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The examples below show that one cannot distinguish a smooth curve from one with corners or from a set of differently oriented lines in an arbitrary configuration based only on histograms of oriented gradients. 
In our earlier work [1] we extended the modeling of object boundary contours beyond the widely used edge orientation histograms by utilizing curvature information to overcome the drawbacks of straightline approximations. However, the cooccurrence of contour curvature can provide even more information about the object boundary and thus increases the distinctiveness (as shown in the examples below). By computing cooccurrences between discriminatively curved boundaries we build a curvature selfsimilarity descriptor that provides a more detailed and accurate object description. 
However, like all descriptors using second order statistics, ours also exhibits a high dimensionality. Although improving discriminability, the high dimensionality becomes a critical issue due to lack of generalization ability and curse of dimensionality. Given only a limited amount of training data, even sophisticated learning alorithms such as the popular kernel methods are not able to suppress noisy or superfluous dimensions of such highdimensional data. Consequently, there is a natural need for feature selection when using presentday informative features and, particularly, curvature selfsimilarity. We therefore suggest an embedded feature selection method for SVMs that reduces complexity and improves generalization capability of object models. 
Method 
Embedded Feature Selection for SVM 
Feature selection methods can be categorized into filters, wrappers and embedded methods. Contrary to filters and wrappers embedded feature selection methods incorporate feature selection as a part of the learning process. The focus of this paper is on embedded feature selection methods for SVMs, since most stateoftheart detection systems use SVM as a classifier. To directly integrate feature selection into the learning process of SVMs sparsity can be enforced on the model parameter . Contrary to linear SVM enforcing sparsity on the model parameter does reduce dimensionality for nonlinear kernel functions in the higher dimensional kernel space rather than in the number of input features. In our work we are following the concept of embedded feature selection and therefore include the feature selection parameter directly in the SVM classifier, where are the training data with labels . The correspondind optimization problem can be expressed in the following way: 
where is the SVM kernel function. The function is typically unknown and represents the mapping of the feature vector into a higher dimensional space. We enforce sparsity of the feature selection parameter by the last constraint, which restricts the L1norm of by a constant . Due to the complexity of this problem we propose to solve two simpler problems iteratively. First we split the training data into a training and validation set. Now we optimize the problem according to and for fixed selection parameter using a standard SVM algorithm on the training set. This Parameter is optimized in a second optimization step on the validation to reduce dimensionality of the feature vector and improve classification performance. 
Representing Curvature SelfSimilarity 
We propose a new holistic selfsimilarity representation based on curvature. As a first step we extract the curvature information and the corresponding 360 degree orientation of all edge pixels in the image. To estimate the curvature we use the distance accumulation method of Han et al. [4], which accurately approximates the curvedness along given 2D line segments. Let be a set of consecutive boundary points, representing one line segment. The perpendicular distance is computed from a chord to the point , using the euclidean distance. The distance accumulation for point is the sum . The distance is positive if is on the lefthand side of the chord, and negative otherwise. The describe procedure is summerized in the figure below. 
To describe complex objects it is not sufficient to build a selfsimilarity descriptor soley based on curvature information, since selfsimilarity of curvature leaves open many ambiguities. To resolve these ambiguities we add 360 degree orientation information to get a more accurate descriptor. To get the 360 degree orientation information we compute the gradient of the edge image and extend the resulting 180 degree gradient orientation to a 360 degree orientation using the sign of the curvature. Next we divide the image into nonoverlapping pixel cells and build histograms over the unsigned curvature values and the 360 degree orientation in each cell. The two histograms are concatenated in each cell and all pairwise similarities between the cells are computed using histogram intersection as a comparison measure. The figure below visualizes the described procedure. 
The examples below show the original images along with their curvature selfsimilarity matrices displaying the similarity between all pairs of curvature histogram cells. While curvature selfsimilarity descriptor is similar for the same object category it looks quite different to other object categories. 
Results 
We evaluate our curvature selfsimilarity descriptor in combination with the suggested embedded
dimensionality reduction algorithm for the object detection task on the PASCAL dataset. To
show the individual strengths of these two contributions we need to perform a number of evaluations.
Since this is not supported by the PASCAL VOC 2011 evaluation server we follow the best practice
guidelines and use the VOC 2007 dataset. Our experiments show, that curvature selfsimilarity
is providing complementary information to straight lines, while our feature selection algorithm is
further improving performance by fulfilling its natural need for dimensionality reduction. The common basic concept shared by many current detection systems are highdimensional, holistic representations learned with a discriminative classifier, mostly an SVM. In particular the combination of HOG and SVM constitutes the basis of many powerful recognition systems and it has laid the foundation for numerous extensions like, part based models, variations of the SVM classifier and approaches utilizing context information. These systems rely on highdimensional holistic image statistics primarily utilizing straight line approximations. In this paper we explore a orthogonal direction to these extensions and focus on how one can improve on the basic system by extending the straight line representation of HOG to a more discriminative description using curvature selfsimilarity. At the same time our aim is to reduce the dimensionality of such highdimensional representations to decrease the complexity of the learning procedure and to improve generalization performance. We provide a structured evaluation of the parts of our final object detection system. We use the HOG of Felzenswalb et al. [2,3] as baseline system, since it is the basis fo many powerful object detection systems. The table below shows the detection performance in terms of average precision of the HOG baseline system, HOG and curvature (Curv) before and afer discarding noisy dimensions using our feature selection method (FS) and our final detection system consisting of HOG, curvature (Curv), the suggested curvature selfsimilarity (CurvSS) with and without feature selection (FS) on the PASCAL VOC 2007 dataset. Note, that we use all data points to compute the average precision as it is specified by the default experimental protocol since VOC 2010 development kit. This yields lower but more accurate average precision measurements. 
PublicationsAngela Eigenstetter and Björn Ommer,Visual Recognition using Embedded Feature Selection for curvature SelfSimilarity, in: NIPS'12, (accepted, in press). References

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Last update: 25.03.2015, 14:05 