Using the Split Bregman Algorithm to Solve the Self-repelling Snakes Model

Preserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

eGHWT: The Extended Generalized Haar–Walsh Transform

Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important, and much research has been done recently. The generalized Haar–Walsh transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh–Hadamard transforms. We propose the extended generalized Haar–Walsh transform (eGHWT), which is a generalization of the adapted time–frequency tilings of Thiele and Villemoes (1996). The eGHWT examines not only the efficiency of graph-domain partitions but also that of “sequency-domain” partitions simultaneously. Consequently, the eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost, $$O(N \log N)$$ O ( N log N ) , where N is the number of nodes of an input graph. While the GHWT best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than $$(1.5)^N$$ ( 1.5 ) N possible orthonormal bases in $$\mathbb {R}^N$$ R N , the eGHWT best-basis algorithm can find a better one by searching through more than $$0.618\cdot (1.84)^N$$ 0.618 · ( 1.84 ) N possible orthonormal bases in $$\mathbb {R}^N$$ R N . This article describes the details of the eGHWT best-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Furthermore, we also show how the eGHWT can be extended to 2D signals and matrix-form data by viewing them as a tensor product of graphs generated from their columns and rows and demonstrate its effectiveness on applications such as image approximation.

Scale-Covariant and Scale-Invariant Gaussian Derivative Networks

This paper presents a hybrid approach between scale-space theory and deep learning, where a deep learning architecture is constructed by coupling parameterized scale-space operations in cascade. By sharing the learnt parameters between multiple scale channels, and by using the transformation properties of the scale-space primitives under scaling transformations, the resulting network becomes provably scale covariant. By in addition performing max pooling over the multiple scale channels, or other permutation-invariant pooling over scales, a resulting network architecture for image classification also becomes provably scale invariant. We investigate the performance of such networks on the MNIST Large Scale dataset, which contains rescaled images from the original MNIST dataset over a factor of 4 concerning training data and over a factor of 16 concerning testing data. It is demonstrated that the resulting approach allows for scale generalization, enabling good performance for classifying patterns at scales not spanned by the training data.

A Multi-view Camera Model for Line-Scan Cameras with Telecentric Lenses

We propose a novel multi-view camera model for line-scan cameras with telecentric lenses. The camera model supports an arbitrary number of cameras and assumes a linear relative motion with constant velocity between the cameras and the object. We distinguish two motion configurations. In the first configuration, all cameras move with independent motion vectors. In the second configuration, the cameras are mounted rigidly with respect to each other and therefore share a common motion vector. The camera model can model arbitrary lens distortions by supporting arbitrary positions of the line sensor with respect to the optical axis. We propose an algorithm to calibrate a multi-view telecentric line-scan camera setup. To facilitate a 3D reconstruction, we prove that an image pair acquired with two telecentric line-scan cameras can always be rectified to the epipolar standard configuration, in contrast to line-scan cameras with entocentric lenses, for which this is possible only under very restricted conditions. The rectification allows an arbitrary stereo algorithm to be used to calculate disparity images. We propose an efficient algorithm to compute 3D coordinates from these disparities. Experiments on real images show the validity of the proposed multi-view telecentric line-scan camera model.

On the Tightness of Semidefinite Relaxations for Rotation Estimation

Why is it that semidefinite relaxations have been so successful in numerous applications in computer vision and robotics for solving non-convex optimization problems involving rotations? In studying the empirical performance, we note that there are few failure cases reported in the literature, in particular for estimation problems with a single rotation, motivating us to gain further theoretical understanding. A general framework based on tools from algebraic geometry is introduced for analyzing the power of semidefinite relaxations of problems with quadratic objective functions and rotational constraints. Applications include registration, hand–eye calibration, and rotation averaging. We characterize the extreme points and show that there exist failure cases for which the relaxation is not tight, even in the case of a single rotation. We also show that some problem classes are always tight given an appropriate parametrization. Our theoretical findings are accompanied with numerical simulations, providing further evidence and understanding of the results.