Fully-Convolutional Measurement Network for Compressive Sensing Image Reconstruction

11/21/2017 ∙ by Xuemei Xie, et al. ∙ Xidian University 0

Recently, deep learning methods have made a significant improvement in com- pressive sensing image reconstruction task. However, it still remains a problem of block effect which degrades the reconstruction results. In this paper, we pro- pose a fully-convolutional network, where the full image is directly measured with a convolutional layer. Thanks to the overlapped convolutional measure- ment, the block effect is removed. In addition, because of the jointly training of the measurement and reconstruction stages, the adaptive measurement can be obtained. Furthermore, to enhance the performance of the network, resid- ual learning is used in the reconstruction network. Experimental results show that the proposed method outperforms the existing methods in both PSNR and visual effect.

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1 Introduction

Compressive sensing (CS) theory baraniuk2010model ; baraniuk2011more ; candes2007sparsity ; haupt2010toeplitz

is able to acquire measurements of signals at sub-Nyquist rates and recover signals with high probability when the signals are sparse in a certain domain. Greedy algorithms

needell2010cosamp ; babacan2010bayesian , convex optimization algorithms candes2005decoding ; figueiredo2007gradient , and iterative algorithms donoho2009message ; daubechies2004iterative have been used for recovering images in conventional CS. However, almost all these methods recover images by solving an optimization problem, which is time-consuming. In order to reduce the computational complexity in the reconstruction stage, convolutional neural networks (CNNs) are applied to replace the optimization process. CNN-based methods mousavi2015deep ; kulkarni2016reconnet ; mousavi2017learning ; mousavi2017deepcodec ; xie2017adaptive use big data stevens2017 to train the networks that speed up the reconstruction stage. Mousavi, Patel, and Baraniuk mousavi2015deep

firstly propose deep learning approach to solve the CS recovery problem. They use stacked denoising autoencoders (SDA) to recover signals from undersampled measurements. ReconNet

kulkarni2016reconnet and DeepInverse mousavi2017learning introduce CNNs to the reconstruction problem, where the random Gaussian measurement matrix is used to generate the measurements. Instead, the methods mousavi2017deepcodec ; xie2017adaptive using adaptive measurement learn a transformation from signals to the measurements. This mechanism allows measurements to retain more information from images. The methods mentioned above divide an image into blocks, which breaks the structure information of the image. This will cause the block effect in the reconstructed image.

In this paper, we propose a fully convolutional measurement network for CS image reconstruction. Instead of block-wise methods, a convolutional layer is applied to obtain the measurement from a full image, which keeps the integrity of structure information of the original image. Furthermore, motivated by the residual learning proposed by ResNet he2016deep

, we apply residual connection block (Resblock) in the proposed network for improvement. Experimental results show that the proposed method outperforms the state-of-the-art method 1

2dB in PSNR at different measurement rates.

The organization of this paper is as follows. The related works using deep learning methods for the CS reconstruction problem are introduced in Section 2. Section 3 presents the proposed fully convolutional measurement network. Section 4 shows experimental results of the proposed method and the previous works. The conclusion of this paper is drawn in Section 5.

2 Related Work

Recently, deep learning methods have been applied in CS image reconstruction tasks and achieve promising results such as mousavi2015deep ; kulkarni2016reconnet ; mousavi2017deepcodec . Among them, CNN-based methods present superior performance. ReconNet kulkarni2016reconnet is a representative work that applies CNNs in reconstructing low-resolution mixed image measured by random Gaussian matrix. The framework is shown in Fig. 1.

Figure 1: Framework of random Gaussian based network.

The training of the network is driven by the error between the label and the reconstructed image. And the loss function is given by

(1)

where is the reconstructed image of ReconNet. is the original signal as well as the label. means the training parameters in ReconNet. is the total number of image blocks in the training dataset. The loss function is minimized by tuning using back propagation.

Based on the way the original image is measured, deep learning methods for CS reconstruction can be divided into two categories: fixed random Gaussian measurement and adaptive measurement.

Fixed random Gaussian measurement

Mousavi et al. mousavi2015deep firstly use SDA to recover signals from undersampled measurements. ReconNet kulkarni2016reconnet and DeepInverse mousavi2017learning utilizes CNNs to recover signals from Gaussian measurements. DR-Net yao2017dr , inheriting ReconNet, adds residual connection blocks (Resblock) to its reconstruction stage and achieves better performance. Instead of learning to directly reconstruct the high-resolution image from the low-resolution one, DR-Net learns the residual between the ground truth image and the preliminary reconstructed image. However, the measurements of these methods are randomly measured, which is not optimally designed for natural images.

Adaptive measurement

In order to keep the information of the training data, the adaptive measurement is proposed. Methods including improved ReconNet lohit2017convolutional , Adp-Rec111Adp-Rec stands for adaptive measurement network for CS image reconstruction, proposed in our previous work. xie2017adaptive , and DeepCodec mousavi2017deepcodec are all based on adaptive measurement. In cases of the improved ReconNet and Adp-Rec, a fully-connected layer is used to measure the signals, which allows for a jointly learning of the measurement and reconstruction stages. With the learned measurement matrix, a significant gain in terms of PSNR is achieved. DeepCodec, closely related to the DeepInverse, learns a transformation for dimensionality reduction. Learning measurements from the original signals helps to preserve more information compared with taking random measurements.

3 Fully Convolutional Measurement

The exsiting CNN-based CS methods always adopt block-wise pattern due to the limitation of GPU memory. The block effect comes accordingly. In order to overcome this shortcoming, we propose a fully convolutional measurement network in which a convolutional layer is used to get the adaptive measurements. It is different from our previous work using fully-connected layers xie2017adaptive . Fig. 2 shows the proposed network which is composed of a convolutional layer, a deconvolutional layer xu2014deep

, and a residual block. The first layer ‘conv’ is used to obtain measurements. The second layer ‘deconv’ is used to recover a low resolution image from the measurements. Furthermore, we apply a residual network (ResNet) to reconstruct the high resolution image. Because batch normalization would get rid of range flexibility from networks

lim2017enhanced

, we remove the batch normalization layer in Resblock. Our framework is different from super-resolution (SR) 

deng2016similarity  fan2017compressed  dong2016image  Ledig_2017_CVPR , since SR just learns a transform form the low-resolution images to high-resolution images, while the proposed framework is jointly trained from the measurement to the recovery part.

Figure 2: Framework of the proposed network.

The loss function of the proposed network is given by

(2)

where is the parameter of the convolutional measurement network, and is the parameters of the reconstruction network. The Euclidian distance between the label and the reconstruction is back propagated to train the whole network.

The main advantage of the proposed network is the use of convolutional layer as the measurement matrix. By means of the overlapped convolutional kernels, this structure can remove block effect of the reconstructed images. In details, one feature map contains several measurements of each pixel, which is similar to the idea proposed by He et al. he2017mask that the feature map preserves the explicit per-pixel spatial correspondence. Another advantage is that the fully convolutional neural network can deal with images of any size, which breaks the limitation that fully-connected layer is only capable of measuring the fixed size of images. Once the network is trained, we can test images with different sizes without changing the structure of the network.

Fig. 3 shows an example of reconstruction results at three kinds of measurements. The original image and those by random Gaussian, Adp-Rec, and the proposed method are shown respectively in Fig. 3(a), (b), (c), and (d). The measurement rate is 10%. We can see that the proposed method performs better than the others.

(a) Origin
(b) ReconNet
(21.49dB)
(c)  Adp-Rec
 (26.65dB)
(d) Proposed
(27.61dB)
Figure 3: The reconstruction results of monarch at measurement rate 10%.

The better performance can be proved through a visualization of the kernels in convolutional layer of the measurement network, as shown in Fig. 4

. Since the original signal in random Gaussian and adaptive measurements is a cloumn vector (Fig.

4(a) and (b)), we reshape the row vectors of measurement matrix to size . Fig. 4(a) shows two reshaped row vectors of the random Gaussian measurement matrix at measurement rates 1% , 10%, and 25% in both time and frequency domain. The content of random Gaussian measurement matrix is obviously irregular. Fig. 4(b) shows two reshaped row vectors of adaptive measurement matrix in Adp-Rec. When measurement rate is set to 1%, low frequency information is already extracted. As the measurement rate increases, much high frequency information is captured. Fig. 4(c) shows two kernels of the proposed measurement matrix. Compared with the adaptive measurements in Adp-Rec, the measurements by the proposed method provide more concentrated energy in the low frequency area at different measurement rates. As for the directional information, when measurement rate is 1%, two extracted typical directions ‘horizontal’ and ‘vertical’ can be easily observed in time domain.

(a) Random Gaussian measurements.
(b) Adaptive measurements in Adp-Rec.
(c) Proposed
Figure 4: Reshaped row vectors of measurement matrix at measurement rates 1%, 10%, and 25% in both time and frequency domain.
Figure 5: Reconstruction image, low-resolution image and residual image at measurement rate 10% in both time and frequency domain.

Fig. 5 shows the reconstruction of image ‘Monarch’, its low-resolution, and the corresponding residual. From residual image in frequency domain, we can see that the high frequency component is mainly learned by the residual network. Rather than ReconNet which reconstructs the high resolution image from the low resolution one directly, ResNet just reconstruct the residual between the low resolution image and the high resolution image, that is the reconstruction image. Thus, all its energy is concentrated on the residual. That is why ResNet has better performance.

4 Experiments

In this section, we perform experiments on the reconstruction of compressive sensing images with existing typical methods. The results show the outstanding performance by the proposed method.

The experiments are conducted on caffe framework

jia2014caffe . Our computer is equipped with Intel Core i7-6700 CPU with frequency of 3.4GHz, 4 NVidia GeForce GTX Titan XP GPUs, 128 GB RAM, and the framework runs on Ubuntu 16.04 operating system. The training dataset consists of 800 pieces of size images downsampled and divided from 800 images in DIV2K dataset timofte2017ntire .

The performance of the proposed method is compared with those by ReconNet and Adp-Rec which are the typical CNN-based CS methods. We give the testing results using image ‘parrots’, ‘flinstones’, and ‘cameraman’ at measurement rates 1%, 10%, and 25%, as shown in Fig. 6, Fig. 7, and Fig. 8, respectively. The proposed method provides the best reconstruction results in terms of PSNR and the results are most visually attractive.

From the results shown in Fig. 6, with measurement rate being 1%, it can be seen that the block effect is removed (Fig 6(d) vs. (b) and (c)). From Fig. 7, when the measurement rate is 10%, the proposed method shows the advantage in reconstructing the image, typically in those smooth areas such as nose, hands, and legs of the man. From Fig. 8, when measurement rate rises to 25%, the proposed method still outperforms other methods, which can be easily seen in the edge of the man’s arm.

(a) Origin
(b) ReconNet
(18.93dB)
(c)  Adp-Rec
 (21.67dB)
(d) Proposed
(22.49dB)
Figure 6: The reconstruction results of parrots at measurement rate 1%.
(a) Origin
(b) ReconNet
(19.04dB)
(c)  Adp-Rec
 (23.83dB)
(d) Proposed
(24.98dB)
Figure 7: The reconstruction results of flinstones at measurement rate 10%.
(a) Origin
(b) ReconNet
(23.48dB)
(c)  Adp-Rec
 (27.11dB)
(d) Proposed
(28.99dB)
Figure 8: The reconstruction results of cameraman at measurement rate 25%.

For an overall look on the performance, the reconstruction results of 11 test images at measurement rates 1%, 10%, and 25% with the methods including ReconNet, DR-Net, Adp-Rec, Fully-Conv222Fully-Conv consists of a convolutional layer and a deconvolutional layer without Resblock, which can be regarded as the tiny model of the proposed network., and the proposed one are shown in Table 1. The mean PSNR is given in the type of blue background. It is obvious that the proposed method shows greatest performance in almost all test images.

MR Samples ReconNet -Net Adp-Rec Fully-Conv Proposed
1% Monarch 15.61dB 15.33dB 17.70dB 17.98dB 18.46dB
Parrots 18.93dB 18.01dB 21.67dB 21.80dB 22.49dB
Barbara 19.08dB 18.65dB 21.36dB 21.61dB 22.06dB
Boats 18.82dB 18.67dB 21.09dB 21.73dB 22.3dB
Cameraman 17.51dB 17.08dB 19.74dB 19.88dB 20.63dB
Fingerprint 15.01dB 14.73dB 16.22dB 16.24dB 16.33dB
Flinstones 14.14dB 14.01dB 16.12dB 16.55dB 16.92dB
Foreman 22.03dB 20.59dB 25.53dB 25.18dB 27.26dB
House 20.30dB 19.61dB 22.93dB 22.93dB 23.67dB
Lena 18.51dB 17.97dB 21.49dB 21.77dB 22.51dB
Peppers 17.39dB 16.90dB 19.75dB 20.80dB 21.38dB
Mean(all) 17.94dB 17.44dB 20.33dB 20.59dB 21.27dB
10% Monarch 21.49dB 23.10dB 26.65dB 25.20dB 27.61dB
Parrots 23.36dB 23.94dB 27.59dB 26.82dB 27.92dB
Barbara 22.17dB 22.69dB 24.28dB 24.39dB 24.28dB
Boats 24.56dB 25.58dB 28.80dB 28.52dB 29.48dB
Cameraman 21.54dB 22.46dB 24.97dB 24.58dB 25.62dB
Fingerprint 20.99dB 22.03dB 26.55dB 26.92dB 27.36dB
Flinstones 19.04dB 21.09dB 23.83dB 23.08dB 24.98dB
Foreman 29.02dB 29.20dB 33.51dB 31.96dB 34.00dB
House 26.74dB 27.53dB 31.43dB 30.81dB 32.36dB
Lena 24.48dB 25.39dB 28.50dB 27.76dB 28.97dB
Peppers 22.72dB 24.32dB 26.67dB 26.69dB 28.72dB
Mean(all) 23.28dB 24.32dB 27.53dB 26.98dB 28.30dB
25% Monarch 24.95dB 27.95dB 29.25dB 28.47dB 32.63dB
Parrots 26.66dB 28.73dB 30.51dB 29.90dB 32.13dB
Barbara 23.58dB 25.77dB 27.40dB 27.11dB 28.59dB
Boats 27.83dB 30.09dB 32.47dB 31.75dB 33.88dB
Cameraman 23.48dB 25.62dB 27.11dB 26.73dB 28.99dB
Fingerprint 26.15dB 27.65dB 32.31dB 30.92dB 32.91dB
Flinstones 22.74dB 26.19dB 27.94dB 27.02dB 30.26dB
Foreman 32.08dB 33.53dB 36.18dB 35.08dB 38.10dB
House 29.96dB 31.83dB 34.38dB 33.63dB 36.22dB
Lena 27.47dB 29.42dB 31.63dB 30.65dB 33.00dB
Peppers 25.74dB 28.49dB 29.65dB 29.71dB 32.90dB
Mean(all) 26.42dB 28.66dB 30.80dB 30.09dB 32.69dB
Table 1: The PSNR results at measurement rates (MR) 1%, 10%, and 25%, where Red is ranked the first and blue is ranked the second.
Samples Original ReconNet -Net Adp-Rec Proposed
MOS Monarch 4.9615 1.0000 1.1538 1.7307 2.4615
Parrots 4.9615 1.0384 1.2307 2.1538 2.9230
Barbara 4.9615 1.0769 1.0769 2.0000 2.6538
Boats 4.9230 1.0769 1.0384 1.5000 2.3846
Cameraman 5.0000 1.1538 1.1923 1.8461 2.7692
Fingerprint 4.8461 1.1538 1.0384 1.4230 1.6823
Flinstones 5.0000 1.1923 1.1538 2.0769 3.1538
Foreman 4.9230 1.1538 1.1538 1.9230 2.7692
House 4.9615 1.0000 1.1153 2.0769 2.7307
Lena 5.0000 1.0384 1.0384 1.8076 2.8461
Peppers 4.9615 1.0000 1.1153 1.8076 2.5769
Mean(all) 4.9545 1.0734 1.1188 1.8496 2.6328
SSIM Monarch 1.0000 0.3801 0.3931 0.4755 0.5058
Parrots 1.0000 0.5328 0.5617 0.6739 0.7135
Barbara 1.0000 0.3729 0.3847 0.4648 0.5007
Boats 1.0000 0.4140 0.4319 0.4888 0.5405
Cameraman 1.0000 0.4516 0.4783 0.5578 0.5867
Fingerprint 1.0000 0.1548 0.1727 0.1628 0.1700
Flinstones 1.0000 0.2502 0.2718 0.3230 0.3801
Foreman 1.0000 0.5647 0.6051 0.6912 0.7396
House 1.0000 0.5278 0.5526 0.6350 0.6624
Lena 1.0000 0.4418 0.4552 0.5554 0.6081
Peppers 1.0000 0.4002 0.4127 0.5053 0.5839
Mean(all) 1.0000 0.4083 0.4291 0.5031 0.5447
Table 2: The SSIM and MOS results. Here measurement rates (MR) 1% is taken as an example. The highest is marked red, while the second is marked blue.

From Table 1, it can be concluded that Adp-Rec beats DR-Net and ReconNet about 3dB in all measurement rates because of its adaptive measurement. Based on the standard ReconNet kulkarni2016reconnet , the improved ReconNet lohit2017convolutional adds several tricks such as adaptive measurement and adversarial loss. Its performance is even lower than Adp-Rec. Despite its promising results, Adp-Rec still divides image into blocks, ignoring the relevance between neighbouring blocks, which causes to the block effect in reconstructed images. For this reason, Fully-Conv uses a convolution layer as measurement matrix to deal with this problem. It achieves comparable results with Adp-Rec even though it contains no additional operation.

To further improve the reconstruction results, we put Resblock after Fully-Conv structure because of the brilliant performance of Resblock in reconstruction task. With this enhancement, the proposed method obtains the best performance in terms of PSNR at all measurement rates, as shown in Table 1.

We also measure the quality of images with Mean Opinion Score (MOS). The test results of different images are shown in Table 2. In this experiment, 26 volunteers take part in ranking the images. The quality of the images is divided into five levels, from 1 to 5, with the quality from low to high. All the test images are randomly ranked before being scored and they are displayed group by group. Each group has four reconstruction images, in different methods, and one original scene image. All participants take this test on the same computer screen, from the same angle and distance. Here the distance from the screen to the tested persons is 50 cm and the eyes of those persons are of the same height of the center of the screen. In addition, we also use structural similarity index (SSIM) to evaluate our method and existing block-wised methods as shown in Table 2. The case of MR = 1% is taken as an example.

In terms of hardware implementation, we follow the approach of the previous work proposed in shi2011high in which sliding window is used to measure the scene. Similarly, we can replace the random Gaussian measurement matrix with the learned pre-defined parameters in the conv layer of the measurement network. The reconstruction part is not on optical device, so only the measurement part needs to be implemented with the approach above.

5 Conclusion

This paper proposes a novel CNN-based deep neural network for high-quality compressive sensing image reconstruction. The network uses a fully convolutional architecture, which removes the block effect caused by block-wise methods. For a further improvement, we add Resblock after the deconvolutional layer, making the network learn the residual information between low and high resolution images. With this enhancement, the network shows best performance in reconstruction task compared with other methods. In future work, we are going to apply perceptual loss into the network for better reconstruction result. And semantics-oriented reconstruction will be also considered.

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