A unified approach to optimal image interpolation problems based on linear partial differential equation models

Abstract

The unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed-form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal-spatial. The unknown image is reconstructed from a finite set of sampled data in such a way that a mean-square error is minimized by first expressing the solution in terms of the reproducing kernel of a related Hilbert space, and then constructing this kernel using the fundamental solution of an induced linear partial differential equation, or the Green's function of the corresponding self-adjoint operator. It is proved that in most cases, closed-form fundamental solutions (or Green's functions) for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first- or second-order linear partial differential operator. An efficient method for obtaining the corresponding closed-form fundamental solutions (or Green's functions) of the operators is presented. A computer simulation demonstrates the reconstruction procedure.

Super-resolution using neural networks based on the optimal recovery theory

“…A lot of image reconstruction algorithms based on the optimal recovery theory arise recently [23,[25][26][27][28]. Although quite different from each other, they benefit from the common strengths intrinsic to the optimal recovery representation: the subjective quality of output images is satisfactory [27].…”

“…(e) Partial Differential Equation (PDE) models [23,24] establish inter-pixel correlation from a more abstract point of view, yet they usually yield images with jagged artifacts and are vulnerable to noises. (f) Optimal recovery interpolators [23,[25][26][27], and (g) Neural-network based methods [29][30][31], which are to be detailed in Section 2.…”

“…Such resolution enhancement techniques may be generally referred to as Super Resolution (SR) image reconstruction, which primarily consists of three categories: (a) Multiple-frame SR by extracting a single HR image from a sequence of Low-Resolution (LR) images aligned in sub-pixel accuracy (see [1][2][3] for excellent surveys) (b) Single-frame example-based SR that increases the resolution of a single image with training based on analogous images [4][5][6][7][8][9][10][11][29][30][31] (c) Single-frame SR that does not depend on any other images [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The latter category of single-frame SR is more commonly referred to as image interpolation or digital zooming.…”

Image interpolation using MLP neural network with phase compensation of wavelet coefficients

“…The PDE-based models can be classified into the time-dependent model and the timeindependent model. The issue of the time-independent model is to find the mathematically optimal interpolation based on the PDE description of the interpolation problem [15][16][17]. The time-dependent model usually exploits the PDEs that delineate physical phenomena such as heat diffusion, curvature evolution, and Brownian motion [18][19][20].…”

Adaptively Quadratic (AQua) Image Interpolation

“…The work of [23] is related to our image interpolation approach. The authors pose the image interpolation problem as one where the image belongs to a fixed quadratic image class.…”