Designing Sparse Sensing Matrix for Compressive

August 24, 2014

Optimization Methods & Software

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To appear in Optimization Methods & Software Vol. 00, No. 00, November 2014, 1–12

Designing Sparse Sensing Matrix for Compressive Sensing to Reconstruct High Resolution Medical Images Vibha Tiwaria∗ , P.P. Bansod

b

and Abhay Kumar

c

a

Medi-Caps Institute of Technology and Management, Indore, M.P., India b Instrumentation Department, S.G.S.I.T.S, Indore, M.P., India c School of Electronics, Devi Ahilya University, Khandawa Road, Indore, M.P., India (Received 00 Month 20XX; final version received 00 Month 20XX) Compressive sensing theory has enabled faithful reconstruction of high resolution images from low resolution i.e. from lesser number (M ) of known coefficients of sparse signal s ∈ RN (M N ). In this respect, the role played by sensing matrix Φ and sparsity matrix Ψ is vital. If the sensing matrix is dense then it takes large storage space and leads to high computational cost. Therefore, in this paper effort is made to design sparse sensing matrix with least incurred computational cost while maintaining quality of reconstructed image. The design approach followed is based on sparse block circulant matrix with few modifications. The other used sparse sensing matrix consists of 15 ones in each column. The low resolution images used are acquired from US, MRI and CT modalities. The image quality measurement parameters are used to compare the performance of reconstructed medical images using various sensing matrices. It is observed that, since Gram matrix of dictionary matrix (ΦΨ) is closed to identity matrix in case of modified SBCM, therefore, it helps to reconstruct the medical images of very good quality, especially that of low resolution CT images.

Keywords: compressive sensing; telemedicine; sensing matrix design; medical image reconstruction.

1.

Introduction

Technological growth has reduced the gap between domestic and global Health Care system. Telemedicine is one such field which promises to provide affordable and quality heathcare facilities to people at large. The successful deployment of Telemedicine depends heavily on storage and transfer of huge size clinical image/video. Various compression techniques have been developed which transforms the signal in some suitable domain and then achieve compression by discarding less significant coefficients. For reconstruction signal must be sampled at the Nyquist rate. However, Compressive Sensing (CS) technique can recover signals from lesser number of samples or measurements of sparse signal [1]. Usually real time signals are sparse in transform domain Ψ such as wavelet or discrete cosine transform. Work done in [2] proves that if the sparse sensing matrix Φ consists of small number (k) of non-zero elements drawn from i.i.d. Gaussian or Bernoulli random variables then it is possible to recover the k-sparse signal of length N , from lesser number of M measurements, using linear program, on condition that k . log(MN ) [3] [4]. However, computational cost and storage space of M using such dense sensing matrix is very high. Therefore, lot of work is going on to design ∗ Corresponding

author. Email: [email protected]

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efficient sensing matrix. Toeplitz and circulant structure sensing matrix was used in [5] [6] [7] [8] [9] [10] , to sucessfully recover original signal and at the same time reduces the storage space and computational time. Various authors have proposed to use deterministic instead of random sensing matrix, consisting of elements based on binary (0, 1), bipolar(±1), ternary (0, 1, -1) values [11] and codes such as Walsh Hadamard, LDPC, BCH and Reed Solomon [12] [13] [14] . It has been proved that if matrices satisfy restricted isometry property (RIP) then signal of good quality may be reconstructed [15]. A sensing matrix which satisfies RIP has minimum mutual coherence between sensing Φ and sparsity Ψ matrix. Considering equivalent dictionary matrix A = ΦΨ, then the coherence between the columns of dictionary matrix must be small [16]. Elad proposes to minimize mutual coherence of equivalent dictionary matrix by minimizing average of off-diagonal elements of the Gram matrix [17]. Duarte et al. [18] have simultaneously optimized sensing matrix and dictionary matrix for specific set of signals using algorithm based on KSVD approach. In [19], sensing matrix is optimized by minimizing a weighted sum of the interblock coherence and subblock coherence of the equivalent dictionary. The nearly orthonormal blocks of equivalent dictionary is obtained by minimizing mostly the subblock coherence resulting in slight increase in total interblock coherence. Equiangular tight frame (ETF) approach to Gram matrix of equivalent dictionary matrix is used in [20] to minimize weighted sum of both interblock and subblock coherence. In [20], best recovery results were obtained when total subblock coherence is minimized in contrast to [19]. Statistical restricted isometry property (STRIP) is used in [21] to prove that deterministic matrices reconstructs signal better comparative to random matrices. The deterministic sensing matrices used were partial FFT matrix, partial Walsh-Hadamard matrix and matrix constructed from kasami code. Deterministic sensing matrix was also used in [22], which was based on chirp signal. The validity of sensing matrix was proved empirically and by bounding the eigenvalues of submatrices. In [23], compressed sensing and recovery algorithm were proposed which performs recovery in O((k log k) logN)-time and provides error bounds as well. The sensing matrix is constructed by randomly subsampling rows from incoherent matrix. The goal of this paper is to design sensing matrix which will help in reconstructing the image faithfully based on sparse block circulant matrix (SBCM) [24]. In this paper, low resolution medical images: US, CT and MRI of size 128x256 have been considered for simulation and analysis. Block Sparse Bayesian Learning Bounded Optimization (BSBL BO) [25] and Smooth approximation to L0 norm (SL0) [26] CS algorithms are used for high resolution image reconstruction. The rest of the paper is organized as follows. In section 2 design of sparse sensing matrix is discussed. Simulation and results are discussed in section 3.

2.

Sparse Sensing Matrix for CS

Real time signals/images are sparse in transform domain. Considering k-sparse signal s ∈ RN sparse in Ψ basis matrix, then a sensing matrix Φ ∈ RM ×N can recover the signal

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with high probability if it satisfies restricted isometric property (RIP) given as [27], (1 − δk ) ksk2 ≤ kΦΨsk2 ≤ (1 + δk ) ksk2

(1)

where δk is restricted isometric constant (RIC) which should be very much smaller than 1. Defining, A = ΦΨ

(2)

(1 − δk ) ksk2 ≤ kAT sk2 ≤ (1 + δk ) ksk2

(3)

Equivalently (1) can be stated as [28],

for every subset T ⊂ {1, .., p}, having no more than k indices, AT a submatrix of size M × |T | obtained from matrix A by considering number of columns corresponding to indices in T . However, evaluating RIP is impractical. The alternative way to find sensing matrix which satisfy RIP and hence guarantee to reconstruct signal is to obtain coherence value between sensing and sparsifying matrix. The two matrices should be incoherent for perfect reconstruction, so the value of coherence should be very small [29]. The coherence can be obtained by taking inner product between two distinct normalized columns of matrix A as [16], µ (A) = max i6=j

hAi , Aj i kAi k kAj k

(4)

Since A is obtained after normalizing columns, so its Gram matrix, given as G = AT A , should have diagonal elements close to one. Further analysis of sensing matrix can be done by obtaining average value of the off-diagonal element of G, which is given as [30], P µav (A) =

i6=j

|Gij |

N (N − 1)

(5)

Ideally, G should be close to identity matrix of size M × M and coherence and average absolute off-diagonal value must be zero [30]. According to Gershgorin’s disc theorem, Gram matrix of sub-matrix AT given as, G = AT T AT , should have eigenvalues bounded in magnitude (1 − δk , 1 + δk ) for A to satisfy RIP(k, δ k ). However, RIP is a satisfactory condition but not necessary condition for perfect reconstruction. Mamaghanian et al. [31] proposed three alternatives to design k−sparse sensing√matrix √ satisfying RIP1 property. The non-zero elements consisted of entries equal to ± k , k or k number of ones in each row. The location of non-zero entries was randomly chosen. Sensing matrix having k number of ones had lower coherence value with Daubechies db10 wavelet sparsity matrix, therefore it is used in this paper. In order to achieve lower computational cost, SBCM for CS, was proposed by Sun et al. [24]. The sensing matrix Φ ∈ RM ×N considered was a sparse block circulant matrix. The matrix Φ, was divided into sub-matrices. There were L number of columns (L = N/d) and D number of rows (D = L × M N ) when n mod L = 0. The sensing matrix Φ was

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divided into number of blocks Φij , given as, 

 Φ11 · · · Φ1L   Φ=  ... . . . ...  ΦD1 · · · ΦDL where Φij is inherently circulant sub-matrix ( i=1,..,D, j=1,..,L). The circulant nature of matrix is given as, b0 b1 · · · bd b b ···b d−1  d 0 Φij =  . . ..  .. .. . b1 b2 · · · b0 

    

Φij is a sparse matrix having elements denoted by bq (q = 1, .., d). Each row and column of Φij had one non-zero element which belong to L set of i.i.d. Gaussian random variables. Each row and column of Φ had different non-zero elements. Thus in a row there were L number of coefficients and in a column D number of coefficients. It had been proved that the diagonal elements of Gram matrix of Φ (G= Φ0 Φ) is close to 1 and off-diagonal elements are bounded in magnitude with high probability and hence satisfies RIP. The SBCM approach for designing sensing matrix is further modified in the paper to reduce computational load without degrading reconstructed image quality. Instead of using i.i.d. Gaussian random variables, non-zero elements are generated by scaling inverse Daubechies db2 wavelet coefficients. The structure of sensing matrix is maintained as earlier and the location of non-zero elements in sensing matrix is kept random. The measurement vector y ∈ RM is obtained by multiplying Φ with s as, y = Φs

(6)

The reduced M number of samples of signal s are scaled, to bound the magnitude, before giving it as an input to the CS algorithm for reconstruction. The sparse column vector s ∈ RN , extracted from image matrix of size N × N is transformed using Daubechies db10 wavelet basis matrix. The CS algorithm used is SL0. The sensing matrix Φ, is divided into sub-matrices each of size d × d, where d is varied from 8, 16, 32, 64 and 128. The mutual coherence value obtained for different sub-matrix size is tabulated in Table 1. As evident from table the coherence value is lowest for sub-matrix size 128 × 128, therefore this is used as a sensing matrix for modified SBCM case. As the block size is increased to 128, the sensing matrix consisted of two blocks and the scaled coefficients of db2 are 1 and 0, thus one block turns out to be identity matrix and other zero matrix. So, the sensing matrix is deterministic matrix given as Φ= [ I 0 ]. To analyze sensing matrix further, Gram matrix G is obtained for three different sensing matrix containing non-zero elements as i) fifteen ones in each column present at random locations, ii) sparse block circulant matrix with i.i.d Gaussian random values, with block-size 32 × 32 iii) modified sparse block circulant matrix with sub-matrix ˆ where Aˆ is a normalized size of 128 × 128. Gram matrix G is given as, G = AˆT A, T ˆ column version of matrix A and A is the transpose. The mutual coherence, average value of absolute off-diagonal elements (µav ) and maximum eigenvalue of G for different

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Table 1. Analysis of Φ Type of Φ Bin Phi SBCM SBCM Mod SBCM Mod SBCM Mod SBCM Mod SBCM Mod

(8) (16) (32) (64) (128)

Mutual Coherence

µav

max eigenvalue

0.5610 0.4889 0.7561 0.7385 0.7024 0.6820 0.7246

0.0989 0.0588 0.0932 0.0756 0.0638 0.0470 0.0105

26.33 4.45 20.08 10.49 5.50 3.56 2.08

sensing matrix is tabulated in Table 1. The sensing matrix with 15 ones is denoted by Bin Phi, SBCM and modified SBCM by SBCM and SBCM Mod respectively in the table. Binary sensing matrix has least coherence value and modified SBCM with 128 × 128 block size has least µav and its eigenvalue is bounded to 2.08. The histogram of off-diagonal elements of G is shown in Figure 1 (b, d & f). It shows that in case of a) sensing matrix having 15 ones, 60% of elements are below 0.1 b) SBCM matrix has 83% elements below 0.1 and c) modified SBCM has 97% of elements lying below 0.1. The distribution of G is shown in Figure 1 (a, c & e) which indicates that for modified SBCM, G has non-zero elements close to unity present in diagonal and just off diagonal position and the rest elements are close to zero. So G, in case of modified SBCM closely resembles identity matrix and this feature is useful for faithful reconstruction of images.

3.

Simulation and Results

Medical images of US, MRI and CT available online in DICOM format are used for simulation [36] [37] [38] [39]. Sparsifying matrix is generated from Daubechies db10 wavelet coefficients. Simulation is performed on 50 different images acquired from each type of modality using three different sensing matrices as defined in section 2. SL0 and BSBL BO CS algorithms are used for reconstruction of medical images. In BSBL BO, input parameters are set to the default values as threshold set for pruning γ is 1e-4, noise variance λ is given by std(y)2 /1e2, block size is set to 16 and stopping criterion δ is set to 1e-8. The input parameter of SL0 algorithm is set to default values as σmin = 0.001 and σdecrease f actor = 0.5. The quality of reconstructed images is evaluated using parameters like Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Structural Similarity Index (SSIM) Normalized Cross-Correlation (NKR) and Normalized Absolute Error (NAE). Computational load is calculated based upon CPU processing time. Column vectors of sparse medical image in wavelet domain are multiplied by sensing matrix of size 128 x 256. Thus, lesser number of measurements of image of size 128 x 256 are obtained. This low resolution image is given as input to CS algorithm for reconstruction. The mean values of different quality indicator parameters are obtained after running simulation over 50 images of each modality. The values obtained for SBCM sensing matrix is obtained over 50 trials. The sensing matrix with 15 ones is denoted by Bin Phi and the CS algorithm used is mentioned within the brackets in the table. In case of SBCM and modified SBCM sensing matrix SL0 CS algorithm is used. US images of resolution 128 x 256 are reconstructed using CS algorithms SL0 and BSBL BO. The mean values of quality indicator parameters for US image are given

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(a) Graphical Representation of G in binary sensing matrix

(b) Off-diagonal elements of G in binary sensing matrix

(c) Graphical Representation of G in SBCM

(d) Off-diagonal elements of G in SBCM

(e) Graphical Representation of G in Modified SBCM

(f) Off-diagonal elements of G in Modified SBCM

Figure 1. Distribution of elements in Gram matrix G for various sensing matrix

in Table 2. PSNR values indicate that the sensing matrix consisting of 15 ones with BSBL BO CS algorithm performs better. However, mean values of other parameters like SSIM, MSE and NKR values are far better in case of modified SBCM with very low variance. Lower value of MSE and NAE with very good SSIM value of about 0.913 and NKR of about 0.9737 indicates that good quality US images are reconstructed using modified SBCM, as also evident from Figure 2. CPU processing time with SL0 algorithm is least as compared to BSBL BO. Total number of non-zero elements in modified SBCM is 128, in SBCM 1024 and 3840 elements in binary sparse sensing matrix, therefore, this is one of the cause of increased CPU processing time in case of binary sparse sensing matrix and reduced timing in modified

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Table 2. Recovered US image quality parameters for different sensing matrix Φ Type of Φ

Time

MSE

SSIM

PSNR

NAE

NKR

SBCM Mod SBCM Bin Phi(BSBL BO) Bin Phi(SL0)

1.21 1.44 26.73 7.97

0.0206 0.5953 0.0481 0.0687

0.913 0.663 0.703 0.685

32.86 23.46 35.57 35.17

0.13 0.58 0.24 0.26

0.9737 0.5158 0.9448 0.9176

Table 3. Recovered MRI image quality parameters for different sensing matrix Φ Type of Φ

Time

MSE

SSIM

PSNR

NAE

NKR


1.23 1.67 28.75 8.07

0.0035 0.6006 0.0077 0.0138

0.8447 0.4282 0.6830 0.6079

37.44 25.10 36.43 34.47

0.05 0.57 0.09 0.11

0.9948 0.4910 0.9901 0.9905

SBCM, as also shown in Figure 5a. In typically 1.21 seconds US image is reconstructed using modified SBCM sensing matrix. MRI images of resolution 128 x 256 are reconstructed very well using modified SBCM as indicated by mean values of quality indicator parameters tabulated in Table 3 with lowest variance. Good quality MRI images are reconstructed as shown in Figure 3, with highest PSNR of about 37.44, 0.9948 NKR, very good SSIM value of about 0.8447 and lowest MSE and NAE values in typically 1.23 seconds. CT images of resolution 128 x 256 are very difficult to reconstruct. As shown in Figure 4, both BSBL BO and SL0 with sparse binary and SBCM sensing matrix are unable to reconstruct the low resolution images. The quality indicator parameter values are shown in Table 4. However, modified SBCM out performs in this case also, reconstructing CT images of good quality having PSNR of about 32.93 dB, 0.9921 NKR, lowest MSE and NAE values and comparatively good SSIM value of about 0.7556. The CPU processing time taken is typically about 1.25 seconds. Overall performance of sensing matrices can be evaluated from graphs of Figure 5. CPU processing time taken is highest for binary sparse sensing matrix and least for modified SBCM as shown in Figure 5a. MSE and NAE values are high (Figure 5b & 5e), PSNR (Figure 5c), SSIM (Figure 5d), and NKR (Figure 5f) values are comparatively lower for SBCM sensing matrix for all type of images. Modified SBCM sensing matrix allows reconstruction of images having lowest MSE and NAE values with comparatively highest PSNR, SSIM and NKR values. Sampling images in appropriate manner is essential for faithful reconstruction of images. Thus sampling or sensing matrix play a significant role in CS. Since Gram matrix G of modified SBCM is closest to identity matrix and then is the G of sparse binary sensing matrix and last is G of SBCM so accordingly sensing matrices reconstructs images having quality in similar order.

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Table 4. Recovered CT image quality parameters for different sensing matrix Φ Type of Φ

Time

MSE

SSIM

PSNR

NAE

NKR


1.25 1.66 19.25 7.60

0.0079 0.6399 0.0157 0.0252

0.7556 0.3241 0.4314 0.3670

32.93 22.45 29.52 28.21

0.05 0.55 0.11 0.13

0.9921 0.5199 0.9814 0.9828

(a) original US image

(b) recovered using BSBL BO

(c) recovered using ModSBCM

(d) recovered using SL0

Figure 2. US original and recovered images

4.

Conclusion

Sparse block circulant matrix of size 128 x 256 is constructed with an aim to reduce mutual coherence between sensing and sparsity matrix. After doing modifications in SBCM matrix, the number of non-zero elements in matrix is only 128 so it takes lesser storage space and reduces computational time. Since the Gram matrix G of dictionary matrix closely resembles with the identity matrix so it is guaranteed that the images are reconstructed with more accuracy. The low resolution medical images of size 128 x 256 acquired from US, MRI and CT modalities are reconstructed using SL0 algorithm to generate good quality 256 x 256 size images. The quality indicator parameters used are PSNR, MSE, SSIM, NAE and NKR. With modified SBCM sensing matrix very good results are obtained which are as follows. US images are reconstructed with mean value of PSNR of about 32.86dB, MSE around 0.0206, very good SSIM value of about 0.913 and 0.9737 NKR. US image is reconstructed in typically 1.21 seconds. In case of MRI images mean value of PSNR is about 37.44dB. NKR is 0.9948, very good SSIM value of about 0.8447 and lowest MSE and NAE values are obtained. MRI image is reconstructed

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(a) original MRI image


(c) recovered using ModSBCM


Figure 3. MRI-ankle original and recovered images of size 256x256

typically in 1.28 seconds. CT images of size 128x256 could not be recovered using other sensing matrices. Using modified SBCM, CT images are very well reconstructed with PSNR of about 32.93 dB, 0.9921 NKR, lowest MSE and NAE values and comparatively good SSIM value of about 0.7556. CT image is reconstructed in typically 1.25 seconds. Thus, low resolution images can be obtained in clinical settings with lesser storage and in lesser time. This enables the medical image/signals to be sent to distant diagnostic center over low data rate transmission channels as well. At remote end, high resolution image/signals can be recovered for better clinical assessment by specialized doctors.

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(a) original CT image


(c) recovered using SL0 with ModSBCM


Figure 4. CT-abdomen original and recovered images of size 256x256

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(a) CPU Processing Time

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(c) PSNR

(d) Structural Similarity Index

11 (e) Normalized Absolute Error

(f) Normalized Cross Correlation

Figure 5. US, CT, MRI Image Quality Measure for various sensing matrix

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