Sparse Gradient Optimization and its Applications in Image Processing

Millions of digital images are captured by imaging devices on a daily basis. The way imaging devices operate follows an integral process from which the information of the original scene needs to be estimated. The estimation is done by inverting the integral process of the imaging device with the use of optimization techniques. This linear inverse problem, the inversion of the integral acquisition process, is at the heart of several image processing applications such as denoising, deblurring, inpainting, and super-resolution. We describe in detail the use of linear inverse problems in these applications. We review and compare several state-of-the-art optimization algorithms that invert this integral process. Linear inverse problems are usually very difficult to solve. Therefore, additional prior assumptions need to be introduced to successfully estimate the output signal. Several priors have been suggested in the research literature, with the Total Variation (TV) being one of the most prominent. In this thesis, we review another prior, the l0 pseudo-norm over the gradient domain. This prior allows full control over how many non-zero gradients are retained to approximate prominent structures of the image. We show the superiority of the l0 gradient prior over the TV prior in recovering genuinely piece-wise constant signals. The l0 gradient prior has shown to produce state-of-the-art results in edge-preserving image smoothing. Moreover, this general prior can be applied to several other applications, such as edge extraction, clip-art JPEG artifact removal, non-photorealistic image rendering, detail magnification, and tone mapping. We review and evaluate several state-of-the-art algorithms that solve the optimization problem based on the l0 gradient prior. Subsequently we apply the l0 gradient prior to two applications where we show superior results as compared to the current state-of-the-art. The first application is that of single-image reflection removal. Existing solutions to this problem have shown limited success because of the highly ill-posed nature of the problem. We show that the standard l0 gradient prior with a modified data-fidelity term based on the Laplacian operator is able to sufficiently remove unwanted reflections from images in many realistic scenarios. We conduct extensive experiments and show that our method outperforms the state-of-the-art. In the second application of haze removal from visible-NIR image pairs we propose a novel optimization framework, where the prior term penalizes the number of non-zero gradients of the difference between the output and the NIR image. Due to the longer wavelengths of NIR, an image taken in the NIR spectrum suffers significantly less from haze artifacts. Using this prior term, we are able to transfer details from the haze-free NIR image to the final result. We show that our formulation provides state-of-the-art results compared to haze removal methods that use a single image and also to those that are based on visible-NIR image pairs

Graph Embedded Nonparametric Mutual Information For Supervised Dimensionality Reduction

In this paper, we propose a novel algorithm for dimensionality reduction that uses as a criterion the mutual information (MI) between the transformed data and their cor- responding class labels. The MI is a powerful criterion that can be used as a proxy to the Bayes error rate. Further- more, recent quadratic nonparametric implementations of MI are computationally efficient and do not require any prior assumptions about the class densities. We show that the quadratic nonparametric MI can be formulated as a kernel objective in the graph embedding framework. Moreover, we propose its linear equivalent as a novel linear dimensionality reduction algorithm. The derived methods are compared against the state-of-the-art dimensionality reduction algorithms with various classifiers and on various benchmark and real-life datasets. The experimental results show that nonparametric MI as an optimization objective for dimensionality reduction gives comparable and in most of the cases better results compared with other dimensionality reduction methods

Extreme Image Completion

It is challenging to complete an image whose 99 percent pixels are randomly missing. We present a solution to this extreme image completion problem. As opposed to existing techniques, our solution has a computational complexity that is linear in the number of pixels of the full image and is real-time in practice. For comparable quality of reconstruction, our algorithm is thus almost 2 to 5 orders of magnitude faster than existing techniques

Single Image Reflection Suppression

Reflections are a common artifact in images taken through glass windows. Automatically removing the reflection artifacts after the picture is taken is an ill-posed problem. Attempts to solve this problem using optimization schemes therefore rely on various prior assumptions from the physical world. Instead of removing reflections from a single image, which has met with limited success so far, we propose a novel approach to suppress reflections. It is based on a Laplacian data fidelity term and an l(0) gradient sparsity term imposed on the output. With experiments on artificial and real-world images we show that our reflection suppression method performs better than the state-of-the-art reflection removal techniques

Laplacian Support Vector Analysis for Subspace Discriminative Learning

In this paper we propose a novel dimensionality reduction method that is based on successive Laplacian SVM projections in orthogonal deflated subspaces. The proposed method, called Laplacian Support Vector Analysis, produces projection vectors, which capture the discriminant information that lies in the subspace orthogonal to the standard Laplacian SVMs. We show that the optimal vectors on these deflated subspaces can be computed by successively training a standard SVM with specially designed deflation kernels. The resulting normal vectors contain discriminative information that can be used for feature extraction. In our analysis, we derive an explicit form for the deflation matrix of the mapped features in both the initial and the Hilbert space by using the kernel trick and thus, we can handle linear and non-linear deflation transformations. Experimental results in several benchmark datasets illustrate the strength of our proposed algorithm

Divergence-free phase contrast MRI

In this work, we extend a separate magnitude and phase regularization framework for Phase Contrast MRI by incorporating the divergence-free condition