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.

In image segmentation, finite mixture modeling has been widely used. In its simplest form, the spatial correlation among neighboring pixels is not taken into account, and its segmentation results can be largely deteriorated by noise in images. We propose a spatially correlated mixture model in which the mixing proportions of finite mixture models are governed by a set of underlying functions defined on the image space. The spatial correlation among pixels is introduced by putting a Gaussian process prior on the underlying functions. We can set the spatial correlation rather directly and flexibly by choosing the covariance function of the Gaussian process prior. The effectiveness of our model is demonstrated by experiments with synthetic and real images.

The mixture modeling framework is widely used in many applications. In this paper, we propose a component reduction technique, that collapses a Gaussian mixture model into a Gaussian mixture with fewer components. The EM (Expectation-Maximization) algorithm is usually used to fit a mixture model to data. Our algorithm is derived by extending mixture model learning using the EM-algorithm. In this extension, a difficulty arises from the fact that some crucial quantities cannot be evaluated analytically. We overcome this difficulty by introducing an effective approximation. The effectiveness of our algorithm is demonstrated by applying it to a simple synthetic component reduction task and a phoneme clustering problem.

One of the advantages of the kernel methods is that they can deal with various kinds of objects, not necessarily vectorial data with a fixed number of attributes. In this paper, we develop kernels for time series data using dynamic time warping (DTW) distances. Since DTW distances are pseudo distances that do not satisfy the triangle inequality, a kernel matrix based on them is not positive semidefinite, in general. We use semidefinite programming (SDP) to guarantee the positive definiteness of a kernel matrix. We present neighborhood preserving embedding (NPE), an SDP formulation to obtain a kernel matrix that best preserves the local geometry of time series data. We also present an out-of-sample extension (OSE) for NPE. We use two applications, time series classification and time series embedding for similarity search, to validate our approach.

We consider an adaptive PCM system, in which the input samples are expanded or compressed by a constant factor c or 1/c each time before quantization. Assuming a stationary Gaussian input with a rational power spectral density, we derive an integral equation for the joint distribution of the input and the state of the system. Its solution provides us with a feasible way to numerical computation. The mean-squared error are computed for the Gauss-Markov input in terms of the constant c, the sampling interval T, bound parameters for the scaler and the number of quantization levels. The numerical results show good performance in comparison with regular PCM.