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Chapter 14: Problem 4

Bayesian face recognition Moghaddam, Jebara, and Pentland \((2000)\) compute separate covariance matrices \(\Sigma_{I}\) and \(\Sigma_{E}\) by looking at differences between all pairs of images. At run time, they select the nearest image to determine the facial identity. Does it make sense to estimate statistics for all pairs of images and use them for testing the distance to the nearest exemplar? Discuss whether this is statistically correct. How is the all-pair intrapersonal covariance matrix \(\mathbf{\Sigma}_{I}\) related to the within-class scatter matrix \(S_{\mathrm{W}}\) ? Does a similar relationship hold between \(\Sigma_{E}\) and \(S_{\mathrm{B}}\) ?

Short Answer

Expert verified
Yes, from a statistical point of view, it is logical to use all pairs of images to estimate statistics in Bayesian face recognition. While this could be computationally expensive, it efficiently accommodates for all possible variations. The covariance matrix \(\mathbf{\Sigma}_{I}\) is related to the within-class scatter matrix \(S_{\mathrm{W}}\) and a similar relationship exists between \(\Sigma_{E}\) and \(S_{\mathrm{B}}\), as both pairs of matrices measure variations within and between classes respectively.

Step by step solution

01

Understanding Bayesian Face Recognition

Bayesian face recognition is a method or a system which recognizes faces by applying Bayes theorem. The faces are first trained in a system and then identified using several image samples.
02

Analyzing the use of Covariance Matrices for Statistics Estimation

In Bayesian face recognition, two separate covariance matrices are calculated: intrapersonal \(\Sigma_{I}\) and extrapersonal \(\Sigma_{E}\). These matrices are computed by looking at differences between all pairs of images. To determine the facial identity, the image that is the closest is selected. Now, the question is if it statistically makes sense to estimate statistics for all pairs of images and use them for testing the distance to the nearest exemplar. In statistics, the covariance matrix is a measure of spread or dispersion. Therefore, it is a plausible method because it computes the relationship between each pair of faces considering all the different variations and possibilities. Yet, this method could be computationally expensive due to the calculation for every possible pair.
03

Relationship between \(\mathbf{\Sigma}_{I}\) & \(S_{\mathrm{W}}\)

The all-pair intrapersonal covariance matrix \(\mathbf{\Sigma}_{I}\) is related to the within-class scatter matrix \(S_{\mathrm{W}}\) as both matrices quantify the total variation within the same class of elements. Hence, they are usually proportional to each other.
04

Relationship between \(\Sigma_{E}\) & \(S_{\mathrm{B}}\)

A similar relationship exists between \(\Sigma_{E}\) and \(S_{\mathrm{B}}\). The covariance matrix \(\Sigma_{E}\) quantifies the variation between elements of different classes, and \(S_{\mathrm{B}}\) measures the between-class scatter. Therefore, they are proportional to each other in most cases.

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Most popular questions from this chapter

Bag of words Adapt the feature extraction and matching pipeline developed in Exercise \(14.8\) to category (class) recognition, using some of the techniques described in Section 14.4.1. 1\. Download the training and test images from one or more of the databases listed in Tables 14.1 and 14.2, e.g., Caltech 101, Caltech 256, or PASCAL VOC. 2\. Extract features from each of the training images, quantize them, and compute the \(t f\)-idf vectors (bag of words histograms). 3\. As an option, consider not quantizing the features and using pyramid matching (14.4014.41) (Grauman and Darrell \(2007 \mathrm{~b}\) ) or using a spatial pyramid for greater selectivity (Lazebnik, Schmid, and Ponce 2006). 4\. Choose a classification algorithm (e.g., nearest neighbor classification or support vector machine) and "train" your recognizer, i.e., build up the appropriate data structures (e.g., \(\mathrm{k}\)-d trees) or set the appropriate classifier parameters. 5\. Test your algorithm on the test data set using the same pipeline you developed in steps \(2-4\) and compare your results to the best reported results. 6\. Explain why your results differ from the previously reported ones and give some ideas for how you could improve your system. You can find a good synopsis of the best-performing classification algorithms and their approaches in the report of the PASCAL Visual Object Classes Challenge found on their Web site (http://pascallin.ecs.soton.ac.uk/challenges/VOC/).

Face recognition using eigenfaces Collect a set of facial photographs and then build a recognition system to re-recognize the same people.1. Take several photos of each of your classmates and store them. 2\. Align the images by automatically or manually detecting the corners of the eyes and using a similarity transform to stretch and rotate each image to a canonical position. 3\. Compute the average image and a PCA subspace for the face images 4\. Take a new set of photographs a week later and use them as your test set. 5\. Compare each new image to each database image and select the nearest one as the recognized identity. Verify that the distance in PCA space is close to the distance computed with a full SSD (sum of squared difference) measure. 6\. (Optional) Compute different principal components for identity and expression, and use them to improve your recognition results.

Determining the threshold for AdaBoost Given a set of function evaluations on the training examples \(x_{i}, f_{i}=f\left(x_{i}\right) \in \pm 1\), training labels \(y_{i} \in \pm 1\), and weights \(w_{i} \in(0,1)\), as explained in Algorithm 14.1, devise an efficient algorithm to find values of \(\theta\) and \(s=\pm 1\) that maximize $$ \sum_{i} w_{i} y_{i} h\left(s f_{i}, \theta\right) $$ where \(h(x, \theta)=\operatorname{sign}(x-\theta) .\)

: Face detection Build and test one of the face detectors presented in Section 14.1.1. 1\. Download one or more of the labeled face detection databases in Table \(14.2\). 2\. Generate your own negative examples by finding photographs that do not contain any people. 3\. Implement one of the following face detectors (or devise one of your own): \- boosting (Algorithm 14.1) based on simple area features, with an optional cascade of detectors (Viola and Jones 2004); \- PCA face subspace (Moghaddam and Pentland 1997); \- distances to clustered face and non-face prototypes, followed by a neural network (Sung and Poggio 1998) or SVM (Osuna, Freund, and Girosi 1997) classifier; \- a multi-resolution neural network trained directly on normalized gray-level patches (Rowley, Baluja, and Kanade 1998a). 4\. Test the performance of your detector on the database by evaluating the detector at every location in a sub-octave pyramid. Optionally retrain your detector on false positive examples you get on non-face images.
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