaccess.thecvf.com/content_ICCV_2017/papers/Chen_Fast_Image_Processing_ICCV_2017_paper.pdf" target="_blank">
[1] Qifeng Chen, Jia Xu, and Vladlen Koltun. Fast image processing with fully-convolutional networks </a><br>
<a style="color:#868686" href="https://arxiv.org/pdf/1505.00853.pdf" target="_blank">
[2] Bing Xu, Naiyan Wang, Tianqi Chen, and Mu Li. Empirical evaluation of rectified activations in convolutional
network </a>-->
</div>
</div>
</div>
</div>
</section>
<div>
<section>
<div class="container">
<div class="table-responsive-sm center">
<table class="table" style="width: 100%">
<thead>
<tr>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Layer</th>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Convolution</th>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Dilation</th>
<th class="text-center" scope="col" style="font-size:1em; font-weight:bold">Receptive Field</th>
</tr>
</thead>
<tbody>
<tr>
<th class="text-center" scope="row">1</th>
<td class="text-center" > 3 X 3</td>
<td class="text-center" > 1</td>
<td class="text-center" > 3 X 3</td>
</tr>
<tr>
<th class="text-center" scope="row">2</th>
<td class="text-center" >3 X 3</td>
<td class="text-center" >2</td>
<td class="text-center" >7 X 7</td>
</tr>
<tr>
<th class="text-center" scope="row">3</th>
<td class="text-center" >3 X 3</td>
<td class="text-center" >4</td>
<td class="text-center" >15 X 15</td>
</tr>
<tr>
<th class="text-center" scope="row">4</th>
<td class="text-center"> 3 X 3</td>
<td class="text-center"> 8</td>
<td class="text-center"> 31 X 31</td>
</tr><tr>
<th class="text-center" scope="row">5</th>
<td class="text-center">3 X 3</td>
<td class="text-center">16</td>
<td class="text-center">63 X 63</td>
</tr>
<tr>
<th class="text-center" scope="row">6</th>
<td class="text-center" >3 X 3</td>
<td class="text-center" >32</td>
<td class="text-center" >127 X 127</td>
</tr><tr>
<th class="text-center" scope="row">7</th>
<td class="text-center" >3 X 3</td>
<td class="text-center" >64</td>
<td class="text-center" >255 X 255</td>
</tr>
<tr>
<th class="text-center" scope="row">8</th>
<td class="text-center" >3 X 3</td>
<td class="text-center" >128</td>
<td class="text-center" >511 X 511</td>
</tr><tr>
<th class="text-center" scope="row">9</th>
<td class="text-center" >3 X 3</td>
<td class="text-center" >1</td>
<td class="text-center" >513 X 513</td>
</tr>
<tr>
<th class="text-center" scope="row">10</th>
<td class="text-center" >1 X 1</td>
<td class="text-center" >1</td>
<td class="text-center" >513 X 513</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
</div>
<!-- <section class="section-header-w" id="sectionCoeffRed">
<div class="os-container">
<div class="section-header">
<h2 class="section-title"> Estimated Coefficient Maps</h2>
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Our network estimated coefficient maps <i> w </i> and <i> b </i> for the red channel</div>
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<table style="width:100%">
<tr>
<th>Company</th>
<th>Contact</th>
<th>Country</th>
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<td>Alfreds Futterkiste</td>
<td>Maria Anders</td>
<td>Germany</td>
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<td>Francisco Chang</td>
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<h2 class="section-title" style="color:#42b129; vertical-align: text-top"> Physics Based Image Deshadowing Using Local Linear Model </h2>
<p style="vertical-align: text-top">Our network estimated coefficient maps Ꞷ and ꞵ for the red channel<br><br><br><br><br><br><br><br><br><br><br><br></p>
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<h2 class="section-title"> Linear shadow model</h2>
<div class="section-desc">We use the physical shadow formation model.
The purpose of this model is to map the connec-tion between a shadowed pixel to its corresponding shadow-free pixel.
The light intensity reflected from a point on adiffusing surface depends on the reflectance of the surfacematerial and the scene illumination.
For a point lying on theshadow-free area, the illumination can be expressed as thesum of direct and ambient illumination components.
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<h2 class="section-title"> Method</h2>
<div class="section-desc">Our network takes asinput a shadow image and optional mask (binary or proba-bility)
Thenetwork outputs6channels that represent the shadow pa-rameters(w,b)per color channel per pixel.
The networkoutputs the predicted shadow free image We use the multi-scale context aggregation network(CAN),
developed in the context of semantic image anal-ysis. The input image and output have the same resolution.We are
using the architecture suggested by Chenet al. [2],with9layers instead of10. The first7layers contain convo-lutions
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<a style="color:#868686" href="https://arxiv.org/pdf/1908.08628.pdf" target="_blank">[1] Hieu Le and Dimitris Samaras. Shadow removal via shadowimage decomposition</a><br>
<a style="color:#868686" href="https://openaccess.thecvf.com/content/CVPR2021/papers/Liu_From_Shadow_Generation_To_Shadow_Removal_CVPR_2021_paper.pdf" target="_blank">[2] Zhihao Liu, Hui Yin, Xinyi Wu, Zhenyao Wu, Yang Mi, and Song Wang. From shadow generation to shadow removal </a><br>
<a style="color:#868686" href="https://www.ecva.net/papers/eccv_2020/papers_ECCV/papers/123560256.pdf" target="_blank">[3] Hieu Le and Dimitris Samaras. From shadow segmentation to shadow removal </a><br>
<a style="color:#868686" href="https://openaccess.thecvf.com/content_cvpr_2018/papers/Hu_Direction-Aware_Spatial_Context_CVPR_2018_paper.pdf" target="_blank">[4] Xiaowei Hu, Chi-Wing Fu, Lei Zhu, Jing Qin and Pheng-Ann Heng. Direction-aware spatial context features for shadow detection and removal </a><br>
<a style="color:#868686" href="https://arxiv.org/pdf/1608.00762.pdf" target="_blank">[5] Han Gong and Darren Cosker. Interactive removal and ground truth for difficult shadow scenes </a><br>
<a style="color:#868686" href="https://arxiv.org/pdf/1911.08718.pdf" target="_blank">[6] Xiaodong Cun, Chi-Man Pun, and Cheng Shi. Towards ghost-free shadow removal via dual hierarchical aggregation network and shadow matting gan</a></div>
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