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About this book

This volume presents the peer-reviewed proceedings of the international conference Imaging, Vision and Learning Based on Optimization and PDEs (IVLOPDE), held in Bergen, Norway, in August/September 2016. The contributions cover state-of-the-art research on mathematical techniques for image processing, computer vision and machine learning based on optimization and partial differential equations (PDEs).

It has become an established paradigm to formulate problems within image processing and computer vision as PDEs, variational problems or finite dimensional optimization problems. This compact yet expressive framework makes it possible to incorporate a range of desired properties of the solutions and to design algorithms based on well-founded mathematical theory. A growing body of research has also approached more general problems within data analysis and machine learning from the same perspective, and demonstrated the advantages over earlier, more established algorithms.

This volume will appeal to all mathematicians and computer scientists interested in novel techniques and analytical results for optimization, variational models and PDEs, together with experimental results on applications ranging from early image formation to high-level image and data analysis.

Table of Contents

Frontmatter

Image Reconstruction from Incomplete Data

Frontmatter

Chapter 1. Adaptive Regularization for Image Reconstruction from Subsampled Data

Abstract
Choices of regularization parameters are central to variational methods for image restoration . In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in Dong et al. (J Math Imaging Vis 40:82–104, 2011), which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach.
Michael Hintermüller, Andreas Langer, Carlos N. Rautenberg, Tao Wu

Chapter 2. A Convergent Fixed-Point Proximity Algorithm Accelerated by FISTA for the ℓ0 Sparse Recovery Problem

Abstract
We propose an approximation model of the original 0 minimization model arising from various sparse signal recovery problems. The objective function of the proposed model uses the Moreau envelope of the 0 norm to promote the sparsity of the signal in a tight framelet system . This leads to a non-convex optimization problem involved the 0 norm. We identify a local minimizer of the proposed non-convex optimization problem with a global minimizer of a related convex optimization problem. Based on this identification, we develop a two stage algorithm for solving the proposed non-convex optimization problem and study its convergence. Moreover, we show that FISTA can be employed to speed up the convergence rate of the proposed algorithm to reach the optimal convergence rate of \(\mathcal {O}(1/k^2)\). We present numerical results to confirm the theoretical estimate.
Xueying Zeng, Lixin Shen, Yuesheng Xu

Chapter 3. Sparse-Data Based 3D Surface Reconstruction for Cartoon and Map

Abstract
A model combining the first-order and the second-order variational regularizations for the purpose of 3D surface reconstruction based on 2D sparse data is proposed. The model includes a hybrid fidelity constraint which allows the initial conditions to be switched flexibly between vectors and elevations. A numerical algorithm based on the augmented Lagrangian method is also proposed. The numerical experiments are presented, showing its excellent performance both in designing cartoon characters, as well as in recovering oriented three dimensional maps from contours or points with elevation information.
Bin Wu, Talal Rahman, Xue-Cheng Tai

Image Enhancement, Restoration and Registration

Frontmatter

Chapter 4. Variational Methods for Gamut Mapping in Cinema and Television

Abstract
The cinema and television industries are continuously working in the development of image features that provide a better visual experience to viewers, increasing spatial resolution, frame rate, contrast, and recently, with emerging display technologies, much more vivid colors. For this reason there is a pressing need to develop fast, automatic and reliable gamut mapping algorithms that can transform the colors of the original content, adapting it to the capabilities of the display or projector system in which it is going to be viewed while at the same time respecting the artistic intent of the creator. In this article we present a review of our work on variational methods for gamut mapping that comply with some basic global and local properties of the human visual system, producing state-of-the-art results that appear natural and are perceptually faithful to the original material.
Syed Waqas Zamir, Javier Vazquez-Corral, Marcelo Bertalmío

Chapter 5. Functional Lifting for Variational Problems with Higher-Order Regularization

Abstract
Variational approaches are an established paradigm in the field of image processing . The non-convexity of the functional can be addressed by functional lifting and convex relaxation techniques, which aim to solve a convex approximation of the original energy on a larger space. However, so far these approaches have been limited to first-order , gradient-based regularizers such as the total variation . In this work, we propose a way to extend functional lifting to a second-order regularizer derived from the Laplacian. We prove that it can be represented efficiently and thus allows numerical optimization. We experimentally demonstrate the usefulness on a synthetic convex denoising problem and on synthetic as well as real-world image registration problems.
Benedikt Loewenhauser, Jan Lellmann

Chapter 6. On the Convex Model of Speckle Reduction

Abstract
Speckle reduction is an important issue in image processing realm. In this paper, we propose a novel model for restoring degraded images with multiplicative noise which follows a Nakagami distribution. A general penalty term based on the statistical property of the speckle noise is used to guarantee the convexity of the denoising model. Moreover, to deal with the minimizing problem, a generalized Bermudez-Moreno algorithm is adopted and its convergence is analysed. The experimental results on some images subject to multiplicative noise as well as comparisons to other state-of-the-art methods are also presented. The results can verify that the new model is reasonable.
Faming Fang, Yingying Fang, Tieyong Zeng

3D Image Understanding and Classification

Frontmatter

Chapter 7. Multi-Dimensional Regular Expressions for Object Detection with LiDAR Imaging

Abstract
Regular expressions are a fundamental technique for pattern matching in textual data and for lexical analysis in compiler design. They are ubiquitous in most systems used today, including operating systems (e.g. grep, awk), computer languages (e.g. Perl, Java, Python), and web search engines (e.g. Google). However, this highly useful way of exploring and mining data has thus far eluded non-textual datasets, such as images and 3D geometric data. Shape-based searching of 3D objects continues to be a core problem in computer vision . We propose a novel extension of traditional finite-automata-based methods to find multi-dimensional objects in spatial data sets. Our approach extends regular expressions and finite automata to multi-dimensional pattern models. While we demonstrate the effectiveness and efficiency of our approach for finding target objects in 3D LiDAR image data sets using an implicit geometry representation of the data, it is important to note that the proposed technique can be applied to any general data set of vertices in 3D space. Non-geometric information, such as material and spectral characteristics from hyperspectral image data can also be discretized and encoded into our approach.
Todd C. Torgersen, V. Paúl Pauca, Robert J. Plemmons, Dejan Nikic, Jason Wu, Robert Rand

Chapter 8. Relaxed Optimisation for Tensor Principal Component Analysis and Applications to Recognition, Compression and Retrieval of Volumetric Shapes

Abstract
The mathematical and computational backgrounds of pattern recognition are the geometries in Hilbert space used for functional analysis and the applied linear algebra used for numerical analysis, respectively. Organs, cells and microstructures in cells dealt with in biomedical image analysis are volumetric data. We are required to process and analyse these data as volumetric data without embedding into higher-dimensional vector spaces from the viewpoint of object-oriented data analysis . Therefore, sampled values of volumetric data are expressed as three-way array data. The aim of the paper is to develop relaxed closed forms for tensor principal component analysis (PCA) for the recognition , classification , compression and retrieval of volumetric data. Tensor PCA derives the tensor Karhunen-Loève transform, which compresses volumetric data, such as organs, cells in organs and microstructures in cells, preserving both the geometric and statistical properties of objects and spatial textures in the space.
Hayato Itoh, Atsushi Imiya, Tomoya Sakai

Machine Learning and Big Data Analysis

Frontmatter

Chapter 9. An Incremental Reseeding Strategy for Clustering

Abstract
We propose an easy-to-implement and highly parallelizable algorithm for multiway graph partitioning . The algorithm proceeds by alternating three simple routines in an iterative fashion: diffusion , thresholding, and random sampling. We demonstrate experimentally that the proper combination of these ingredients leads to an algorithm that achieves state-of-the-art performance in terms of cluster purity on standard benchmark data sets. We also describe a coarsen, cluster and refine approach similar to Dhillon et al. (IEEE Trans Pattern Anal Mach Intell 29(11):1944–1957, 2007) and Karypis and Kumar (SIAM J Sci Comput 20(1):359–392, 1998) that removes an order of magnitude from the runtime of our algorithm while still maintaining competitive accuracy.
Xavier Bresson, Huiyi Hu, Thomas Laurent, Arthur Szlam, James von Brecht

Chapter 10. Ego-Motion Classification for Body-Worn Videos

Abstract
Portable cameras record dynamic first-person video footage and these videos contain information on the motion of the individual to whom the camera is mounted, defined as ego. We address the task of discovering ego-motion from the video itself, without other external calibration information. We investigate the use of similarity transformations between successive video frames to extract signals reflecting ego-motions and their frequencies. We use novel graph-based unsupervised and semi-supervised learning algorithms to segment the video frames into different ego-motion categories. Our results show very accurate results on both choreographed test videos and ego-motion videos provided by the Los Angeles Police Department.
Zhaoyi Meng, Javier Sánchez, Jean-Michel Morel, Andrea L. Bertozzi, P. Jeffrey Brantingham

Chapter 11. Synchronized Recovery Method for Multi-Rank Symmetric Tensor Decomposition

Abstract
Symmetric tensor decomposition is of great importance in applications. In this paper, we design a synchronized multi-rank symmetric Tensor Decomposition alternating minimization method. In this algorithm, we start from a careful initialization for the non-convex symmetric tensor decomposition and then perform an alternating minimization algorithm. Our contributions are as follows: (1) Our method is synchronized and there is no need for a greedy algorithm to get the multi-rank tensor decomposition . (2) Initialization is an important part in our proposed method. With a careful initialization, our proposed algorithm can converge to the global minimizer of the non-convex objective function. (3) The designed alternating minimization algorithm can give a highly accurate result. In numerical results, our proposed algorithm is much better than the simple gradient descent method from the same initialization. Moreover, our results show that with eigenvectors of random projection as initialization, we can quickly get the global solution by using simple alternating minimization algorithm, though finding the global minimum of this non-convex minimization problem is NP-hard .
Haixia Liu, Lizhang Miao, Yang Wang

Backmatter

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