3D Pose-by-Detection of Vehicles via Discriminatively Reduced Ensembles of Correlation Filters: Supplementary Material

In this supplementary material we show a full derivation of our Reduced Ensemble of Correlation Filters and provide more quantitative results that are not shown in the paper due to space constraints. 1 Ensemble of Exemplar Classifiers for Pose-by-Detection 1.1 Exemplar Correlation Filters Exemplar classifiers are suited to the task of pose-by-detection. For each one of the V viewpoint renders we train an Exemplar Correlation Filter (ECF) using the rendered image as the single positive, and N− 1 image patches selected randomly from a background set of images that do not contain the object instance. Each ECF is trained to detect the object from a specific viewpoint. Let {xi}i=1 be a set of Histogram of Oriented Gradients (HOG) representations of the training examples, consisting of one positive exemplar rendering of the v-th view and N−1 negative bounding boxes. Also, define { gv , · · · ,gv } as the ECF for a viewpoint v, where C is the number of channels of the HOG feature representation (commonly 32). The response of an image xi to the filter is defined as C ∑ c=1 xi ⊗gv = Correlation Output, (1) where ⊗ denotes the 2D convolution operator. The ECF design is posed as: min gv ,··· ,gv N ∑ i=1 ∥∥∥∥ C ∑ c=1 xi ⊗gv− ri ∥∥∥∥ 2 2 +λ C ∑ c=1 ‖gv‖ 2 2 , (2) c © 2014. The copyright of this document resides with its authors. It may be distributed unchanged freely in print or electronic forms. 2 MOVSHOVITZ-ATTIAS, BODDETI, WEI, SHEIKH: 3D POSE-BY-DETECTION where ri is the matrix holding the desired correlation output of the i-th training image, and λ moderates the degree of regularization. The desired correlation output ri is set to a positively scaled Gaussian for the positive exemplar and to a negatively scaled Gaussian for the negative patches. This choice of the desired output correlation shape also implicitly calibrates the different exemplar classifiers. The minimization problem can be equivalently posed in the frequency domain to derive a closed form expression, which in turn lends itself to an efficient solution [1]. It should be noted that, as a complete set, each view v ∈ V is trained independently, and that increase in the desired precision d increases the size of the ensemble (linearly for one axis of rotation, quadratically for two, and cubically for all three). Figure 1 (A) shows the training configuration for one exemplar correlation filter. For visualization clarity we do not show negative images. 1.2 Discriminative Reduction of Ensembles of Correlation Filters The procedure described in Section 1 produces a large set of exemplar classifiers, one per view that needs to be resolved. Let G ∈ RD×V be the matrix of all V filters arranged as column vectors, where D is the dimensionality of the feature. This set is an exhaustive representation of the object’s appearance from many views, but applying all the filters during test time is computationally expensive. It is also highly redundant as many views of the object are similar in appearance. Our reduced Ensemble of Exemplar Correlation Filter (EECF) approach is designed to jointly learn a set of K exemplar correlation filters F = [f1, . . . , fK ] (each with C channels) and a set of V sparse coefficient vectors A = [α1, . . . ,αV ] such that a detector gv for any viewpoint v of the object is defined by