We will present the state of the art energy minimization algorithms that are used to perform inference in modern artificial vision models: that is, efficient methods for obtaining the most likely interpretation of a given visual input. We will also cover the popular max-margin framework for estimating the model parameters using inference.

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**Lecture 1: Introduction to artificial vision with discrete graphical models**: In this lecture, the interdisciplinary nature of computational vision is briefly introduced along with its potential use in different application domains. Subsequently, the concept of discrete modeling of artificial vision tasks is introduced from theoretical view point along with short examples demonstrating the interest of such an approach in low, mid and high-level vision. Examples refer to blind image deconvolution, knowledge-based image segmentation, optical flow, graph matching, 2d-to-3d view-point invariant detection and modeling and grammar-driven image based reconstruction.**Lecture 2: Reparameterization and dynamic programming**: In this lecture, we provide a brief introduction to undirected graphical models. We also provide a formal definition of the problem of inference (specifically, energy minimization). We introduce the concept of reparameterization, which forms the building block of all the inference algorithms discussed in the course. We describe a simple inference algorithm known as dynamic programming, which consists of a series of reparameterization. We show how dynamic programming can be used to perform exact inference on chains.

**Lecture 3:**In this lecture, we introduce the concept of functions on arcs of a directed graph. We focus on a special function known as the flow function. Associated with this function is the combinatorial optimization problem of computing the maximum flow of a directed graph. We also introduce the concept of a cut in a directed graph, and prove that the minimum cost cut is equivalent to the maximum flow. We describe a simple algorithm for solving the maximum flow, or equivalent the minimum cut, problem.**Maximum flow and minimum cut**:**Lecture 4:****Minimum cut based inference**: In this lecture, we show how the problem of inference for undirected graphical models with two labels can be formulated as a minimum cut problem. We characterize the energy function that can be minimized optimally using the minimum cut problem. We show examples using the image segmentation and texture synthesis problems, which can be formulated using two labels. We consider the multi-label problem, and devise approximate algorithms for inference based on the minimum cut algorithms. We show examples using the stereo reconstruction and the image denoising problems.

**Lecture 5: Belief propagation:**In this lecture we present the basic concepts of message passing and belief propagation networks. The concept is initially demonstrated using chains, extended to the case of trees and then eventually to arbitrary graphs. The strengths and the limitations of such an optimization framework are presented. The image completion and texture synthesis problems are considered as examples to demonstrate the interest of such a family of optimization algorithms.

**Lecture 6: Linear programing and duality:**In this lecture, discrete inference is addressed through concepts coming from linear programming relaxations. In particular, we explain how a graph-optimization problem can be expressed as a linear programing one and then how one can take benefit of the duality theorem to develop efficient optimization methods. The problem of optical flow and its deformable registration variant in medical image analysis is considered as an example to demonstrate the interest of such optimization algorithms.**Lecture 7: Dual decomposition and higher order graphs:**In this lecture, we introduce the dual decomposition framework for the optimization of low rank and higher order graphical models. First, we demonstrate the concept of the method using a simple toy example and then we extend to the most general optimization problem case. Three different examples are considered in the context of higher order optimization, the problem of linear mapping between images, the case of dense deformable graph matching and the development of pose invariant object segmentation methods in the context of medical imaging.

**Lecture 8: Parameter learning:**In this lecture, we introduce two frameworks for estimating the parameters of a graphical model using fully supervised training data. The first framework maximizes the likelihood of the training data while regularizing the parameters. The second framework minimizes the empirical risk, as measured by a user-defined loss function, while regularizing the parameters. We provide a brief description of the algorithms required to solve the related optimization problems. We show the results obtained on standard machine learning datasets.

- Pawan Kumar
- Nikos Paragios - Applied Mathematics

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