A Low Patch-Rank Interpretation of Texture

We propose a novel cartoon-texture separation model using a sparse low-rank decomposition. Our texture model connects the separate ideas of robust principal component analysis (PCA) [E. J. Candes, X. Li, Y. Ma, and J. Wright, J. ACM, 58 (2011), 11], nonlocal methods [A. Buades, B. Coll, and J.-M. Morel, Multiscale Model. Simul., 4 (2005), pp. 490-530], [A. Buades, B. Coll, and J.-M. Morel, Numer. Math., 105 (2006), pp. 1-34], [G. Gilboa and S. Osher, Multiscale Model. Simul., 6 (2007), pp. 595-630], [G. Gilboa and S. Osher, Multiscale Model. Simul., 7 (2008), pp. 10051028], and cartoon-texture decompositions in an interesting way, taking advantage of each of these methodologies. We define our texture norm using the nuclear norm applied to patches in the image, interpreting the texture patches to be low-rank. In particular, this norm is easier to implement than many of the weak function space norms in the literature and is computationally faster than nonlocal methods since there is no explicit weight function to compute. This norm is used as an additional regularizer in several image recovery models. Using total variation as the cartoon norm and our new texture norm, we solve the proposed variational problems using the split Bregman algorithm [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-343]. Since both of our regularizers are of L 1 type, a double splitting provides a fast algorithm that is simple to implement. Based on experimental results, we demonstrate our algorithm's success on a wide range of textures. Also, our particular cartoon-texture decomposition model has the advantage of separating noise from texture. Our proposed texture norm is shown to better reconstruct texture for other applications such as denoising, deblurring, sparse reconstruction, and pattern regularization.