A two-dimensional shape is generally represented with line drawings or object contours in a digital image. Shapes can be divided into two types, namely ordered and unordered shapes. An ordered shape is an ordered set of points, while an unordered shape is an unordered set. As a result, each type typically uses different attributes to define the local descriptors involved in representing the local distributions of points sampled from the shape. Throughout this paper, we focus on unordered shapes. Since most local descriptors of unordered shapes are not scale-invariant, we usually make the shapes in an image data set the same size through scale normalization, before applying shape matching procedures. Shapes obtained through scale normalization are suitable for such descriptors if the original whole shapes are similar. However, they are not suitable if parts of each original shape are drawn using different scales. Thus, in this paper, we present a scale-invariant descriptor constructed by von Mises distributions to deal with such shapes. Since this descriptor has the merits of being both scale-invariant and a probability distribution, it does not require scale normalization and can employ an arbitrary measure of probability distributions in matching shape points. In experiments on shape matching and retrieval, we show the effectiveness of our descriptor, compared to several conventional descriptors.

Scale-invariant features are widely used for image retrieval and shape classification. The curvature of a planar curve is a fundamental feature and it is geometrically invariant with respect it the coordinate system. The curvature-based feature varies in position when multi-scale analysis is performed. Therefore, it is important to recognize the scale in order to detect the feature point. Numerous shape descriptors based on contour shapes have been developed in the field of pattern recognition and computer vision. A curvature scale-space (CSS) representation cannot be applied to a contour fragment and requires the tracking of feature points. In a gradient-based curvature computation, although the gradient computation considers the scale, the curvature is normalized with respect to not the scale but the contour length. The scale-invariant feature transform algorithm that detects feature points from an image solves similar problems by using the difference of Gaussian (DoG). It is difficult to apply the SIFT algorithm to a planar curve for feature extraction. In this paper, an automatic scale detection method for a contour fragment is proposed. The proposed method detects the appropriate scales and their positions on the basis of the difference of curvature (DoC) without the tracking of feature points. To calculate the differences, scale-normalized curvature is introduced. An advantage of the DoC algorithm is that the appropriate scale can be obtained from a contour fragment as a local feature. It then extends the application area. The validity of the proposed method is confirmed by experiments. The proposed method provides the most stable and robust scales of feature points among conventional methods such as curvature scale-space and gradient-based curvature.

This paper presents a scale invariant face detection and classification method which uses shift invariant features extracted from a Log-Polar image. Scale changes of a face in an image are represented as shift along the horizontal axis in the Log-Polar image. In order to obtain scale invariant features, shift invariant features are extracted from each row of the Log-Polar image. Autocorrelations, Fourier spectrum, and PARCOR coefficients are used as shift invariant features. These features are then combined with simple classification methods based on Linear Discriminant Analysis to realize scale invariant face detection and classification. The effectiveness of the proposed face detection method is confirmed by experiments using face images captured under different scales, backgrounds, illuminations, and dates. To evaluate the proposed face classification method, we performed experiments using 2,800 face images with 7 scales under 2 different backgrounds and face images of 52 persons.