Originally, mathematical morphology was a theory of signal transformations which are invariant under Euclidean translations. An interest in the extension of mathematical morphology to spatially-variant (SV) operators has emerged due to the requirements imposed by numerous applications in adaptive signal (image) processing. This
paper presents a general theory of spatially-variant mathematical morphology in the Euclidean space. We define the binary and gray-level spatially-variant basic morphological operators (i.e., erosion, dilation, opening and closing) and study their properties. We subsequently derive kernel representations for a large class of binary
and gray-level SV operators in terms of the basic SV morphological operators. The theory of SV mathematical morphology is used to extend and analyze two important image processing applications: morphological image restoration and skeleton representation of binary images. For morphological image restoration, we obtain new realizations of adaptive median filters in terms of the basic SV morphological operators. For skeleton representation, we develop an algorithm to construct the optimal structuring elements, in the sense of minimizing the cardinality of the spatially-variant morphological skeleton representation. Experimental results show the power of
the proposed theory of spatially-variant mathematical morphology in practical image processing applications.
We propose an optimal framework for active surface extraction from video sequences. An active surface is a collection of active contours in successive frames such that the active contours are constrained by spatial and temporal energy terms. The spatial energy terms impose constraints on the active contour in a given frame. The temporal energy terms relate the active contours in different frames to preserve desired internal and external properties of the active surface. For computational efficiency, we reduce the 3-D active surface ((x,y,t) coordinates) optimization problem to a 2-D model ((φ,t) coordinates) by considering only point indices along normal lines φ of each contour and define the energy terms in a causal way. We develop an n-D dynamic tree programming algorithm to find the optimum of n-D semi-causal functions. We prove that the n-D dynamic tree programming algorithm converges to the global optimum. In particular, the classical 1-D dynamic programming algorithm is a special case of the n-D dynamic tree programming algorithm. The optimal active surface is subsequently obtained by using the 2-D dynamic tree programming algorithm. Simulation results show the efficiency and robustness of the proposed approach in active surface extraction for video tracking and segmentation of the human head in real-world video sequences.
Recent advances in multimedia and communication require techniques for accurately tracking objects in video sequences. We propose a complete system for head tracking and contour refinement. Our tracking approach is based on particle filtering framework. However, unlike existing methods that use prior knowledge or likelihood functions as proposal densities, we use a motion-based proposal. Adaptive Block Matching (ABM) algorithm is the motion estimation technique used. Several advantages arise from this choice of proposal. (i) Only few samples are propagated. (ii) The tracking is adaptive to different categories of motion (iii) Off-line motion learning is not needed. Following the tracking is the contour refinement step. We want to transform the parametric estimate representing the tracked head at a given time instant into an elastic contour delineating the head’s boundaries. We use an active contour framework based on a dynamic programming scheme. However active contours are very sensitive to parameter assignment and initial condition. Using the tracked parametric estimate, we create a set of randomly perturbed initial conditions. The optimal contour is then the one corresponding to the lowest energy. Our system demonstrates tracking a person’s head in complex environments and delineates its boundaries for future use.