Optimal Sampling in Compressed Sensing

Transcription

1 Optimal Sampling in Compressed Sensing Joyita Dtta Introdction Compressed sensing allows s to recover objects reasonably well from highly ndersampled data, in spite of violating the Nyqist criterion. In the context of MRI, this techniqe is appealing since a smaller dataset implies faster imaging. This work seeks to explore the mathematics of compressed sensing and to look for ways to generate optimal sampling patterns. The content of this report incldes the following: A simplified overview of compressed sensing Optimality measres for sampling patterns Methodology sed for optimally sampling 1x1 images Analyses and simlation reslts based on optimal sampling of 1x1 images Overview In order to recover an object x R N from a measrement set f R M is where M << N, we have to solve an extremely ill-posed problem: Φ x = f, (1) where Φ R MxN is the measrement system (the ndersampled Forier basis in the case of MRI). This is an nderdetermined system and has infinitely many soltions. So how do we make this problem tractable? The sccess of compressed sensing rests on two key tenets. The first key idea of compressed sensing is to add more information to the inverse problem by exploiting the idea that we can find a basis Ψ R NxN, in which, the objects we intend to image will have a sparse representation. In other words, x = Ψs, () where s R N and has at most K non-zero elements. How this extra information is added will be explained shortly, in the context of the inverse problem formlation. Before that, we are still to find a way to prevent aliasing artifacts in the reconstrcted image de to severe ndersampling in the dataset f. This leads s to the second key idea, which is to make sre that the systems Φ and Ψ are incoherent. This ensres that the aliasing artifacts will appear noise-like in the Ψ domain and are spread ot over all the bases. Ths, the K dominant elements of s are less likely to be corrpted by aliasing. Now, let s look at the formlation of the complete inverse problem:

2 minimize s.t. Ψ x 1 Φ x f < ε (3) The first condition involving the minimization of the L 1 norm of s = Ψ x imposes the sparseness condition as illstrated in Figre 1 [1]. The second condition, where ε is a small positive nmber, ensres that the soltion is consistent with the data f. Figre 1: (a) A sparse vector s lies on a K-dimensional hyperplane aligned with the coordinate axes in R N and ths close to the axes. (b) L minimization gives s the point where the largest hypersphere toches the nllspace (green hyperplane), (c) L 1 minimization will give s a point near the coordinate axes, which is where the sparse vector s is located. [1] Optimal Sampling The primary qestion this work is aiming to tackle is this: How do we choose the best sampling pattern and hence the ndersampling measrement operator, Φ? The existing literatre point at two main approaches: Correlation-based approach Assming Φ, Ψ R NxN are orthobases, the coherence between the two is given by eqation () [,3]. ( Φ, Ψ) = N max φk ψ j µ, 1 k, j N () The idea is to choose Φ for a fixed Ψ sch that they are maximally incoherent, for reasons mentioned before. It can be shown that µ ( Φ, Ψ) [ 1, N ]. Also, random matrices are largely incoherent with any given basis Φ. Herein lies the sccess of random sampling schemes. Another way to measre the coherence is to look at the maximm side-lobe-to-peak ratio in the transform point spread fnction defined in [] as:

3 TPSF( i, j) = e j Ψ Φ ΦΨe i, (5) The two measres have similar connotations and similar comptational costs (as we shall see later). Combinatorial approach As illstrated in Figre, we may look at the measrement, f, as a linear combination of K colmns of Θ =ΦΨ, where Θ, Φ R MxN, Ψ R NxN. If we knew beforehand which of the K elements of s are non-zero, we cold have constrcted a stable, well-conditioned system Θ, by making sre it preserves the lengths of the particlar K-sparse vectors, as given by the restricted isometry property [] in eqation (): Θs δ s = s () In real life, we do not know the particlar K non-zero locations. Ths we wold have to verify eqation () for every one of the grops of colmns in every possible Φ N CK MxN R. This appears to be a danting task considering the prohibitively expensive comptations involved! f f Figre : Representation of f as a linear combination of K colmns of Θ = ΦΨ, where Θ,Φ R MxN, Ψ R NxN, and s is K-sparse. [1] Methodology Let s now look at the reslts presented so far in the context of MRI. To simplify comptation, let s assme we are trying to recover a small 1x1 image (a total of N = 5 pixels) from ndersampled Forier (k-space) data. Or focs is on a DFT Cartesian readot scheme, widely sed in MRI, in which ndersampling is restricted to the phase-encodes, while the readots are flly sampled. The D Forier and Wavelet transforms F and S of the image X R nxn (n = 1) may be written as: F = Φ XΦ S = Ψ XΨ (7) Here Φ R nxn and Ψ R nxn are the Forier and Wavelet operators respectively. In order to be able to apply eqations 1- to or problem, we mst first represent the D

4 transforms in the eqations (7) as linear transformations of the stacked image vector x R N (N=5). We se the fact that the separable D transform matrix can be written as the Kronecker tensor prodct of the transforms: Φ =Φ Φ Ψ =Ψ Ψ () Once these operators have been constrcted, we can formlate or problem in the form described in Figre. We pose or analysis as a series of qestions, which will be answered in the next section: 1. For given N and Φ (the fll Forier operator in this case), what Ψ shold we chose?. What are the comptational costs involved in sing the two different approaches for finding optimal patterns for a given object size N? 3. For given N, Φ, and Ψ, and for M samples, what Φ shold be optimal? Reslts The Wavelet Transform We look for an answer to qestion 1 assming N = 5 and Φ is the fll (orthonormal) Forier matrix. The answer will decide two key things: the sparsity, K, of s and the natral coherence µ ( Φ, Ψ) between the two bases. The first of these is primarily determined by the coarseness level. Figre 3 shows or 1x1 test object, and its level Haar transform. Figre shows plots of the level, 3, and Haar coefficients (the level coefficients being the pixel intensities in the object itself). Figre 5 shows the level Coiflet- transform coefficients and the image reconstrcted from the 15 largest of these indicating the compressibility of or test object. Test Object Level Haar Transform of the Test Object Figre 3: The 1x1 test object (left) and its level Haar transform (right)

5 (a) (b) (c) Figre : Haar Coefficients for levels (a), (b) 3, and (c) (actal pixel intensities) respectively Level Coiflet Transform of the Test Object Image Constrcted from the 15 Largest Coiflet Coefficients Figre 5: (a) Level transform sing Coiflet- wavelets (b) Image constrcted from the 15 largest coefficients indicating compressibility

6 The coefficients were compted for different wavelets generated sing WaveLab 50, and it was observed that the fraction of nonzero coefficients was abot 7% for level and abot 0% with level 3. Next we se eqation to compte the coherence of different wavelet bases with the Forier bases. The reslts are shown in tables I and II for N = 5 and N = respectively. Table I Coherence measres between wavelet bases and the Forier bases for N = 5 Level K Haar- Coiflet- Dabechies- Symmlet- ~7% ~0% Table II Coherence measres between wavelet bases and the Forier bases for N = Level Haar- Coiflet- Dabechies- Symmlet The analysis gives s similar reslts for different wavelet types. Also, it indicates a tradeoff between sparsity and incoherence. So we shall be looking at optimal sampling patterns for both levels and 3. Also, we choose the Haar bases for their inherent simplicity and ease of implementation. Comptational Costs Next, we address qestion and look at the comptational costs involved in sing the two different approaches for finding optimal patterns for a given object size N. Table III lists the nmber of iterations in the combinatorial approach and the correlation-based approaches. Table III Nmber of iterations reqired by different approaches Nmber of ways to pick m samples from n: n C m 1 C = 10 3 C = Combinatorial Coherence SPR 5 C 1 ~ C 3 ~ In order to handle larger images and more phase encodes, we wold have to resort to Monte Carlo simlations as was done in []. Crrently, we are restricting orselves to a 1x1 image and phase encodes. Optimal Sampling Patterns The coherence comptations revealed a minimm coherence level of 1. There was a set of patterns corresponding to this coherence level. Masks for the corresponding

7 sampling patterns were generated and the Forier matrix, Φ, was zero-padded. This zero-padded matrix is then sed to generate ndersampled k-space data and to reconstrct the object. The Sparse MRI V0. set of Matlab fnctions was sed for reconstrction. Figre shows the first test object. Figre 7 shows a set of phase encode levels that lead to minimm coherence and lastly the reconstrcted image Figre : The first test object (same as the one sed before for analyzing wavelets) (a) (b) Figre 7: (a) A set of phase encode levels that lead to minimm coherence, and (b) the reconstrcted image. The reconstrcted image does not look too good obviosly becase data arond center of k-space is missing. To get arond this, we pick a different sampling pattern that gives the same minimm coherence, bt incldes the central phase encode levels. Figre shows this sampling pattern, the initial gess generated from a backward projection of the zero-padded Forier matrix on the ndersampled data, and lastly the reconstrcted image. A second test object is shown in Figre 9. The reconstrction reslts for this object for the sampling pattern in Figre (a) is shown in Figre.

8 (a) (b) (c) Figre : (a) Optimal sampling pattern that incldes central phase encode levels, (b) the initial gess generated from a backward projection of the zero-padded Forier matrix on the ndersampled data, and (c) the mch smoother reconstrcted image Figre 9: Second test object: Another 1x1 image derived from a web icon.

9 Figre : (a) The blrry initial gess generated from a backward projection of the zeropadded Forier matrix on the ndersampled data and (b) the reconstrcted image. Conclsion The main goals of this project were to develop an nderstanding of compressed sensing in the context of its application in MRI, to explore optimal sampling, and to develop optimal sampling patterns for small images with 1x1 pixels. The reconstrcted images look reasonably good and certainly better than those generated by back-projecting the zero-padded Forier matrix on the ndersampled data. Overall, this project was a great learning experience, and I wold like to stdy ahead and frther explore the intricacies of the mathematics behind compressed sensing. References [1] R. G. Baranik, Compressive Sensing [Lectre Notes], IEEE SPM, vol., no., pp , Jly, 007. [] E. Candès and M. B. Wakin, People Hearing Withot Listening: An Introdction To Compressive Sampling, (To appear in IEEE SPM). [3] E. Candès, Compressive sampling, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 00. [] M. Lstig, D. Donoho, and J. M. Paly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Reson Med., vol. 5, no., pp , Dec., 007.