Efficient classification using the Euler characteristic

This paper introduces an object descriptor for classification based on the Euler characteristic of subsets created by thresholding a function at multiple levels (sub-level filtration). We demonstrate the effectiveness of this basic topological invariant of sets, the Euler characteristic, and use it to compute descriptors in two different domains – images and 3D mesh surfaces. The descriptors used as input to linear SVMs achieve state of the art classification results on various public data sets. Moreover, these descriptors are extremely fast to compute. We present linear time methods to calculate the Euler characteristic for multiple threshold values and to compute the Euler characteristic in a sliding window. Supervised object classification entails two main elements – features (descriptors) and learning algorithms. In recent years, much effort has been invested in developing features that yield good classification. Different features are developed for different domains such as images and 3D objects. Some features are even object specific, e.g. faces or texture. For good classification, features should be rich, descriptive and discriminative, and on the other hand, invariant to different transformations and robust enough to allow intra-class variation. The focus in recent years shifted from global features that describe the object as a whole, to statistical descriptors of local features. The statistical descriptor of low-dimensional local features is simply their distribution (e.g. color histogram). For more complex local features (e.g. SIFT), a Bag-of-Words (BoW) scheme is used. The main criticism for statistical descriptors of local features is the loss of all spatial information: ''because these methods disregard all information about the spatial layout of the features, they have severely limited descriptive ability'' [14]. For example, in one object class, the different values of the local feature might be evenly distributed, and in another class, the different values are clustered. Several approaches for putting the local features into some spatial (or spatio-temporal) context have been suggested. Some examples are spatial pyramids [14] and hierarchical neighborhood features [12]. This paper presents a new descriptor for supervised classification , which is based on simple local features, but instead of using their distribution, we propose to threshold the feature at multiple levels and calculate the Euler characteristic (EC) values of the resulting subsets of the domain. The vector of Euler Numbers – the Euler Characteristic Graph (or EC Graph) is then used as an object descriptor. The EC is a global topological invariant which in the case of …