Introduction. Wavelets, Curvelets [4], Surfacelets [5].

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1 Introduction Signal reconstruction from the smallest possible Fourier measurements has been a key motivation in the compressed sensing (CS) research [1]. Accurate reconstruction from partial Fourier data is possible when the signal is sparse or sparsely represented in a transform domain [2,3]. Examples of effective transforms: Wavelets, Curvelets [4], Surfacelets [5].

2 Sparse Representation in gradient domain is more effective for bounded variation (BV) signals. TV-based CS reconstruction is based on minimizing total variation (gradient magnitude). Individual gradient components are sparser than the gradient magnitude, and can be recovered more accurately from fewer Fourier measurements [6], but the resulting gradient field may not be integrable [6]. Gradient Magnitude Horizontal Derivative Vertical Derivative

3 Objective Image reconstruction from Partial Fourier data, by simultaneous recovery of its sparse partial derivatives. Leverage higher sparsity of individual gradient components Enforce integrability of the gradient field at the reconstruction stage Reconstruct final image via Poisson equation [7]

4 Method Notation F f p D x, D y f x, f y p x, p y Description partial Fourier operator unknown image with N pixels partial Fourier measurements of the image, p = Ff partial Derivative operators partial derivatives of the image: f x =D x f, f y =D y f partial Fourier measurements corresponding to f x, f y TV minimization problem: argmin Ff p λ( f x 1 + f y ) 1 f Horizontal and vertical partial Fourier measurements: p x = 1 e 2πiωx/N Ff, p y = (1 e 2πiωy/N )Ff

5 Recover image gradient components separately [6]: To ensure recovered field [f x, f y ] is a gradient field, hence integrable, it has to be curl-free (zero curl): D x f y D y f x = 0 Recover an integrable gradient field instead of two individual components: argmin Ff x p 2 x 2 + Ff y p y 2 f x, f y argmin Ff x p 2 x 2 + λ f x 1 f x 2 argmin Ff y p y 2 + λ fy 1 f y + λ f y 1 + μ D x f y D y f x λ fx 1

6 Define: G = F 0 0 F & p = μd y μd x p x p y 0 & v = f x f y v is the gradient vector field New ι1 minimization formulation argmin Gv p λ v 1 v Given f x, f y, image f is reconstructed through Poisson equation: 2 f = D x f x + D y f y

7 Results The proposed method, called Sparse Partial Derivatives (SPD), outperforms TV and separate recovery of gradient components (GradRec) [6]. Near optimal recovery of Shepp-Logan phantom is possible with only 2.5% of data. Outperforms TV reconstruction with fewer number of samples, since individual partial derivatives are sparser than TV image.

8 GradRec SNR: 7.35dB TV SNR: 14.61dB Proposed SNR: 37.72dB Shepp-Logan phantom reconstructed from 2.5% of Fourier Data.

9 Error in (a) Ground Truth (a) Separate Gradient Recovery SNR : db Error in (b) (b) TV reconstruction SNR : db (c) Proposed SNR : db Error in (c) Brain dataset [8] (and the amplified error) recovered from partial Fourier data at a 17% sampling rate.

10 The Marschner-Lobb dataset sampled at , and recovered from 18.5% partial Fourier data Results (cntd.) Ground Truth[9] Separate Gradient Recovery SNR : 8.63 db TV reconstruction SNR : db Proposed SNR : db

11 Conclusion Proposed a novel formulation for CS image reconstruction from incomplete Fourier data that exploits the interdependency of image partial derivatives. The formulation benefits from the integration of curl-free constraints into the sparse approximation problem. Results show that at very low sampling rates the proposed method achieves significantly higher accuracy, compared to TV and gradient-based methods that rely on post-processing.

12 References 1) D. Donoho, Compressed sensing, IEEE Trans. on Info. Theory, vol. 52, no. 4, pp , ) A. M. Bruckstein, D. L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM review, vol. 51, no. 1, pp , ) E. J. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. on Info. Theory, vol. 52, no. 2, pp , ) E. Candès, L. Demanet, D. Donoho, and L. Ying, Fast discrete curvelet transforms, Multiscale Modeling & Simulation, vol. 5, no. 3, pp , Jan ) Y. M. Lu and M. N. Do, Multidimensional directional filter banks and surfacelets, IEEE Trans. on Image Processing, vol. 16, no. 4, pp , ) V. M. Patel, R. Maleh, A. C. Gilbert, and R. Chellappa, Gradient-based image recovery methods from incomplete Fourier measurements, Image Processing, IEEE Trans. on, vol. 21, no. 1, pp , ) P. Pérez, M. Gangnet, and A. Blake, Poisson image editing, in ACM Trans. on Graphics (TOG). ACM, 2003, vol. 22, pp ) M. Guerquin-Kern et al., Realistic analytical phantoms for parallel magnetic resonance imaging, IEEE Trans. On Med. Imaging, vol. 31, no. 3, pp , ) S. R. Marschner and R. J. Lobb, An evaluation of reconstruction filters for volume rendering, in Proceedings of the conference on Visualization 94. IEEE Com. Society Press, 1994, pp