Object tracking is a challenging problems in real-time computer vision due to variations of lighting condition, pose, scale, and view-point over time. However, it is exceptionally difficult to model appearance with respect to all of those variations in advance; instead, on-line update algorithms are employed to adapt to these changes. We present a new on-line appearance modeling technique which is based on sequential density approximation. This technique provides accurate and compact representations using Gaussian mixtures, in which the number of Gaussians is automatically determined. This procedure is performed in linear time at each time step, which we prove by amortized analysis. Target model is based on the probabilistic template, and color and rectangular feature for each pixel are modeled together by the proposed sequential density approximation algorithm, and the target model is updated in scale robustly.


Figure 1: Tracking results of football sequence for severe pose variations and fast motion (a)-(d) results in each time step (e) target appearance model (f) number of components in each pixel (blue) and variation (black)

Figure 2: Tracking results of car1 sequence for severe scale change (a)-(c) results in each time step (d) target appearance model

Kernel-based Bayesian filtering is motivated to ameliorate typical problems in conventional particle filtering such as degeneracy or loss of diversity, especially in high dimensional cases. Every relevant density function is Gaussian mixture, and the analytical representation of density overcome the problems of particle filters and ccontribute to reducing the number of samples.

This framework is applied to build general object tracking algorithm, our method shows better performance than the SIR particle filtering. The following figure shows the comparative results between our kernel-based Bayesian filtering and conventional particle filtering. The state space is 8-dimensional -- two rectangles, 4D for each rectangle, and same process/measurement model is used. The number of samples (=50) is also same, and according to our experiment, one would need to run the conventional particle filter using about 200 particles to obtain a comparable result with our method using 50 particles.


(a) results by kernel-based Bayesian filtering

(b) results by conventional particle filtering
Figure 3: Tracking results of person sequence at t=1,94, 140, 192, 236, 300.

We consider modeling the background based on intensity or color at each pixel by kernel density approximation. The first step for background subtraction is to estimate the feature density of each pixel with training sequence; then, the background model of a pixel is composed of a mixture of Gaussians through kernel density approximation. The initial covariance associated with a each kernel has been computed using the median of the magnitude of differences between consecutive measurements. Once the initial background model is constructed, background subtraction is performed for the individual pixel. We consider a pixel as foreground if the feature of the unit is far (e.g. outside 99.9% confidence ellipsoid) from every mode in the underlying density function. This decision making process is much simpler than kernel density estimation since we maintain only a small number of Gaussians in each pixel. While performing the background subtraction, we update the model by sequential kernel density approximation. Note that the density function are updated selectively only if the new data is classified as background.


(a)

(b)
Figure 4: Background subtraction results of (a) subway and (b) water sequence. (left) original image (middle) intensity modeling (1D) (right) color modeling (3D)

Various clustering methods have been proposed for the analysis of gene expression data, but conventional clustering algorithms have several critical limitations; how to set parameters such as number of clusters, initial cluster centers, and so on. We propose a semi-parametric model-based clustering algorithm in which the underlying model is a mixture of Gaussian. Each gene expression data builds a Gaussian kernel, and the uncertainty of microarray data is naturally integrated in the data representation. Our algorithm provides a principled method to automatically determine parameters -- number of components in the mixture, mean, covariance, and weight of each Gaussian -- by mean-shift procedure and curvature fitting. After the initialization, Expectation Maximization (EM) algorithm is employed for clustering to achieve Maximum Likelihood (ML).

When I was in Seoul National University, I studied genome rearrangement problems for my MS thesis. This problem is about how one sequence can be transformed into another by several operations such as reversal, translocation, and transposition, and is very important since it can be used to evaluate the relationship between two organisms and find evolutionary phylogeny. We addressed a simple algorithm to empirically improve the performance of sorting by reversal and transposition.


(a) DNA Microarray clustering

(b) Genome arrangement problem
Figure 5: Bioinformatics applications