Highly Corrupted Image Inpainting Through Hypoelliptic Diffusion

Abstract

We present a new bio-mimetic image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in (U. Boscain et al. SIAM J. Imaging Sci., 7(2):669-695, 2014) and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on highly corrupted images (i.e., where more than the 80% of the image is missing), for which we obtain high-quality reconstructions.

Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion

“…It is based on previous work by Hubel-Wiesel [24] and Hoffman [23], then it was refined by Citti et al [13,14], Duits et al [17][18][19][20]. and by the authors of the present paper in [4][5][6][7]. It was also studied by Hladky and Pauls in [22].…”

“…In [4,5,7] we presented a sophisticated and highly parallelizable numerical scheme, based on the noncommutative Fourier transform on a suitable semidiscretization of the group SE(2), i.e., the semidiscrete group of roto-translations SE(2, N ) for N ∈ N. This is the semi-direct product SE(2, N ) = R 2 Z N , where Z N is the cyclic group of order N and the action of n ∈ Z N on R 2 is given by the rotation R n of angle 2πn/N . As pointed out in [5], considering a discrete number of orientations seems to be in accordance with experimental evidence.…”

“…In view of the many well-studied examples of contour completion operated by the human brain (see Figure 1), one of the main problems that was attacked from this point of view has been that of image reconstruction, also known as image inpainting [3,12,15,28]. In this survey, we present the results we obtained in a series of works [4][5][6][7]33] in this context. The paper is divided in three parts, corresponding to different methods and purposes of image reconstruction.…”

Total Variation and Mean Curvature PDEs on the Space of Positions and Orientations

“…The PDE system (13) can be re-expressed by a left-invariant PDE on SE(d) as done in related previous works by several researchers [16,21,9,12,6,10]. For d = 2 this is straightforward as SE(2) ≡ R 2 S 1 .…”

“…In the last decade, many PDE-based image analysis techniques for tracking and enhancement of curvilinear structures took advantage of lifting the image data to the homogeneous space M = R d S d−1 of d-dimensional positions and orientations, cf. [14,10,34,6,4,8]. The precise definition of this homogeneous space follows in the next subsection.…”