COMPRESSIVE IMAGE RECOVERY USING RECURRENT GENERATIVE MODEL

Akshat Dave, Anil Kumar Vadathya, Kaushik Mitra∗

Department of Electrical EngineeringIIT Madras

Original Image

* =

Compressive Sensing using Single Pixel Camera(15% measurements)

OursPSNR: 29.68 dB

TVAL3PSNR: 24.70 dB

Reconstructed Image

D-AMPPSNR: 26.76 dB

Fig. 1: We propose to use a deep generative model, RIDE [1] , as an image prior for compressive signal recovery. Since RIDEmodels long-range dependency in images using spatial LSTM, image recovery is better than other competing methods.

ABSTRACT

In this paper we leverage the recurrent generative model,RIDE, for applications like image inpainting and compres-sive image reconstruction. Recurrent networks can modellong range dependencies in images and hence are suitableto handle global multiplexing in reconstruction from com-pressive imaging. We perform MAP inference with RIDEas prior using back-propagation to the inputs and projectedgradient method. We propose a entropy thresholding basedapproach for preserving texture well. Our approach showscomparable results for image inpainting task. It shows supe-rior results in compressive image reconstruction compared totraditional methods D-AMP and TVAL3 which uses globalprior of minimizing TV norm.

Index Terms— Compressive imaging, recurent genera-tive models, image inpainting,

1. INTRODUCTION

It is a well known fact that natural images are sparse in sometransformed basis like DCT and Wavelet etc. Compressivesensing (CS) theory states that such sparse signals can bereconstructed from much lower compressed measurementsthan sampling at Nyquist rate []. This cuts down the cost ofsensing, especially for imaging in the non-visible region ofthe spectrum where sensors are much costlier. Imaging in

∗All the authors acknowledge the Qualcomm Innovation Fellowship2016, India by Qualcomm India.

non-visible spectrum has applications in industrial inspec-tion, surveillance, anti-counterfeiting and many more [2].Given the potential advantages, CS is a promising solutionfor its practical viability. The single-pixel camera (SPC) isa classical example of CS framework [3]. In SPC, a singlephoto diode is used to capture compressive measurementsand then reconstruct back the whole scene. It is extended forSWIR imaging []. SPC is commercially made available fromInView Corporation.

In SPC, we take M linear measurements of the scenex ∈ RN , with M < N reconstruction of the signal x is ill-posed making data priors essential. Initially, priors based onempirical observations of natural image statistics were pro-posed for CS reconstruction. For example, priors assumingsparsity in the domain of wavelets [4], DCT coefficients andgradients [5]. Most widely used minimum total-variation (TVnorm) prior is based on smooth local variations in natural im-ages [6, 7]. However, using these priors at low measurementrates results in low quality reconstructions (see TVAL3 re-construction in fig. 1). This is due to their inability to capturethe complexity of natural image statistics. On the other hand,data driven approaches based on dictionary learning [8] anddeep neural nets [9, 10] have also been proposed. Althoughthey capture the complexity both the approaches handle onlylocal multiplexing on image patches and are not suitable forglobal multiplexing scenario as in case of SPC. To addressthese problems, in this work we prospose to use a generativemodel, RIDE, proposed by Theis et al. [1] as the prior for CSimage recovery. RIDE is both data driven and because of re-

current nature handles the global multiplexing in SPC well. Aline about LSTM and modeling the long term dependencies.Our contributions are as follows:

• We utilize RIDE’s ability as an image prior to modellong term dependencies for reconstructing compres-sively sensed images.

• We use backpropagation to inputs while doing gradientascent for MAP inference.

• We formulate random image inpainting as a specialcase of compressive sensing recovery problem andusing our MAP framework, show better results thanmultiscale KSVD approach.

• We hypothesize that the model’s uncertainty in predic-tion can be related to the entropy of component pos-terior probabilities. By thresholding the entropy, weenhance texture preserving ability of the model.

2. RELATED WORK

Single Pixel Camera: Signal reconstruction from CS mea-surements is an ill-posed problem and hence we need to usesignal priors. Initially algorithms were proposed to look fora sparse solution in l1 norm. They assume sparsity in thedomain of wavelet coefficients, DCT coefficients or gradi-ents. Later class of algorithms known as approximate mes-sage passing (AMP) algorithms [11, 12] use off-the-shelf de-noiser to iteratively refine their solution. ReconNet is anotherrecent method using CNNs [9]. But as mentioned eariler, itcan only handle local multiplexing since it is a patch basedapproach. Here we propose to do compressive image recon-struction with recurrent generative model, RIDE as the imageprior. Since it is not patch limited, we can handle global mul-tiplexing.

Deep Generative Models: Recently advances with deepneural nets have led to powerful deep generative models.These include Generative Adversarial Nets (GAN) [13], Vari-ational Auto Encoders (VAE) [14], Pixel Recurrent NeuralNetworks (PixelRNN) [15] and Recurrent Image Density Es-timator (RIDE) [1]. Among these contemporary models wefind RIDE particularly suitable as low level image prior forour tasks involving Bayesian inference. GANs don’t modelthe data distribution and VAE doesn’t provide the exact like-lihood measure. PixelRNN although models the distribution,it discretizes the distribution of a pixel resulting in optimiza-tion difficulties. RIDE models continuous distribution andgives exact likelihood thus facilitating gradient ascent foroptimization. Also, RIDE being auto regressive isn’t limitedto patch size, as is the case with discriminative and even nondeep generative models like dictionary learning. This is veryuseful particularly in cases like SPC where the reconstructionhas to take account of global multiplexing and patch basedmethods can’t be used directly.

3. BACKGROUND

Let x be a gray scale-image and xij be the pixel intensity atlocation ij then x<ij describes the causal context around thatpixel containing all xmn such thatm ≤ i and j < n. Now thejoint distribution over the image can be factorized as,

p(x) =∏ij

p(xij |x<ij), (1)

which is a valid factorization involving conditional ditribu-tions. Natural images in general exhibit long range correla-tions. In order to model such dependencies Theis et al. [1]have proposed to use two dimensional Spatial Long ShortTerm Memory (LSTMs) units [16]. Spatial LSTMs summa-rize the entire causal context x<ij through their hidden rep-resentation hij , as hij = f(x<ij ,hi−1,j ,hi,j−1), where f isa complex non linear function. f has memory elements anal-ogous to physical read, write and erase operations thus en-abling LSTMs to model long term dependencies in sequences.Now each factor in the above joint distribution is modeled us-ing conditional Gaussian Scale Mixtures, thus the completedistribution is given by,

p(x) =∏ij

p(xij |hij ,θ), (2)

p(xij |hij ,θ) =∑c,s

p(c, s|hij ,θ)p(xij |hij , c, s,θ).(3)

For more details we recommend the reader to go through [1].

21.21 dB, 0.811 22.07 dB, 0.813

Fig. 2: Inpainting comparisons: From left to right, originalimage, masked image, multiscale KSVD [17], ours. Thenumbers mentioned below the figures are PSNR(left) andSSIM(right).

4. COMPRESSIVE IMAGE RECOVERY USING RIDE

Here we consider the problem of image restoration, x ∈ RN ,from linearly compressed measurements, y ∈ RM as y =Φx+n, where Φ ∈ RM×N is the sensing matrix withM < Nand n ∈ RM is noise in the observation with known statistics.

4.1. MAP Inference via Backpropagation

Compressive image recovery: Here, we use Maximum-A-Posteriori principle to find the desired image as x̂ =argmaxx p (x) p (y|x). For SPC, we formulated the MAPinference as,

x̂ = argmaxx

log p (x) s.t. y = Φx. (4)

Here we do reconstruction for the noise less case. The log-likelihood and gradients are given by the model as in Eqns.2, 3 and 7. To optimize the above we use projected gradientsmethod, where after each gradient update solution is projectedback on to the affine solution space for y = Φx. Every k-thiteration consists of the following two steps,

x̂k = xk−1 + η∇xk−1log p (x) , (5)

xk = x̂k − ΦT(ΦΦT

)−1(Φx̂k − y) . (6)

The gradient with respect to the log prior is given by,

∂ log p(x)

∂xij=

∑k≥i,l≥j

∂ log p(xkl|hkl,θ)

∂xij, (7)

due to the recurrent nature of the model, each pixel throughits hidden representation contributes to the likelihood of allthe pixels that come after it in forward pass. Hence, duringbackward pass the gradient from each pixel propagates to allthe pixels prior to it in the sequence.

Image inpainting: In image inpainting our goal is torecover the missing pixels from a randomly masked image.Here, we consider this as the special case of compressiveimaging, where Φ is a binary matrix depending on the mask.Φ has a single 1 in every row and a column either has single1 or all zeros. The masked image can be written as ΦTy. Theiterative update (6) and (7) here simplifies to gradient ascentof the prior over missing pixels, while keeping the observedpixels constant. We have included proof for this in our sup-plementary material.

In all of our experiments we consider row orthonormal-ized Φ and the term

(ΦΦT

)−1reduces to identity matrix.

4.2. Tricks used for inference

4.2.1. Four directions

Joint distribution in eqn. (1) can be factorized in multipleways, for example along each of the four diagonal directions

of an image, i.e., top-right, top-left, bottom-right and bottom-left. Gradients from different factorizations are consideredat each iteration of the inference, by flipping the image inthe corresponding direction. This leads to faster convergenceas compared to just considering one direction. While doinginference on crops from BSDS test images, we observe thatthe convergence rate is twice faster with flipping.

4.2.2. Entropy-based Thresholding

While solving the MAP optimization, we observed that wecan recover the edges quite well but texture regions areblurred. This happens because the RIDE model may nothave the right mixture component (see Eqn. (3)) to explainthe latent texture. In such cases, all the mixture componentscan be chosen with almost uniform probability, resulting inblurred texture. To detect such cases, in each iteration, weconsider entropy of the posterior probability of scales andcomponents in RIDE at each point as a metric to understandhow confident the model is in modeling the distribution atthat point. This is mathematically given as,

H(i, j) = −∑c,s

p(c, s|x<ij , xij) log(p(c, s|x<ij , xij)),

if the point ij lies on an edge, the entropy is low as thereare only certain selected components which can explain thatedge. Whereas, if it lies in a flat or textured patch, the en-tropy is high and the point is equiprobable to come from dif-ferent components and scales. Therefore, to reduce blurringwe maintain a threshold on the entropy above which we clipthe gradients to zero. Figure 3 shows the effect of entropyconstraint on the texture reconstruction.

Original Image

w/o threshold, 28.4 dB

w/ 3.0, 26.97 dB

w/ 3.5, 29.90 dB

Fig. 3: Compressive sensing image reconstructions from 30%measurements obtained by varying entropy thresholds. Thetexture of the magnified patch is recovered better with thethreshold 3.5. The numbers mentioned in dBs are PSNR ofwhole image.

5. EXPERIMENTS

For training the RIDE model we have used publicly avail-able Berkeley Segmentation dataset (BSDS300). Followingthe instincts from [1], we trained the model with increas-ing patch size in each epoch. Starting with 8x8 patchwe go till 22x22 in steps of 2 for 8 epochs. We usedthe code provided by authors of RIDE in caffe availablehere(https://github.com/lucastheis/ride/). We start with a verylow learning rate (0.0001) and decrease it to half the previ-ous value after every epoch. We used Adam optimization[18] for training the model. We observe that models withmore than one spatial LSTM layer don’t result in much ofimprovement for our tasks of interest. Hence we proceedwith a single layer RIDE model for all the inference tasksmentioned in this paper. Also we have used entropy basedgradient thresholding described above (sec. 4.2.2) to avoidblurring the texture region in all the experiments. In orderto accommodate for boundary issues we remove a two pixelneighbourhood around the image for PSNR and SSIM calcu-lations in all the experiments. For a fair comparison, we alsodo the same for the results of other methods.

5.1. Single Pixel Camera

In general, the SPC framework involves global multiplexingof the scene. But the recently proposed state-of-the-art meth-ods for signal reconstruction, like ReconNet, are designed forlocal spatial multiplexing and can’t handle the global mul-tiplexing case directly. Our model, using Spatial LSTMs,can reason for long term dependencies in image sequencesand is preferable for such kind of tasks. We show SPC re-construction results on some randomly chosen images fromthe BSDS300 test set which were cropped to 160 × 160 sizefor computational feasibility, see figures on top of table 1.We generate compressive measurements from them usingrandom Gaussian measurement matrix with orthonormalizedrows. We take measurements at four different rates 0.4, 0.3,0.25 and 0.15. Using the projected gradient method, weperform gradient ascent for 200 iterations for 0.4 and 0.3measurement rates. For lower measurement rates, we rungradient ascent for 400 iterations. Also, we follow the en-tropy thresholding procedure mentioned in section 4.2.2 witha threshold value of 3.5 which we empirically found to begood for preserving textures. In all the cases, we start witha random image uniformly sampled from (0, 1). Reconstruc-tion results for five images are shown in Table 1 and Figure 1.We were able to show improvements both in terms of PSNRand SSIM values for different measurement rates. Even atlow measurement rates, our method preserves the sharp andprominent structure in the image. D-AMP has the tendencyto over-smooth the image, whereas TVAL3 adds blotches toeven the smooth parts of the image.

5.2. Image Inpainting

For image inpainting, we randomly removed 70% of pixelsand estimated them using aforementioned inference method.We compared our approach with the multiscale adaptive dic-tionary learning approach [17], which is an improvement overthe KSVD algorithm, see Figure 2. It is clear from the figurethat our approach is able to recover the sharp edges better thanthe multiscale KSVD approach. This is because our methodis based on global image prior as compared to the patch-basedmultiscale KSVD approach.

6. CONCLUSIONS AND FUTURE WORK

We demonstrate that deep recurrent generative image modelssuch as RIDE can be used effectively for solving compres-sive image recovery problems. The main advantages of usingsuch deep generative models is that they are global priors andhence can model long term image dependencies. Also usingthe proposed MAP formulation we can solve many other im-age restoration tasks such as image deblurring, superresolu-tion, demosaicing and computational photography problemssuch as coded aperture and exposure. Another direction offuture work would be to adapt the trained generative modelto the specific image that we are interested in restoring. We

Sr. No Method M.R.=30% M.R.=25% M.R.=15%PSNR SSIM PSNR SSIM PSNR SSIM

1TVAL3 29.54 0.833 28.40 0.802 26.76 0.736D-AMP 31.59 0.867 29.70 0.836 24.70 0.716

Ours 33.87 0.898 32.72 0.872 29.68 0.792

2TVAL3 26.65 0.726 25.86 0.6810 24.51 0.575D-AMP 26.08 0.674 25.44 0.633 23.67 0.497

Ours 30.03 0.835 28.85 0.791 25.59 0.612

3TVAL3 26.20 0.789 25.08 0.745 22.63 0.638D-AMP 31.66 0.923 29.09 0.883 24.5 0.757

Ours 33.02 0.919 31.70 0.897 25.32 0.754

4TVAL3 30.20 0.849 29.10 0.820 25.44 0.719D-AMP 30.17 0.858 28.06 0.796 25.81 0.70

Ours 34.70 0.919 33.37 0.900 25.88 0.798

4TVAL3 30.20 0.849 29.10 0.820 25.44 0.719D-AMP 30.17 0.858 28.06 0.796 25.81 0.70

Ours 34.70 0.919 33.37 0.900 25.88 0.798

MeanTVAL3 27.858 0.783 26.90 0.745 24.69 0.646D-AMP 29.07 0.787 27.51 0.7464 24.48 0.633

Ours 31.57 0.859 30.54 0.818 27.09 0.700

Table 1: Comparisons of compressive imaging reconstruc-tions at different measurement rates for the images shownabove. Our method outperforms the existing global priorbased methods in most of the cases.

can use our entropy thresholding step to detect which part ofthe image is not modeled well by the generic model and thenadapt the model accordingly.

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