Sung Ha Kang

Euler's Elastica and Curvature-Based Inpainting

Image inpainting is a special image restoration problem for which image prior models play a crucial role. Eulers elastica was first introduced to computer vision by Mumford [Algebraic Geometry and its Applications, Springer-Verlag, New York, 1994, pp. 491--506] as a curve prior model. By functionalizing the elastica energy, Masnou and Morel [Proceedings of the 5th IEEE International Conference Image Processing, 3 (1998), pp. 259--263] proposed an elastica-based variational inpainting model. The current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester [SIGGRAPH 2000, ACM Press, New York, 2000] and Chan and Shen [J. Visual Comm. Image Rep., 12 (2001), pp. 436--449]. A computational scheme based on numerical PDEs is presented, which allows the automatic handling of topologically complex inpainting domains.

Total Variation Denoising and Enhancement of Color Images Based on the CB and HSV Color Models

Most denoising and enhancement methods for color images have been formulated on linear color models, namely, the channel-by-channel model and vectorial model. In this paper, we study the total variation (TV) restoration based on the two nonlinear (or nonflat) color models: the chromaticity?brightness model and hue?saturation?value model. These models are known to be closer to human perception. Recent works on the variational/PDE method for nonflat features by several authors enable us to denoise the chromaticity and hue components directly. We present both the mathematical theory and digital implementation for the TV method. Comparison to the traditional TV restorations based on linear color models is made through various experiments.

MULTIPHASE IMAGE SEGMENTATION VIA MODICA-MORTOLA PHASE TRANSITION ∗

We propose a novel multiphase segmentation model built upon the celebrated phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term that complements it. The proposed sine-sinc model outputs a single multiphase distribution from which each individual segment or phase can be easily extracted. Theoretical analysis is developed for the

Color image decomposition and restoration

\Gamma

Total variation minimizing blind deconvolution with shock filter reference

-convergence behavior of the proposed model and the existence of its minimizers. Since the model is not quadratic nor convex, for computation we adopted the convex-concave procedure (CCCP) that has been developed in the literatures of both computational nonlinear PDEs and neural computation. Numerical details and experiments on both synthetic and natural images are presented.

Variational Models for Image Colorization via Chromaticity and Brightness Decomposition

Meyer has recently introduced an image decomposition model to split an image into two components: a geometrical component and a texture (oscillatory) component. Inspired by his work, numerical models have been developed to carry out the decomposition of gray scale images. In this paper, we propose a decomposition algorithm for color images. We introduce a generalization of Meyer’s G norm to RGB vectorial color images, and use Chromaticity and Brightness color model with total variation minimization. We illustrate our approach with numerical examples.

Error Analysis for Image Inpainting

We present a preconditioned method for blind image deconvolution. This method uses a pre-processed reference image (via the shock filter) as an initial condition for total variation minimizing blind deconvolution. Using the shock filter gives good information on location of the edges, while using the variational functionals such as Chan and Wongs [T.F. Chan, C.K. Wong, Total variation blind deconvolution, IEEE Transactions on Image Processing 7 (1998), 370-375] allows robust reconstruction of the image and the blur kernel. Comparison between using the L^1 and L^2 norms for the fidelity term is presented, as well as an analysis on the choice of the parameter for the kernel functional. Numerical results indicate the method is robust for both black and non-black background images while reducing the overall computational cost.

Image and Video Colorization Using Vector-Valued Reproducing Kernel Hilbert Spaces

Colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given. We propose a couple of variational models using chromaticity color components to colorize black and white images. We first consider total variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Second, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.

Inpainting from multiple views

Image inpainting refers to restoring a damaged image with missing information. In recent years, there have been many developments on computational approaches to image inpainting problem [2, 4, 6, 9, 11–13, 27, 28]. While there are many effective algorithms available, there is still a lack of theoretical understanding on under what conditions these algorithms work well. In this paper, we take a step in this direction. We investigate an error bound for inpainting methods, by considering different image spaces such as smooth images, piecewise constant images and a particular kind of piecewise continuous images. Numerical results are presented to validate the theoretical error bounds.

Unsupervised Multiphase Segmentation: A Phase Balancing Model

Motivated by the setting of reproducing kernel Hilbert space (RKHS) and its extensions considered in machine learning, we propose an RKHS framework for image and video colorization. We review and study RKHS especially in vectorial cases and provide various extensions for colorization problems. Theory as well as a practical algorithm is proposed with a number of numerical experiments.