Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting

Abstract

Abstract. Finding optimal inpainting data plays a key role in the field of image compression with partial differential equations (PDEs). In this paper, we optimise the spatial as well as the tonal data such that an image can be reconstructed with minimised error by means of discrete homogeneous diffusion inpainting. To optimise the spatial distribution of the inpainting data, we apply a probabilistic data sparsification followed by a nonlocal pixel exchange. Afterwards we optimise the grey values in these inpainting points in an exact way using a least squares approach. The resulting method allows almost perfect reconstructions with only 5% of all pixels. This demonstrates that a thorough data optimisation can compensate for most deficiencies of a suboptimal PDE interpolant.

Compression of Depth Maps with Segment-Based Homogeneous Diffusion

“…Then, as long as we have at least one mask pixel in each segment, there exists an unique solution of the discrete problem (cf. [18]). …”

“…We accept the larger coding costs of these free points and place them at locations where the quality can be improved most. Therefore, we perform in a first step a so-called probabilistic densification, which is similar to the probabilistic sparsification process described in [18]. We consecutively select points as additional mask points, starting with the ones from the hexagonal mask as an initialisation.…”

“…Moreover, it has been demonstrated that it pays off to apply a so called nonlocal pixel exchange [18], which tries to find better combinations for the mask points not lying on the hexagonal grid. Thereby a small number k of non-mask points is randomly selected, and the local error is computed.…”

“…Thus, we can only improve our result. For more details we refer to [18]. We use the parameter k = 10, as this choice turns out to yield good results.…”

“…Instead of being fully exact at the selected pixels, it makes sense to allow for some deviation at these locations to achieve an overall smaller global error. In order to find optimal values we make use of a least squares approach as suggested in [18]. Note that, as we are using SBHD, the computations can be speeded up by computing the optimal grey values for the individual segments in parallel.…”