Gerry Melnikov

A review of the minimum maximum criterion for optimal bit allocation among dependent quantizers

In this paper, we review a general framework for the optimal bit allocation among dependent quantizers based on the minimum maximum (MINMAX) distortion criterion. The pros and cons of this optimization criterion are discussed and compared to the well-known Lagrange multiplier method for the minimum average (MINAVE) distortion criterion. We argue that, in many applications, the MINMAX criterion is more appropriate than the more popular MINAVE criterion. We discuss the algorithms for solving the optimal bit allocation problem among dependent quantizers for both criteria and highlight the similarities and differences. We point out that any problem which can be solved with the MINAVE criterion can also be solved with the MINMAX criterion, since both approaches are based on the same assumptions. We discuss uniqueness of the MINMAX solution and the way both criteria can be applied simultaneously within the same optimization framework. Furthermore, we show how the discussed MINMAX approach can be directly extended to result in the lexicographically optimal solution. Finally, we apply the discussed MINMAX solution methods to still image compression, intermode frame compression of H.263, and shape coding applications.

Operationally optimal vertex-based shape coding

In this article, we address the issue of operationally optimal shape encoding, which is a step in the direction of globally optimal resource allocation in object-oriented video. After an overview of shape-based coding and algorithms, we define the problem mathematically, introduce the necessary notation, and then present the basic idea behind the proposed algorithms. We then discuss the constraints imposed on the code used to encode the approximation. We then introduce a definition of distortion that fits into the proposed framework and introduce the directed acyclic graph (DAG) formulation of the problem, which results in a fast solution approach. We also show how the DAG algorithm can be used to find the approximation with the minimum-maximum segment distortion for a given rate as well as to find the approximation with the smallest total distortion for a given rate. We then present experimental results and point out directions for future research.

Joint optimal object shape estimation and encoding

A major problem in object-oriented video coding and MPEG-4 is the encoding of object boundaries. Traditionally this problem is treated separately from the texture encoding problem. In this paper, we present a vertex-based shape coding method which is optimal in the operational rate-distortion sense and takes into account the texture information of the video frames. This is accomplished by utilizing a variable-width tolerance band whose width is a function of the texture profile. As an example, this width is inversely proportional to the magnitude of the image gradient. Thus, in areas where the confidence in the estimation of the boundary is low and/or coding errors in the boundary will not affect the application (e.g., object-oriented coding and MPEG-4) significantly, a larger boundary approximation error is allowed. We present experimental results which demonstrate the effectiveness of the proposed algorithm.

Simultaneous optimal boundary encoding and variable-length code selection

This paper describes efficient and optimal encoding and representation of object contours. Contours are approximated by connected second-order spline segments, each defined by three consecutive control points. The placement of the control points is done optimally in the rate-distortion (RD) sense and jointly with their entropy encoding. We utilize a differential scheme for the rate and an additive area-based metric for the distortion to formulate the problem as a Lagrangian minimization. We investigate the sensitivity of the resulting operational RD curve on the variable length codes used and propose an iterative procedure arriving at the entropy representation of the original boundary for any given rate-distortion tradeoff.

Shape coding using temporal correlation and joint VLC optimization

This paper investigates ways to explore the between frame correlation of shape information within the framework of an operationally rate-distortion (ORD) optimized coder. Contours are approximated both by connected second-order spline segments, each defined by three consecutive control points, and by segments of the motion-compensated reference contours. Consecutive control points are then encoded predictively using angle and run temporal contexts or by tracking the reference contour. We utilize a novel criterion for selecting global object motion vectors, which improves the efficiency. The problem is formulated as a Lagrangian minimization and solved using dynamic programming. Furthermore, we employ an iterative technique to remove dependency on a particular variable length code and jointly arrive at the ORD globally optimal solution and an optimized conditional parameter distribution.

A jointly optimal fractal/DCT compression scheme

In this paper a hybrid fractal and discrete cosine transform (DCT) coder is developed. Drawing on the ability of DCT to remove inter-pixel redundancies and on the ability of fractal transforms to capitalize on long-range correlations within the image, the hybrid coder performs an operationally optimal, in the rate-distortion sense, bit allocation among coding parameters. An orthogonal basis framework is used within which an image segmentation and a hybrid block-based transform are selected jointly. The selection of coefficients in the DCT component of the overall block transform is made a part of the optimization procedure. A Lagrangian multiplier approach is used to optimize the hybrid transform parameters together with the segmentation. Differential encoding of the DC coefficient is employed, with the scanning path based on a 3rd-order Hilbert curve. Simulation results show a significant improvement in quality with respect to the JPEG standard, an approach based on optimization of DCT basis vectors, as well as, the purely fractal techniques.

Jointly optimal coding of texture and shape

A major problem in object oriented video coding and MPEG-4 is the encoding of object boundaries. Traditionally, and within MPEG-4, the encoding of shape and texture information are separate steps (the extraction of shape is not considered by the standards). We present a vertex-based shape coding method which is optimal in the operational rate-distortion sense and takes into account the texture information of the video frames. This is accomplished by utilizing a variable-width tolerance band which is proportional to the degree of trust in the accuracy of the shape information at that location. Thus, in areas where the confidence in the estimation of the boundary is not high and/or coding errors in the boundary will not affect the application (object oriented coding, MPEG-4, etc.) significantly, a larger boundary approximation error is allowed. We present experimental results which demonstrate the effectiveness of the proposed algorithm.

Inter mode vertex-based optimal shape coding

This paper investigates the problem of optimal lossy encoding of object contours in the inter mode. Contours are approximated by connected second-order spline segments, each defined by three consecutive control points. Taking into account correlations in the temporal direction, control points are chosen optimally in the rate-distortion sense. Applying motion to contours in the reference frame followed by the temporal context extraction, we predict the next control point location, given the previously encoded one. Based on the chosen differential encoding scheme and an additive MPEG-4 based distortion metric, the problem is formulated as Lagrangian minimization. We utilize an iterative procedure to jointly find the optimal solution and the associated DPCM parameter probability mass functions.

Rate-distortion optimal boundary encoding using an area distortion measure

In this paper an optimal boundary encoding algorithm in the rate-distortion sense is proposed. Second-order B-spline curves are used to model object boundaries. An additive area distortion measure between the original boundary and its approximation is employed in the optimization process. The problem is formulated in a Directed Acyclic Graph (DAG) paradigm, and the shortest path solution is used to optimally select control point locations of the B-spline curve approximation based on the desired rate-distortion tradeoff.

A non uniform segmentation optimal hybrid fractal/DCT image compression algorithm

In this paper a hybrid fractal and discrete cosine transform (DCT) coder is developed. Drawing on the ability of DCT to remove inter-pixel redundancies and on the ability of fractal transforms to capitalize on long-range correlations in the image, the hybrid coder performs an optimal, in the rate-distortion sense, bit allocation among coding parameters. An orthogonal basis framework is used within which an image segmentation and a hybrid block-based transform are selected jointly. A Lagrangian multiplier approach is used to optimize the hybrid parameters and the segmentation. Differential encoding of the DC coefficient is employed, with the scanning path based on a 3rd-order Hilbert curve. Simulation results show a significant improvement in quality with respect to the JPEG standard.