Accelerated Optimization in the PDE Framework: Formulations for the Active Contour Case

Following the seminal work of Nesterov, accelerated optimization methods have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios where second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it also performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with a basis of attraction large enough to contain the initial overshoot. This behavior has made accelerated and stochastic gradient search methods particularly popular within the machine learning community. In their recent PNAS 2016 paper, Wibisono, Wilson, and Jordan demonstrate how a broad class of accelerated schemes can be cast in a variational framework formulated around the Bregman divergence, leading to continuum limit ODE's. We show how their formulation may be further extended to infinite dimension manifolds (starting here with the geometric space of curves and surfaces) by substituting the Bregman divergence with inner products on the tangent space and explicitly introducing a distributed mass model which evolves in conjunction with the object of interest during the optimization process. The co-evolving mass model, which is introduced purely for the sake of endowing the optimization with helpful dynamics, also links the resulting class of accelerated PDE based optimization schemes to fluid dynamical formulations of optimal mass transport

Quickest Moving Object Detection

We present a general framework and method for simultaneous detection and segmentation of an object in a video that moves (or comes into view of the camera) at some unknown time in the video. The method is an online approach based on motion segmentation, and it operates under dynamic backgrounds caused by a moving camera or moving nuisances. The goal of the method is to detect and segment the object as soon as it moves. Due to stochastic variability in the video and unreliability of the motion signal, several frames are needed to reliably detect the object. The method is designed to detect and segment with minimum delay subject to a constraint on the false alarm rate. The method is derived as a problem of Quickest Change Detection. Experiments on a dataset show the effectiveness of our method in minimizing detection delay subject to false alarm constraints

SurfCut: Surfaces of Minimal Paths From Topological Structures

We present SurfCut, an algorithm for extracting a smooth, simple surface with an unknown 3D curve boundary from a noisy 3D image and a seed point. Our method is built on the novel observation that certain ridge curves of a function defined on a front propagated using the Fast Marching algorithm lie on the surface. Our method extracts and cuts these ridges to form the surface boundary. Our surface extraction algorithm is built on the novel observation that the surface lies in a valley of the distance from Fast Marching. We show that the resulting surface is a collection of minimal paths. Using the framework of cubical complexes and Morse theory, we design algorithms to extract these critical structures robustly. Experiments on three 3D datasets show the robustness of our method, and that it achieves higher accuracy with lower computational cost than state-of-the-art

Accelerated Optimization in the PDE Framework: Formulations for the Manifold of Diffeomorphisms

We consider the problem of optimization of cost functionals on the infinite-dimensional manifold of diffeomorphisms. We present a new class of optimization methods, valid for any optimization problem setup on the space of diffeomorphisms by generalizing Nesterov accelerated optimization to the manifold of diffeomorphisms. While our framework is general for infinite dimensional manifolds, we specifically treat the case of diffeomorphisms, motivated by optical flow problems in computer vision. This is accomplished by building on a recent variational approach to a general class of accelerated optimization methods by Wibisono, Wilson and Jordan, which applies in finite dimensions. We generalize that approach to infinite dimensional manifolds. We derive the surprisingly simple continuum evolution equations, which are partial differential equations, for accelerated gradient descent, and relate it to simple mechanical principles from fluid mechanics. Our approach has natural connections to the optimal mass transport problem. This is because one can think of our approach as an evolution of an infinite number of particles endowed with mass (represented with a mass density) that moves in an energy landscape. The mass evolves with the optimization variable, and endows the particles with dynamics. This is different than the finite dimensional case where only a single particle moves and hence the dynamics does not depend on the mass. We derive the theory, compute the PDEs for accelerated optimization, and illustrate the behavior of these new accelerated optimization schemes

Matching Through Features and Features Through Matching

This paper addresses how to construct features for the problem of image correspondence, in particular, the paper addresses how to construct features so as to maintain the right level of invariance versus discriminability. We show that without additional prior knowledge of the 3D scene, the right tradeoff cannot be established in a pre-processing step of the images as is typically done in most feature-based matching methods. However, given knowledge of the second image to match, the tradeoff between invariance and discriminability of features in the first image is less ambiguous. This suggests to setup the problem of feature extraction and matching as a joint estimation problem. We develop a possible mathematical framework, a possible computational algorithm, and we give example demonstration on finding correspondence on images related by a scene that undergoes large 3D deformation of non-planar objects and camera viewpoint change

Accelerated PDE's for efficient solution of regularized inversion problems

We further develop a new framework, called PDE Acceleration, by applying it to calculus of variations problems defined for general functions on $\mathbb{R}^n$, obtaining efficient numerical algorithms to solve the resulting class of optimization problems based on simple discretizations of their corresponding accelerated PDE's. While the resulting family of PDE's and numerical schemes are quite general, we give special attention to their application for regularized inversion problems, with particular illustrative examples on some popular image processing applications. The method is a generalization of momentum, or accelerated, gradient descent to the PDE setting. For elliptic problems, the descent equations are a nonlinear damped wave equation, instead of a diffusion equation, and the acceleration is realized as an improvement in the CFL condition from $\Delta t\sim \Delta x^{2}$ (for diffusion) to $\Delta t\sim \Delta x$ (for wave equations). We work out several explicit as well as a semi-implicit numerical schemes, together with their necessary stability constraints, and include recursive update formulations which allow minimal-effort adaptation of existing gradient descent PDE codes into the accelerated PDE framework. We explore these schemes more carefully for a broad class of regularized inversion applications, with special attention to quadratic, Beltrami, and Total Variation regularization, where the accelerated PDE takes the form of a nonlinear wave equation. Experimental examples demonstrate the application of these schemes for image denoising, deblurring, and inpainting, including comparisons against Primal Dual, Split Bregman, and ADMM algorithms

Coarse-to-Fine Segmentation With Shape-Tailored Scale Spaces

We formulate a general energy and method for segmentation that is designed to have preference for segmenting the coarse structure over the fine structure of the data, without smoothing across boundaries of regions. The energy is formulated by considering data terms at a continuum of scales from the scale space computed from the Heat Equation within regions, and integrating these terms over all time. We show that the energy may be approximately optimized without solving for the entire scale space, but rather solving time-independent linear equations at the native scale of the image, making the method computationally feasible. We provide a multi-region scheme, and apply our method to motion segmentation. Experiments on a benchmark dataset shows that our method is less sensitive to clutter or other undesirable fine-scale structure, and leads to better performance in motion segmentation

Shape Tracking With Occlusions via Coarse-To-Fine Region-Based Sobolev Descent

We present a method to track the precise shape of an object in video based on new modeling and optimization on a new Riemannian manifold of parameterized regions. Joint dynamic shape and appearance models, in which a template of the object is propagated to match the object shape and radiance in the next frame, are advantageous over methods employing global image statistics in cases of complex object radiance and cluttered background. In cases of 3D object motion and viewpoint change, self-occlusions and dis-occlusions of the object are prominent, and current methods employing joint shape and appearance models are unable to adapt to new shape and appearance information, leading to inaccurate shape detection. In this work, we model self-occlusions and dis-occlusions in a joint shape and appearance tracking framework. Self-occlusions and the warp to propagate the template are coupled, thus a joint problem is formulated. We derive a coarse-to-fine optimization scheme, advantageous in object tracking, that initially perturbs the template by coarse perturbations before transitioning to finer-scale perturbations, traversing all scales, seamlessly and automatically. The scheme is a gradient descent on a novel infinite-dimensional Riemannian manifold that we introduce. The manifold consists of planar parameterized regions, and the metric that we introduce is a novel Sobolev-type metric defined on infinitesimal vector fields on regions. The metric has the property of resulting in a gradient descent that automatically favors coarse-scale deformations (when they reduce the energy) before moving to finer-scale deformations. Experiments on video exhibiting occlusion/dis-occlusion, complex radiance and background show that occlusion/dis-occlusion modeling leads to superior shape accuracy compared to recent methods employing joint shape/appearance models or employing global statistics.Comment: Extension of ICCV paper, added coarse-to-fine optimization based on new Riemannian manifold of parameterized region

Minimum Delay Object Detection From Video

We consider the problem of detecting objects, as they come into view, from videos in an online fashion. We provide the first real-time solution that is guaranteed to minimize the delay, i.e., the time between when the object comes in view and the declared detection time, subject to acceptable levels of detection accuracy. The method leverages modern CNN-based object detectors that operate on a single frame, to aggregate detection results over frames to provide reliable detection at a rate, specified by the user, in guaranteed minimal delay. To do this, we formulate the problem as a Quickest Detection problem, which provides the aforementioned guarantees. We derive our algorithms from this theory. We show in experiments, that with an overhead of just 50 fps, we can increase the number of correct detections and decrease the overall computational cost compared to running a modern single-frame detector.Comment: ICCV 201

Optical, Structural and Elemental Analysis of New Nonlinear Organic Single Crystal of Urea L-Asparagine

Single crystal of urea L-asparagine was synthesized and crystallized by slow evaporation solution growth \ud method at room temperature. The bright, transparent and colourless crystal was obtained with average \ud dimension of 11 × 0.7 × 0.3 cm3\ud . The elemental analysis of the compound confirms the stoichiometric ratio of \ud the compound. The sharp and well defined Bragg peaks obtained in the powder X-ray diffraction pattern \ud confirm the crystalline nature of the compound. The optical property of the compound was ascertained through \ud UV-visible spectral analysis. The various characteristics absorption bands in the compound were assigned \ud through fourier transform infra-red (FTIR) spectroscopy. The single crystal unit cell parameters of the \ud compound show that the grown crystal belongs to orthorhombic system with space group P. The nonlinear \ud optical property study indicates that the compound has SHG efficiency 0.5 times greater than that of standard \ud potassium dihydrogen phosphate (KDP)