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This book covers recent advances in image processing and imaging sciences from an optimization viewpoint, especially convex optimization with the goal of designing tractable algorithms. Throughout the handbook, the authors introduce topics on the most key aspects of image acquisition and processing that are based on the formulation and solution of novel optimization problems. The first part includes a review of the mathematical methods and foundations required, and covers topics in image quality optimization and assessment. The second part of the book discusses concepts in image formation and capture from color imaging to radar and multispectral imaging. The third part focuses on sparsity constrained optimization in image processing and vision and includes inverse problems such as image restoration and de-noising, image classification and recognition and learning-based problems pertinent to image understanding. Throughout, convex optimization techniques are shown to be a critically important mathematical tool for imaging science problems and applied extensively.

Convex Optimization Methods in Imaging Science is the first book of its kind and will appeal to undergraduate and graduate students, industrial researchers and engineers and those generally interested in computational aspects of modern, real-world imaging and image processing problems.



1. Introduction

Images and videos are ubiquitous in our multimedia-rich environment today. We can capture photographs on a variety of devices, from inexpensive mobile phones to highly sophisticated cameras, spanning an impressive range in sensor pixel-count and picture resolution. We watch video content on displays ranging in size from handheld devices to gigantic movie screens. The resolution on these videos ranges from the grainy CIF (352 × 288) in CCTVs all the way up to 8K ultra high-definition (7680 × 4320) and beyond. If that is not impressive enough, devices routinely support high-quality streaming video content, while preserving the richness of spectral or color information. By some estimates, the global consumer electronics market is poised to be worth a mind-boggling trillion US dollars by 2020.

Vishal Monga

2. Optimizing Image Quality

The fact that multimedia services have become the major driver for next generation wireless networks underscores their technological and economic impact. A vast majority of these multimedia services are consumer-centric and therefore must guarantee a certain level of perceptual quality. Given the massive volumes of image and video data in question, it is only natural to adopt automatic quality prediction and optimization tools. The past decade has seen the invention of several excellent automatic quality prediction tools for natural images and videos. While these tools predict perceptual quality scores accurately, they do not necessarily lend themselves to standard optimization techniques. In this chapter, a systematic framework for optimization with respect to a perceptual quality assessment algorithm is presented. The Structural SIMilarity (SSIM) index, which has found vast commercial acceptance owing to its high performance and low complexity, is the representative image quality assessment model that is studied. Specifically, a detailed exposition of the mathematical properties of the SSIM index is presented first, followed by a discussion on the design of linear and non-linear SSIM-optimal image restoration algorithms.

Dominique Brunet, Sumohana S. Channappayya, Zhou Wang, Edward R. Vrscay, Alan C. Bovik

3. Computational Color Imaging

Color quality and fidelity are fundamental considerations in today’s digital imaging systems. Optimization of a color imaging system is a multifaceted problem involving deep understanding of device physics, light-surface interactions, human visual perception, and computational mathematics. The design of a successful color imaging system that meets the desired performance, reliability and cost evokes many interesting and challenging optimization problems. This chapter explores a variety of optimization frameworks that have been developed for color capture, display, and printing. For each device genre, a broad introduction to challenges in color imaging is first presented, followed by a detailed exposition of selected optimization problems and their solutions. Practical considerations such as computational cost, noise containment, and power consumption are introduced as mathematical constraints into the given optimization problem. The chapter concludes with suggestions for future work in this domain.

Raja Bala, Graham Finlayson, Chul Lee

4. Optimization Methods for Synthetic Aperture Radar Imaging

We review recent developments in Synthetic Aperture Radar (SAR) image formation from an optimization perspective. Majority of these methods can be viewed as constrained least squares problems exploiting sparsity. We reviewed analytic and large scale numerical optimization based approaches in both deterministic and Bayesian frameworks. These methods offer substantial improvements in image quality, suppression of noise and clutter. Analytic methods also have the advantage of computational efficiency.

Eric Mason, Ilker Bayram, Birsen Yazici

5. Computational Spectral and Ultrafast Imaging via Convex Optimization

Multidimensional optical imaging, that is, capturing light in more than two-dimensions (unlike conventional photography), has been an emerging field with widespread applications in diverse domains. Due to the intrinsic limitation of two-dimensional detectors in capturing inherently higher-dimensional data, multidimensional imaging techniques conventionally rely on a scanning process, which renders them inefficient in terms of light throughput and unsuitable for dynamic scenes. In this chapter, we present recent multidimensional imaging techniques for spectral and temporal imaging, which overcome the temporal, spectral, and spatial resolution limitations of conventional scanning-based systems. Each development is based on the computational imaging paradigm, which involves distributing the imaging task between a physical and a computational system and then digitally forming the image datacube of interest from multiplexed measurements by means of solving an inverse problem via convex optimization techniques.

Figen S. Oktem, Liang Gao, Farzad Kamalabadi

6. Discriminative Sparse Representations

In recent years, Sparse Representation (SR) and Dictionary Learning (DL) have emerged as powerful tools for efficient processing image and video data in non-traditional ways. An area of promise for these theories is object recognition. In this chapter, we review the role of algorithms based on SR and DL for object recognition. In particular, supervised, unsupervised, weakly supervised, nonlinear kernel-based, convolutional sparse coding and analysis DL methods are reviewed.

He Zhang, Vishal M. Patel

7. Sparsity Based Nonlocal Image Restoration: An Alternating Optimization Approach

There are many rich connections between the theory of mathematical optimization and the practice of image restoration. However, several fundamental questions remain open—e.g., how to translate some physical insight into an appropriate mathematical objective/cost functional? what kind of optimization tools should be called on first? The objective of this chapter is to stress the difference between the theory and the practice—namely, in the practice of image restoration, the objective is often not to solve the formulated optimization problem correctly but to obtain a nicely-restored image through the process of optimization. In other words, we advocate the termination of an iterative optimization algorithm before it reaches the convergence for various practical considerations (e.g., computational constraints, regularization purpose). Meanwhile, we will show that strategies such as relaxation and divide-and-conquer—even though they do not help the pursuit of a globally optimal solution—are often sufficient for the applications of image restoration. We will use two specific applications—namely image denoising and compressed sensing—to demonstrate how simultaneous sparse coding and nonlocal regularization both admit a nonconvex optimization-based formulation, which can lead to novel insights to our understanding why BM3D and BM3D-CS can achieve excellent performance.

Xin Li, Weisheng Dong, Guangming Shi

8. Sparsity Constrained Estimation in Image Processing and Computer Vision

Over the past decade, sparsity has emerged as a dominant theme in signal processing and big data applications. In this chapter, we formulate and solve new flavors of sparsity-constrained optimization problems built on the family of spike-and-slab priors. First, we develop an efficient Iterative Convex Refinement solution to the hard non-convex problem of Bayesian signal recovery under sparsity-inducing spike-and-slab priors. We also offer a Bayesian perspective on sparse representation-based classification via the introduction of class-specific priors. This formulation represents a consummation of ideas developed for model-based compressive sensing into a general framework for sparse model-based classification.

Vishal Monga, Hojjat Seyed Mousavi, Umamahesh Srinivas

9. Optimization Problems Associated with Manifold-Valued Curves with Applications in Computer Vision

A commonly occurring need in many computer vision applications is the need to represent, compare, and manipulate manifold-valued curves, while allowing for enough flexibility to operate in resource constrained environments. We address these concerns in this chapter, by proposing a dictionary learning scheme that takes geometry and time into account, while performing better than the original data in applications such as activity recognition. We are able to do this with the use of the transport square-root velocity function, which provides an elastic representation for trajectories on Riemannian manifolds. Since these operations can be computationally very expensive, we also present a geometry-based symbolic approximation framework, as a result of which low-bandwidth transmission and accurate real-time analysis for recognition or searching through sequential data become fairly straightforward. We discuss the different optimization problems encountered in this context—learning a sparse representation for actions using extrinsic and intrinsic features, solving the registration problem between two Riemannian trajectories, and learning an optimal clustering scheme for symbolic approximation.

Rushil Anirudh, Pavan Turaga, Anuj Srivastava
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