Computer Vision: Models, Learning, and Inference
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This modern treatment of computer vision focuses on learning and inference in probabilistic models as a unifying theme. It shows how to use training data to learn the relationships between the observed image data and the aspects of the world that we wish to estimate, such as the 3D structure or the object class, and how to exploit these relationships to make new inferences about the world from new image data. With minimal prerequisites, the book starts from the basics of probability and model fitting and works up to real examples that the reader can implement and modify to build useful vision systems. Primarily meant for advanced undergraduate and graduate students, the detailed methodological presentation will also be useful for practitioners of computer vision. - Covers cutting-edge techniques, including graph cuts, machine learning, and multiple view geometry. - A unified approach shows the common basis for solutions of important computer vision problems, such as camera calibration, face recognition, and object tracking. - More than 70 algorithms are described in sufficient detail to implement. - More than 350 full-color illustrations amplify the text. - The treatment is self-contained, including all of the background mathematics. - Additional resources at www.computervisionmodels.com.
, Σk ] . (7.14) k=1 Unfortunately, if we take the derivative with respect to the parameters θ and equate the resulting expression to zero, it is not possible to solve the resulting equations in closed form. The sticking point is the summation inside the logarithm, which precludes a simple Figure 7.6 Mixture of Gaussians model in 1D. A complex multimodal probability density function (black solid curve) is created by taking a weighted sum or mixture of several constituent normal distributions.
Regression which we deal with in subsequent sections. The linear regression model with maximum likelihood learning is overconfident, and hence we develop a Bayesian version. It is unrealistic to always assume a linear relationship between the data and the world and to this end, we introduce a nonlinear version. The linear regression model has many parameters when the data dimension is high, and hence we consider a sparse version of the model. The ideas of Bayesian estimation, nonlinear functions.
Similar to Equation 8.52 except that instead of every datapoint having the same prior variance σp2 , they now have individual variances that are determined by the hidden variables that form the elements of the diagonal matrix H. In relevance vector regression, we alternately (i) optimize the marginal likelihood with respect to the hidden variables and (ii) optimize the marginal likelihood with respect to the variance parameter σ 2 using hnew = i 1 − hi Σii + ν µ2i + ν (8.55) and (σ 2 )new =.
Calculated the function for each sample, and took the average of these values, the result would be the expectation. More precisely, the expected value of a function f [•] of a random variable x is defined as E[f [x]] = f [x]P r(x) x E[f [x]] = f [x]P r(x) dx, (2.12) Discussion 15 Function f [•] x xk (x − µx )k (x − µx )2 (x − µx )3 (x − µx )4 (x − µx )(y − µy ) Expectation mean, µx th k moment about zero k th moment about the mean variance skew kurtosis covariance of x and y Table 2.1.
Maximize the log likelihood by empirically trying a number of different threshold values and choosing the one that gives the best result. We then perform this same procedure recursively; the data that pass to the left branch have a new randomly chosen classifier applied to them, and a new threshold that splits it again is chosen. This can be done without recourse to the data in the right branch. When we classify a new data example x∗ , we pass it down the tree until it reaches one of the 9.11.