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Two -dimensional cubic convolution for image interpolation, restoration, and super -resolution
Cubic convolution has been used for image interpolation since the 1970's and provides a good compromise between computational complexity and interpolation accuracy. Cubic convolution can be parameterized and then optimized either for general performance characteristics or for optimal fidelity over an image ensemble with specific characteristics. Traditionally, the cubic kernel has been derived in one dimension with one parameter and applied to two-dimensional images in a separable fashion. However, images typically are statistically non-separable. ^ This dissertation develops three new non-separable cubic convolution kernels. The first kernel, with three parameters (designated 2D-3PCC), is the most general two-dimensional, piecewise-cubic interpolator defined on [−2, 2] × [−2, 2] with constraints for biaxial symmetry, diagonal symmetry, continuity, and smoothness. The second kernel, with five parameters (designated 2D-5PCC), relaxes the constraint of diagonal symmetry, based on the observation that many images have rotationally asymmetric statistics. The new interpolation kernels can improve image interpolation accuracy by incorporating a priori knowledge of scenes. Relaxing the interpolation constraint on 2D-3PCC yields the third kernel, with five parameters (designated 2D-5PCC-R), for one-pass image restoration and reconstruction. ^ This dissertation also develops a closed-form solution for determining the optimal parameter values for 2D-3PCC and 2D-5PCC with respect to ensembles of scenes characterized by autocorrelation (or power spectrum) and for 2D-5PCC-R with respect to scene autocorrelation (or power spectrum) and the continuous-discrete-continuous (CDC) imaging system. The spatial-domain formulation for optimal parameterization is computationally efficient and establishes the basis for locally adaptive filters. The closed-form solution supports any approximation of scene autocorrelation. This dissertation develops a Markov random field (MRF) model with affine transformation to model scene autocorrelation and a closed-form solution to fit the model. ^ Quantitative fidelity analysis and visual experiments indicate that 2D-3PCC and 2D-5PCC can outperform several popular interpolation methods. An analysis of the error budgets for reconstruction error associated with blurring and aliasing illustrates that the methods improve interpolation fidelity for images with aliased components. For images with little or no aliasing, the methods yield results similar to other popular methods. 2D-5PCC-R can perform as well as traditional two-pass image restoration and reconstruction methods. Cubic convolution kernels are low-order polynomials with small spatial support and so are easy to implement and efficient to apply. ^ Finally, this dissertation demonstrates an important application of 2D-5PCC-R in super-resolution imaging. Super-resolution imaging restores and reconstructs an enhanced image from two or more overlapping images. This dissertation formulates and solves the super-resolution problem based on the CDC imaging system model. Experimental results from simulation and real images indicate that super-resolution with 2D-5PCC-R significantly improves image visual quality. ^
Shi, Jiazheng, "Two -dimensional cubic convolution for image interpolation, restoration, and super -resolution" (2005). ETD collection for University of Nebraska - Lincoln. AAI3186881.