Image reconstruction based on back propagation learning in Compressed Sensing theory

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1 Image reconstruction based on back propagation learning in Compressed Sensing theory Gaoang Wang Project for ECE 539 Fall 2013

2 Abstract Over the past few years, a new framework known as compressive sampling has been developed for simultaneous sampling and compression. It can significantly reduce the number of measurements required for a given signal in traditional compression methods. One way to increase the compression ratio during the sampling is that we can apply different compression ratio to different part of the image. Thus we can increase the compression ratio to the background of an image in order to increase the total compression ratio. However, we know nothing of the original image before we start sampling. So we can use sampling data to judge which part of the image belongs to the background then we apply second-time sampling to these parts of the image. Before sampling an image, we should use a lot of images to be the training data to compute the weights of the classification. Then in the second-sampling, we use these weights to decide which part belongs to the background. In the first-sampling, I apply the algorithm of block based Compressed Sensing and use the sampling data as feature vectors. In the construction step, I use OMP method and DCT matrix to reconstruct the image.

3 Content Introduction... 4 Compressed Sensing Background Algorithm of SPL-BCS... 5 MLP and Back-Propagation Introduction of Back-Propagation Traning method in image reconstruction Feature vectors in training BP learning program Sampling testing data Image reconstruction algorithm Results The confusion rate of BP learning... 14

4 Introduction Over the past few years, a new framework known as compressive sampling has been developed for simultaneous sampling and compression. It can significantly reduce the number of measurements required for a given signal in traditional compression methods. Compressed sensing (CS), built upon the groundbreaking work by Candes et al. [1] and Donoho [2], aims at exactly reconstructing the original signals while sampling at sub-nyquist rate. Unlike traditional theories, CS theory greatly reduces the signal sampling rate, signal processing time, data storage and transmission costs, leading signal processing into a new revolutionary era. Due to its great practical potentials, CS has been intensively studied and used both in academia and industries in the past few years [3, 4]. The field of CS is related to other topics in signal processing and computational mathematics, such as underdetermined linear-systems, group testing, heavy hitters, sparse coding, multiplexing, sparse sampling, and finite rate of innovation. Imaging techniques having a strong affinity with CS include coded aperture and computational photography. There are many algorithms of image reconstruction based on CS, like block-based CS sampling (BCS) [5]. It is a quite efficient method, which can solve the artifact problem among block edges. Before sampling since we know little about the original images, so few algorithm can take consideration of image characteristics during the reconstruction. As we all know, most parts of national images are smooth. Therefore we could take a high compression ratio in the sampling. However, since some parts of images have complicated texture, these parts of image can be hardly reconstructed well with a high compression ration. Thus we have to reduce the total compression ratio even if most parts of image are very smooth. Fortunately, we can use some learning method in the training data. In this way, we will know which parts of image are smooth than other parts after sampling. Then we could sample these parts of image in a second time with a higher ratio. Therefore the total compression ratio will increase.

5 Compressed Sensing 2.1 Background Consider a real-valued, N-length, one-dimensional, discrete-time signal x, which can be viewed as an N 1 column vector in RN with elements x[n], n = 1, 2,..., N. Suppose that we are allowed to take M (M<<N) linear non-adaptive measurement of x through the following linear transformation [1, 2]: y=фx, (1.1) where y represents an M 1sampled vector and Φ is an M N measurement matrix. Since M<<N, the reconstruction of x from y is generally ill-posed. However, the CS theory is based on the fact that x has a sparse representation in a known transform domain Ψ. In other words, the transform-domain signal f = Ψx can be well approximated using only d<m<<n non-zero entries. It was proved in [1, 2] that when Φ and Ψ are incoherent, x can be well recovered from M measurements. In the study of CS, a couple of the most important issues include: (a) the design of measurement matrix Φ; (b) the selection of transform Ψ; (c) the reconstruction algorithm. Random matrix is always selected as measurement matrix since incoherence can be achieved with a high probability. As for transform, there are DCT, wavelet, grouplet, bandlet and curvelet [6], Dual-tree discrete wavelet transform (DDWT) [7], contourlet [8] and so on. For the reconstruction methods, orthogonal matching pursuit (OMP) and basis pursuit (BP) are classical ones. For 2D images, another well known reconstruction algorithm is through the minimization of total variation (TV) [9]. Other algorithms include iterative soft-thresholding and projection onto convex sets. [10] 2.2 Algorithm of SPL-BCS In BCS, an image is divided into B B blocks and sampled using an appropriately-sized measurement matrix. That is, suppose that x j is a vector representing, in raster-scan fashion, block j of input image x. The corresponding y j is then yj =Φ B x j, where Φ B is an M B B 2 orthonormal measurement matrix with M B =(M/N)B 2. Using BCS rather than random sampling applied to the entire image x has several merits [11]. First, the measurement operator Φ B is conveniently stored and employed because of its compact size. Second, the encoder does not need to wait until the entire image is measured, but may send each block after its linear projection. Last, an initial approximation x (0) with minimum mean squared error can be feasibly calculated due to the small size of Φ B [11]. In [11], Wiener filtering was incorporated into the basic PL framework in order

6 to remove blocking artifacts. In essence, this operation imposes smoothness in addition to the sparsity inherent to PL. Specifically, in [11], a Wiener-filtering step was interleaved with the PL projection of (2) (3); thus, the approximation to the image at iteration i + 1, x (i+1), is produced from x (i) as: Here, Wiener( ) is pixelwise adaptive Wiener filtering using a neighborhood of 3 3, while Threshold( ) is a thresholding process as discussed below. In our use of SPL, we initialize with x (0) = Φ T y and terminate when D (i+1) D (i) < 10 4, where ( i) 1 ( i) ( i 1) D x xˆ N ( i+1) ( i) function x = SPL( x, y, B, p1, p2) 2 x = Wiener( x ) ( i) ( i) For each block j xˆ xˆ p ( y xˆ ) ( i) ( i) T ( i) j j 1 B j B j x D xˆ ( i) ( i) ( i) ( i) ( i) x Threshold( x, ) ( i) ˆ ( i) ( i) x D x For each block j x x p ( y x ) ( i) ( i) T ( i) j 2 B j B j

7 MLP and Back-Propagation 3.1 Introduction of Back-Propagation Multilayer perceptrons have been applied successfully to solve some difficult and diverse problems by training them in a supervised manner with a highly popular algorithm known as the error back-propagation algorithm. This algorithm is based on the error-correction learning rule. As such, it may be viewed as generalization of an equally popular adaptive filtering algorithm: the ubiquitous least-mean-square (LMS) algorithm for the special case of a single linear neuron model. Basically, the error back-propagation process consists of two passes through the different layers of the network: a forward pass and a backward pass. In the forward pass, an activity pattern (input vector) is applied to the sensory nodes of the network, and its effect propagates through the network, layer by layer. Finally, a set of outputs is produced as the actual response of the network. During the forward pass the synaptic weights of the network are all fixed. During the backward pass, on the other hand, the synaptic weights are all adjusted in accordance with the error-correction rule. Specifically, the actual response of the network is subtracted from a desired (target) response to produce an error signal. Then this error signal is propagated backwards through the network, against the direction of synaptic connections-hence the name error back-propagation. The synaptic weights are adjusted so as to make the actual response of the network move closer to the desired response. 3.2 Traning method in image reconstruction I use a large amount of sampling blocks from images as training data. The outputs determine whehter these blocks can bare a higher sampling ratio. After training, I take the final weights into sampling. Given an original image, I take two times measurement. In the first measurement, since we know nothing about the characteristics of the image, we use a general lower compression ratio for all the image blocks. Then the weights from training come into use. They decide whether these parts of image can have a higher compression ratio. If satisfied, then these parts will proceed a second sampling. With this method, the total compression ratio will increase Feature vectors in training In real time processing, we know nothing about the original image before sampling. We could do something after the first sampling step. This requiresus that we could only take sampling data as traning data in BP learning method. We know that

8 sampling data is Φ B v, where Φ B is a Gaussian random matrix and v is a vectorized block from original image. Since Φ B is a random matrix, if we take Φ B v as feature vectors, the BP learning can hardly work. Therefore, I take two measurement as pre-processing: (1) Fix measurement matrix. In the whole process, I use identical random matrices in each sampling step. If sampling a new image, I won t generate a new random matrix again. (2) To reduce the randomness of the measurement matrix, I times the pseudo inverse of the measurement matrix, i.e. I take Φ B -1 Φ B v as the feature vectors. In this project, the training data comes from 10 images ( ). Since it will save the computing time with gray level images. Therefore all these 10 images are gray level. The block size is 8 8. Thus for each image, there are /8 2 =1024 training data. So there are training data in total. The size of M B is 16 64, i.e. equals 4 compression ratio. The training images are given as below:

9 Each image is BP learning program As mentioned above, each feature vector is For decreasing the computing comlexity. Each time, I random select 1024 vectors from vectors as training data. I run the training program hundreds of times. I use 3 layer and 4 layer separately. In addition, I set the Epoch=1000, μ=0.8 and η=0.01. For the first training, I use the random values as the initial weights. For the times afterwards, I use the weights generated from last time as the initial weights. The confusion rate is ploted as below:

10 3 layer 4 layer

11 From the diagram we can see the variance of rates of 4 layer configuration is much higher than 3 layer configuration Sampling testing data After BP learing algorithm, we get the weights of MLP. This step we will use these weights to deal with the testing data. I use 12 images ( ) as the testing example. All these 12 images are shown below:

12 Each Image is From upper left to lower right, we denote the images as 1 to 12. In the first step, we divide each image into block of size 8 8. Then we sample each block with compression ratio of 4. In this step, we don t need to consider the characteristics of images. When having obtained the sampled images, we times the weights and then we know which parts of images need to take a second step of compressed sensing. In the second step, all the satisfied blocks are resampled by the measurement matrix of 4 64, i.e. the compression ratio is 16. Thus the total compression ratio will increase.

13 3.2.4 Image reconstruction algorithm In the reconstruction, we apply the SPL-BCS algorithm by James E. Fowler. Since in block based CS, it is easy to have artifacts among block edges in the final reconstruction image. In this algorithm, we use Wiener filter to remove blocking artifacts. The pseudocodes are given below: ( i+1) ( i) function x = SPL( x, y, B, p1, p2) x = Wiener( x ) ( i) ( i) For each block j xˆ xˆ p ( y xˆ ) ( i) ( i) T ( i) j j 1 B j B j x D xˆ ( i) ( i) ( i) ( i) ( i) x Threshold( x, ) ( i) ˆ ( i) ( i) x D x For each block j x x p ( y x ) ( i) ( i) T ( i) j 2 B j B j Since there are two kinds of Φ B in the algorithm (one has the size of 16 by 64, and another is 4 by 64). So we should modify this algorithm into two parts. For different parts of image we apply different measurement matrix to them. Then combine them together.

14 Results 4.1 The confusion rate of BP learning If we apply the obtained weights to the testing examples. The confusion rates are given below: Image 3 layer Con rate 4 layer Con rate We see the confusion rates in the form are not too high, especially for 4 layer configuration. However, these rates are only the referrence since the label values I give to the testing images are subjective. In other words, if I think this part of image is smooth enough, then I give it the value 1. If not, I give 0 value. This is how the target label comes from. The most interesting part results are the total compression rates and the PSNR of the reconstructed images. The results are shown below: Image 3 layer BP 4 layer BP PSNR Com.Rate PSNR Com.Rate

15 Image SPL-BCS PSNR Com.Rate We can see in the form that the algorithm with using 3 layer BP learing is efficient. On one hand, it increase the total compression ratio. On the other hand, the reconstructed images can get a higher PSNR than SPL-BCS. However, with 4 layer configuration, since the confusion rates are very low (which have been shown in last chapter), most smooth parts of images have been justified as unsmooth. So there are few blocks take a second sampling, which leads to the low compression ratio. Since the compression ratio is low, the PSNR is much higher than 3 layer configuration. Furthermore, we can see the difference of using BP method and without BP method in the reconstructed images.

16

17

18 The left-hand-side are reconstructed images using 3 layer BP learning and the right-hand-side are images without BP learning. From the reconstructed images, we find that the edges with BP learning are much clear than the edges without BP learning. Because of time limit, I haven t compare much of the different MLP configures. In the future research, I would find which configuration in MLP is better for image reconstruction.

19 Referrence [1] E. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, vol. 52, no. 2, pp , Feb [2] D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, vol. 52, no. 4, pp , Apr [3] Y.Tsaig and D. L. Donoho, Extensions of compressed sensing, Signal Processing, vol. 86, pp , July [4] D. L. Donoho, Y.Tsaig, I. Drori, and J.-L. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit, Mar [5] Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, Orthogonalmatching pursuit: Recursive function approximation with applications to wavelet decomposition, in Conf. Rec. 27th Asilomar Conf. Signals, Syst. Comput, vol.1, pp , [6] E. Pennec and S. Mallat, Bandelet image approximation and compression, Multiscale Modeling & Simulation, vol.4, no. 4, pp , [7] N. G. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, Journal of Applied Computational Harmonic Analysis, vol. 10, pp , May [8] M. N. Do and M. Vetterli, The contourlet transform: An efficient directional multiresolution image representation, IEEE Transactions on Image Processing, vol. 14, no. 12, pp , December [9] E. Cand`es, J. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics, vol. 59, no. 8, pp , August [10] E. Candes and J. Romberg, Practical signal recovery from random projections, 2005,[Online].Available: [11] L. Gan, Block compressed sensing of natural images, in Proceedings of the International Conference on Digital Signal Processing, Cardiff, UK, pp , July 2007.

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