Compressed Sensing Image Reconstruction Based on Discrete Shearlet Transform

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1 Sensors & Transducers 04 by IFSA Publishing, S. L. Compressed Sensing Image Reconstruction Based on Discrete Shearlet Transorm Shanshan Peng School o Inormation Science and Engineering, Hunan International Economics University, Changsha, 4005, China Tel.: matlab_w@6.com Received: July 04 /Accepted: 30 September 04 /Published: 3 October 04 Abstract: The two-dimensional wavelet transorm or magnetic resonance imaging (MRI) images does not sparsely represent curve singularity characteristics, which can only capture the limited direction inormation. Pointing at this problem, this paper presents a new method based on discrete Shearlet transorm or compressed sensing MRI (CS-MRI). Frequency coeicients can be got at all scales and in all directions ater perorming discrete shearlet transorm to MRI image. Then adopting orthogonal matching pursuit algorithm to recover the sparsing coeicients. Finally, the reconstructed image is getting by inverse shearlet transorm. Experimental results show that, compared with wavelet transorm, discrete shearlet transorm or CS-MRI improves quality o reconstructed image and preserves more inormation about texture and edge. Copyright 04 IFSA Publishing, S. L. Keywords: Discrete shearlet transorm, Compressed sensing (CS), MRI image s reconstruction, Sparse.. Introduction The traditional signal sampling process must ollow the Nyquist sampling requency, which will undoubtedly increase the sample volume, it is a huge challenge to transport and store. The compressed sensing theory [, ] is as a new signal acquisition theory, as long as the signal is compressible or is sparse in a transorm domain, it can be optimized by solving related problems, the original signal is restored by random sampling transorm coeicients. Magnetic resonance imaging is an important means o clinical imaging, MRI image has a sparse representation in a transorm domain (such as spatial inite dierence and wavelet transorm domain, etc.), in order to meet the compressed sensing sparse image reconstruction requirements, a number o scholars' researches show that the theory o MRI images compressed sensing reconstruction is an important ield o application [3-6]. In compressed sensing, the sparsity o the reconstructed image quality has a signiicant impact, the current traditional two-dimensional wavelet transorm base is usually used to sparse sparse MRI images. However, two-dimensional wavelet decomposition are only horizontal, vertical, and diagonal direction, and the transorm ilter is isotropic, the singular point eature in image can preerably be sparsed, a characteristic curve o singularities can not represented in the sparsity [7]. MRI images oten contain a lot o curves and edges, and thereore two-dimensional wavelet transorm can not provide the best representation o the image. Shear waves (Shearlet) is the latest development o multi-scale geometric analysis, ater an synthesized and expansion aine system is used by 7

2 Guo et al [8, 9], multi-dimensional unctions are constructed close to the optimal representation, it has a more simple math structure, in the image represents, there are multi-resolution, non-linear approximation and optimal directionality and so on. The shear waves has excellent characteristics, it provides a new research ideas or solving diicult problems in image processing. Based on the above ideas, this paper analyzes the MRI image signal sparse characteristics in shear wave transormation, or the irst time, discrete shear waves transorm is used to sparse MRI image processing, and orthogonal matching pursuit algorithm is used as a compressed sensing reconstruction algorithm, the compressed sensing MRI image reconstruction is achieved.. Shearlet Principle.. Continuous Shear Wave Transormation Shear wave transormation [8, 9] theory is based on the synthesis o wavelet theory. Synthetic wavelet theory provides an eective method or geometry multiscale analysis through aine systems. When the dimension is, the aine system with a synthetic expansion is as ollows: / l AB ψ ψ, k, l x A ψ B A x k Ψ ( ) = { ( ) = det ( )}, () Which meet l, Z, k Z, ψ L ( ℵ ). A and B are invertible matrices, det B =. I Ψ AB ( ψ ) meets tight rame structure, its elements are called synthetic wavelets. Among them, matrix A is associated with the scale transormation, B l is associated with the constant holding area geometric transormation. Shear wave transorm is a special case o the synthesis wavelet, i ψ L ( ℵ ) satisies the ollowing conditions: ξ ) ˆ ψξ ( ) = ˆ ψξ (, ξ) = ˆ ψ( ξψ ) ˆ ( ), which ˆψ is ξ ψ s Fourier transorm. ) ψ continuous wavelet, ˆ ψ C ( ℵ ), supp ˆ ψ [, / ] [ /, ]. 3) ˆ ψ C ( ℵ ) and supp ˆ ψ [, ], in the interval (,), ψ > 0 and ψ =, the ollowing a 0 system is generated by the ψ, Aa = and 0 a s Bs = ψ ast ( x) = { a ψ ( Aa Bs ( x t)),, () + a ℵ, s ℵ, t ℵ } ( ) S a, s, t =, ψ ast, (3) Which is continuous shear-wave system, and ψ ( x) is called continuous shear waves. ast.. Discrete Shear Wave Transormation The scale parameter a and shear parameter s are discreted, and discrete shear wave transormation is enabled. Localization properties o discrete shear wave is very good, supporting regional basis unctions satisy the parabolic scaling, with changes in scale, the singularity o the characteristic unction can be accurately described [0]. Usually shear matrix B and anisotropic expansion o matrix A are taken. 4 0 A =, B =, (4) 0 0 Function o discrete shear waves is as ollows: Which 3 ( 0),, l ( ) ψ ( ) ψ lk x = BAx k, (5) ( 0) ( 0) ξ ˆ ψ () ξ = ˆ ψ ( ξ, ξ ) = ˆ ψ( ξ) ˆ ψ ( ), ξ ˆ ˆ ψ, ˆ ψ C ( ℵ ), supp ˆ ψ,,, 6 6 supp ˆ ψ,. I it is assumed I 0 0 = ˆ ψ ( ) ω = ω = or ω, (6) 8 ˆ ψ ( ) or ω, (7) From the above assuming, unction k ℵ ( 0) ˆ ψ,, k ( 0,, ) constitutes a split shear waves in the requency domain, which is ξ D = ( ξ, ξ ): ξ,, it is shown in { ξ } 0 8 ˆ ( 0) Fig. (a). Each element ψ,, k is supported on the trapezoid, the approximate size is, the direction slope is l straight line, it is shown in Fig. (b). Similarly understood, we can construct a unction () ˆ (), it satisies ψ x,, k () i { ˆ ψ : 0, < l <, k ℵ }, lk,, 8

3 Which shear wave is split in requency domain, it ξ is D = {( ξ, ξ): ξ, 8 ξ }, it is shown in Fig. (a). The appropriate ϕ L ( ℵ ) is selected to meet { ϕ k ( x) = ϕ( x k) : k ℵ }, which is L ([, ] ) tight rame. Thus, a collection 6 6 ( d ) i { ϕ k, ψ, l, k : 0, < l <, k ℵ, d = 0,} o shear waves is L ( ℵ ) tight rame. Fig.. Shear wave requency domain subdivision map and cross-sectional area geometry [8]. (a) shear waves requency domain subdivision graph; (b) shear waves requency domain support. In summary, shear wave has the ollowing good properties: ) very good localization characteristics; ) to satisy the parabolic scaling characteristics; 3) there is good directional sensitivity; 4) shear waves can represent a rich direction close to the optimal image inormation on a variety o scales and directions. 3. Image Reconstruction Based on Compressed Sensing x Compressed sensing theory is a new theoretical ramework o which a sampling is combined with compression process. I the signal A is compressible in orthogonal basis or on a tight rame B, the signal nonlinear reconstruction is made rom the relatively small measurement, and it is precise or approximate recovery []. Speciic steps are as ollows: ) Sparse signals. Sparse transorm coeicient T vector a = ψ x is obtained, which is only K nonzero transorm coeicients (K << N). M N ) The proection matrix φ R ( M < N ) is used to proect signals to obtain the observed values y = φx = φψa =Θ a o the signal. Theory shows that i Θ meets the limited equidistant (RIP) nature [], the observation y is used to reconstruct the signal x. 3) Reconstructed signal can be expressed as solving the problem l 0 norm, it is expressed as T min ψ x 0 s.t. y = φ x, minimal solution o nonzero element number is looked rom transorm coeicient a, which is most sparse x solution. The problem is a NP-Hard problem, its solving approach is not unique, we use the most widely application and the most representative Orthogonal Matching (OMP) algorithm [3], the MRI image compression are sampled and reconstructed. 4. Compressed Sensing MRI Image Reconstruction Based on Discrete Shear Waves Sparse representation o the signal is compressed sensing priori conditions. In the MRI image reconstruction based on compressed sensing, the twodimensional wavelet transorm is commonly used as a sparse group, while the wavelet transorm image highly anisotropic edge and texture direction inormation can not give the optimal representation, these aects the quality o the reconstruction. There are multi-directional and anisotropic strengths in discrete shear wave, this paper presents that discrete shear waves transorm is selected as MRI image sparse transormation. Discrete shear waves transorm o MRI images can be divided into two steps in the requency domain, which are namely multi-scale subdivision and localized direction. The image ield is developed and represented in the orm o a inite group ℵ, a discrete MRI image l ( ℵn ) is given, and is decomposed into L layer [4]. Step. Multiscale subdivision. Symmlet wavelet basis is used as a mother unction, discrete domain image is decomposed, low requency coeicients and high requency coeicients a are obtained under various scales, where is the decomposed scales. Step. Localized Direction. In order to obtain high-requency components in dierent directions, in each scale actor, the high-requency is split under the band direction and scale changes o tapered unction, it is as shown in Fig. (a). The above steps are repeated until = L stops. The two scales ( L = ) discrete shear waves low chart shows in Fig.. Fig.. Discrete shear waves low chart. d 9

4 Among them, the scale vector scale can be set as scale = [ s s... sl ], the support size o the scale ( L s ) ( L ) is in the horizontal cone direction, ( L ) ( L s ) the support size o the scale is in the vertical cone direction. The direction vector ndir is ndir = n n n, scale has both set as [ ]... L n + directions in horizontal cone and vertical cone. Fig. 3 is MRI images exploded view, when L =, scale = [ 3], ndir = [ ], there are both ive directions in the horizontal cone and vertical cone. 5) By shear-wave inverse transorm the approximation image is restored ater reconstruction, and the reconstructed image is inally gotten. 5. Experimental Results and Analysis 5.. Obective Evaluation Methods o the Image Reconstruction Quality In order to obectively evaluate the quality o image reconstruction, we use three kinds o obective assessment methods: ) MSE (Mean Square Error, MSE). Let the size o the original MRI image be M N, where the original image is represented by, image reconstructions represented by ˆ, the MSE is deined as: MSE = ˆ MN ) PSNR (Peak Signal-to-Noise Ratio, PSNR) is deined as ollows: 0lg M PSNR = N MSE 3) Structural similarity (Structural Similarity, SSIM) is a measure o the similarity o two images, it is a new index, the more large the value, the better the value, its maximum value is. The original image is represented by, image reconstructions represented by ˆ, SSIM is deined as: ( ) ( ) ( ) γ η μ SSIM(, ˆ) = l, ˆ c, ˆ s, ˆ (a) MRI image; (b) decomposition level is, the coeicient o shear wave in horizontal cone direction; (c) decomposition level is, the coeicient o shear wave in vertical cone direction Fig. 3. Two layer exploded view o MRI images in discrete shear waves. In summary, based on discrete shear waves in this paper, compressed sensing MRI reconstruction algorithm is described as ollows: ) MRI images are decomposed in discrete shear wave, shear coeicients are obtained in each scale, directional sub bands. ) The sampling rate i set, the measurement matrix is constructed. 3) Wave requency subbands o discrete shear transorm coeicients were measured, decomposition coeicients is retained or the low requency approximation subband. Thus one can eectively reduce the amount o data which is required or image reconstruction, other hand, it eectively improves the quality o the reconstructed image. 4) OMP algorithm is used as a compressed sensing reconstruction algorithm, sparse coeicient is recovered ater treatment. Which γ, η, μ is used, respectively, to adust the brightness, the contrast and the heavy right o structural inormation, l(, ˆ ) is brightness comparison unction, c(, ˆ ) is or contrast comparison unction, s (, ˆ ) is or structure comparison unction, their deinitions are μμˆ+ c δδ (, ˆ ˆ + c l ) =, μ + μ ˆ + c ( ˆ c, ) = δ + δ ˆ + c δ ˆ + c 3 and (, ˆ s ) =. Among them, μ δδ ˆ + c 3, μ ˆ is respectively the brightness mean o, ˆ, δ, is respectively the standard deviation o, ˆ, δ ˆ δ ˆ is covariance, c, c, c 3 are generated by adding a constant, when the denominator is close to zero in order to prevent instability. 5.. Experimental Analysis This article select brain MRI to perorm experiments, compressed sensing image 30

5 reconstruction algorithm OMP is used, the sparsity o the discrete shear waves are analyzed, the advantages o MRI compressed sensing reconstruction are veriied based on discrete shear wave. In this paper, the discrete wavelet transorm and shear wave transormation are as sparse transorm, the reconstruction quality o MRI images were compared, and the image reconstruction quality eects o two types o transormation are analyzed at dierent sampling rate. Ater a comparative analysis, to select 5 scale = decomposition scale. L =, [ ] 5... Sparsity Analysis o Discrete Shear Wave To test the shear wave sparsity, we irst do shear wave transormation to test image, the absolute value o the larger shear wave coeicients are retaied only part, while other actors are set to 0, the retention percentage was 30 %, 5 %, 4 % and %, then compressed sensing reconstruction process are done, and inally the reconstructed image is obtained in this part o shear waves inverse transorm coeicients. Fig. 3 is with sampling rate 0.5, the reconstructed image is obtained in the retention percentages o dierent coeicients, SSIM values and the PSNR o the reconstructed image are given in Table. Table. SSIM and PSNR values o the reconstructed image in reserved dierent percentage values o the coeicient. Retention percentage o coeicient PSNR SSIM 30 % % % % The oregoing analysis shows that when the retention actor are 30 % and 5 % coeicient, their reconstruction results are close, although retaining only % o the coeicients, the reconstruction eect is variation, but the PSNR and SSIM values are obtained, and the reconstructed image can be accepted, thereore, there is good sparsity in shear waves transorm, the accuracy o reconstruction can be ensured by compressed sensing The Experimental Comparison Between Wavelet Transorm and Shearlet Transorm Two-dimensional wavelet transorm and shear wave conversion are done to test images respectively, some larger absolute coeicient are retained, while the other coeicients are set to 0, retention percentages are 6.73 %, the images o the two sparse transorm base are shown in Fig. 4. Fig. 5. Sparse images based on two transorm base: (a) shear wave discrete transorm base; (b) the wavelet transorm base. When the sampling rate is 0.5, the reconstructed results are shown in Fig. 6, the corresponding value o MSE, PSNR and SSIM are given in Table. Table. MSE, SSIM and PSNR values o reconstructed images based on wavelet transorm and shearlet transorm. Fig. 4. The reconstructed image in dierent retention rate coeicients: (a) o MRI picture; (b) 30 % or the retention coeicient o the reconstructed image; (c) 5 % or the retention coeicient o the reconstructed image; (d) 4 % coeicient o retention reconstructed image; (e) % to preserve the image reconstruction coeicient. MSE PSNR SSIM Wavelet base.987e Shearlet base.8993e Meanwhile, in order to better illustrate the advantages o the reconstructed image edge detail 3

6 based on the shearlet transorm, Fig. 7 shows the results o comparison o the edge map. reconstructed image is also signiicantly better visual eect, shear waves transorm overcomes the nonsensitive detail o image edge in wavelet transorm, there is an advantage or the reconstruction o edge details. Experiment shows that compressed sensing MRI reconstruction based on shear waves can improve the quality o image reconstruction Quality Comparison o Image Reconstruction at Dierent Sampling Rates At dierent sampling rates, the quality o image reconstruction are compared and to analyzed between wavelet transorm and discrete shearlet transorm. At dierent sampling rates, the corresponding SSIM values are shown in Fig. 8, where the blue line represents the reconstruction based on wavelet transorm, the red line represents reconstruction based on shearlet transorm. Fig. 6. The results o experimental comparison between wavelet transorm and shearlet transorm: (a) or the MRI original; (b) wavelet-based image reconstruction; (c) reconstructed image based on shear waves. Fig. 8. At dierent sampling rates, the SSIM values0 o the image reconstruction between two transorm methods. Fig. 7. Reconstruction eect contrast o edge detail: (a) original edge portions; (b) edge portions o the reconstruction o wavelet-based; (c) edge portions o the shear-wave reconstruction based. The oregoing analysis shows that the MRI reconstruction o compressed sensing is compared based on shearlet transorm and wavelet transorm, a smaller MSE value, higher PSNR, and SSIM are obtained based on shearlet transorm, theoretically reconstruction result is better. Meanwhile, the Experimental results show that with the increase o the sampling rate, the reconstructed image structural similarity increases based on the two transorm, the image quality is improvement. When the sampling rate is lower, SSIM value based on discrete shear waves is signiicantly higher than the reconstruction based on wavelet, the quality o its reconstruction has obvious advantages. When sampling rate is higher, the dierence between the two methods is not obvious, the purpose o the compressed sensing is achieved by ewer low dimensional signal sample values to reconstruct the original high-dimensional signal, the sampling rate is typically selected to be less than 0.5, so the compressed sensing reconstruction based on discrete shearlet transorm is superior to the conclusion based on wavelet reconstruction. 6. Conclusions and Outlook Shear wave transorm is a new multi-scale geometric analysis tool, we use multi-directional and 3

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