A simple mistake in a formula given in the original paper [1] resulted in errors associated with the results given in that paper; the corrections to that formula and the subsequent results are presented below. The given impulsive eigenfunctions still defined over the same manifolds as in the original paper: the coordinate axes of ${\mathbb {R}}^{N}$ are arbitrarily partitioned into two sets; each set defines a hyperplane passing through the origin and these two hyperplanes are orthogonal.

The coordinate axes of RNare arbitrarily partitioned into two sets; each set defines a hyperplane passing through the origin and these two hyperplanes are orthogonal. After a review of impulse functions over such hyperplanes and their Fourier transforms, it is shown that an impulse function over the union of these two hyperplanes is an eigenfunction of the N-dimensional Fourier transform. Furthermore, based on the simple rotation property of the Fourier transform, it is also shown that impulse functions over unions of finite number of arbitrarily rotated versions of those two hyperplane sets are also eigenfunctions of the N-dimensional Fourier transform.

Holography aims to record and regenerate volume filling light fields to reproduce ghost-like 3-D images that are optically indistinguishable from their physical 3-D originals. Digital holographic video displays are pixelated devices on which digital holograms can be written at video rates. Spatial light modulators (SLMs) are used for such purposes in practice; even though it is desirable to have SLMs that can modulate both the phase and amplitude of the incident light at each pixel, usually amplitude-only or phase-only SLMs are available. Many laboratories have reported working prototypes using different designs. Size and resolution of the SLMs are quite demanding for satisfactory 3-D reconstructions. Space–bandwidth product (SBP) seems like a good metric for quality analysis. Even though moderate SBP is satisfactory for a stationary observer with no lateral or rotational motion, the required SBP quickly increases when such motion is allowed. Multi-SLM designs, especially over curved surfaces, relieve high bandwidth requirements, and therefore, are strong candidates for futuristic holographic video displays. Holograms are quite robust to noise and quantization. It is demonstrated that either laser or light-emitting diode (LED) illumination is feasible. Current research momentum is increasing with many exciting and encouraging results.

Although the impulse (Dirac delta) function has been widely used as a tool in signal processing, its more complicated counterpart, the impulse function over higher dimensional manifolds in RopfN, did not get such a widespread utilization. Based on carefully made definitions of such functions, it is shown that many higher dimensional signal processing problems can be better formulated, yielding more insight and flexibility, using these tools. The well-known projection-slice theorem is revisited using these impulse functions. As a demonstration of the utility of the projection-slice formulation using impulse functions over hyperplanes, the scalar optical diffraction is reformulated in a more general context

Diffraction and holography are fertile areas for application of signal theory and processing. Recent work on 3DTV displays has posed particularly challenging signal processing problems. Various procedures to compute Rayleigh-Sommerfeld, Fresnel and Fraunhofer diffraction exist in the literature. Diffraction between parallel planes and tilted planes can be efficiently computed. Discretization and quantization of diffraction fields yield interesting theoretical and practical results, and allow efficient schemes compared to commonly used Nyquist sampling. The literature on computer-generated holography provides a good resource for holographic 3DTV related issues. Fast algorithms to compute Fourier, Walsh-Hadamard, fractional Fourier, linear canonical, Fresnel, and wavelet transforms, as well as optimization-based techniques such as best orthogonal basis, matching pursuit, basis pursuit etc., are especially relevant signal processing techniques for wave propagation, diffraction, holography, and related problems. Atomic decompositions, multiresolution techniques, Gabor functions, and Wigner distributions are among the signal processing techniques which have or may be applied to problems in optics. Research aimed at solving such problems at the intersection of wave optics and signal processing promises not only to facilitate the development of 3DTV systems, but also to contribute to fundamental advances in optics and signal processing theory.

The set of six papers that we invited to this part of the Special Section present extensive reviews of the state-of-the-art in functional building blocks of 3DTV systems.

Two important signal processing problems in the display side of a holographic 3DTV are the computation of the diffraction field of a 3D object from its abstract representation, and determination of the best display configuration to synthesize some intended light distribution. To solve the former problem, we worked on the computation of 1D diffraction patterns from discrete data distributed over 2D space. The problem is solved using matrix pseudo-inversion which dominates the computational complexity. Then, the light field synthesis problem by a deflectable mirror array device (DMAD) is posed as a constrained linear optimization problem. The formulation makes direct application of common optimization algorithms quite easy. The simulations indicate that developed methods are promising.

A novel video-object segmentation algorithm is proposed, which takes the previously estimated 2-D dense motion vector field as input and uses the generalized recursive shortest spanning tree method to approximate each component of the motion vector field as a piecewise planar function. The algorithm is successful in capturing 3-D planar objects in the scene correctly, with acceptable accuracy at the boundaries. The proposed algorithm is fast and requires no initial guess about the segmentation mask. Moreover, it is a hierarchical scheme which gives finest to coarsest segmentation results. The only external parameter needed by the algorithm is the number of segmented regions that essentially control the level at which the coarseness the algorithm would stop. The proposed algorithm improves the "analysis model" developed in the European COST211 framework.

As an alternative to current two-dimensional (2-D) motion models, a robust three-dimensional (3-D) motion estimation method is proposed to be utilized in object-based video coding applications. Since the popular E-matrix method is well known for its susceptibility to input errors, a performance indicator, which tests the validity of the estimated 3-D motion parameters both explicitly and implicitly, is defined. This indicator is utilized within the RANSAC method to obtain a robust set of 2-D motion correspondences which leads to better 3-D motion parameters for each object. The experimental results support the superiority of the proposed method over direct application of the E-matrix method.

Flexibility and efficiency of coding, content extraction, and content-based search are key research topics in the field of interactive multimedia. Ongoing ISO MPEG-4 and MPEG-7 activities are targeting standardization to facilitate such services. European COST Telecommunications activities provide a framework for research collaboration. At present a significant effort of the COST 211/sup ter/ group activities is dedicated toward image and video sequence analysis and segmentation-an important technological aspect for the success of emerging object-based MPEG-4 and MPEG-7 multimedia applications. The current work of COST 211 is centered around the test model, called the analysis model (AM). The essential feature of the AM is its ability to fuse information from different sources to achieve a high-quality object segmentation. The current information sources are the intermediate results from frame-based (still) color segmentation, motion vector based segmentation, and change-detection-based segmentation. Motion vectors, which form the basis for the motion vector based intermediate segmentation, are estimated from consecutive frames. A recursive shortest spanning tree (RSST) algorithm is used to obtain intermediate color and motion vector based segmentation results. A rule-based region processor fuses the intermediate results; a postprocessor further refines the final segmentation output. The results of the current AM are satisfactory.

Intensity prediction along motion trajectories removes temporal redundancy considerably in video compression algorithms. In three-dimensional (3-D) object-based video coding, both 3-D motion and depth values are required for temporal prediction. The required 3-D motion parameters for each object are found by the correspondence-based E-matrix method. The estimation of the correspondences-two-dimensional (2-D) motion field-between the frames and segmentation of the scene into objects are achieved simultaneously by minimizing a Gibbs energy. The depth field is estimated by jointly minimizing a defined distortion and bit-rate criterion using the 3-D motion parameters. The resulting depth field is efficient in the rate-distortion sense. Bit-rate values corresponding to the lossless encoding of the resultant depth fields are obtained using predictive coding; prediction errors are encoded by a Lempel-Ziv algorithm. The results are satisfactory for real-life video scenes.

Object segmentation and tracking are problems within the scope of MPEG-4 and MPEG-7 standardization activities. A novel algorithm for both object segmentation and tracking is presented. The algorithm fuses motion, color, and accumulated previous segmentation data at 'region level', in contrast to conventional 'pixel level' approaches. The information fusion is achieved by a rule-based region processing unit which intelligently utilizes the motion information to locate the objects in the scene, the color information to extract the true boundaries, and the segmentation result of the previous frame for tracking the objects. The algorithm is generic in the sense that the modules prior to the rule-based region processor can independently be replaced by alternative units which can achieve the same tasks. In the proposed algorithm, while the recursive-shortest-spanning-tree (RSST) algorithm is used for segmentation purposes, hierarchical-block-matching (HBM) is utilized for estimating motion between frames. The simulation results are very promising for this novel object segmentation approach.

The extended versions of common Laplace and Fourier transforms are given. This is achieved by defining a new function f/sub e/(p), p/spl epsiv/C related to the function to be transformed f(t), t/spl epsiv/R. Then f/sub e/(p) is transformed by an integral whose path is defined on an inclined line on the complex plane. The slope of the path is the parameter of the extended definitions which reduce to common transforms with zero slope. Inverse transforms of the extended versions are also defined. These proposed definitions, when applied to filtering in complex ordered fractional Fourier stages, significantly reduce the required computation.

3-D motion models can be used to remove temporal redundancy between image frames. For efficient encoding using 3-D motion information, apart from the 3-D motion parameters, a dense depth field must also be encoded to achieve 2-D motion compensation on the image plane. Inspired by rate-distortion theory, a novel method is proposed to optimally encode the dense depth fields of the moving objects in the scene. Using two intensity frames and 3-D motion parameters as inputs, an encoded depth field can be obtained by jointly minimizing a distortion criteria and a bit-rate measure. Since the method gives directly an encoded field as an output, it does not require an estimate of the field to be encoded. By efficiently encoding the depth field during the experiments, it is shown that the 3-D motion models can be used in object based video compression algorithms.

The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.

Motion analysis is the most crucial part of object-based coding. Motion in a 3-D environment can be analyzed better by using a 3-D motion model compared to its 2-D counterpart and hence may improve coding efficiency. Gibbs formulated joint segmentation and estimation of 2-D motion not only improves the performance, but also generates robust point correspondences which are necessary for linear 3-D motion estimation algorithms. Estimated 3-D motion parameters are used to find the structure of the previously segmented objects by minimizing another Gibbs energy. Such an approach achieves error immunity compared to linear algorithms. Experimental results are promising and hence the proposed motion and structure analysis method is a candidate to be used in object-based (or even knowledge-based) video coding schemes.

Spatial interpolation is an important technique that is often used to recover an image from its downsampled version, or to simply perform image expansion. Many conventional linear techniques exist, however, these often perform rather poorly in a subjective manner. In this paper, image interpolation is performed using a binary-based Gibbs random field (GRF) model. Images are interpolated from their downsampled versions along with a number of texture parameters that are estimated within smaller image blocks. These iterative GRF methods are subsequently approximated by a non-iterative nonlinear filtering operation, thereby reducing the computational complexity of the interpolation process. Experimental results indicate that the statistical GRF approaches adapt to textured regions as well as the smooth areas within an image, and thus, can achieve better results than the conventional linear schemes.

We address the problem of 3-D motion estimation in the context of knowledge-based coding of facial image sequences. The proposed method handles the global and local motion estimation and the adaptation of a generic wire-frame to a particular speaker simultaneously within an optical flow based framework including the photometric effects of motion. We use a flexible wire-frame model whose local structure is characterized by the normal vectors of the patches which are related to the coordinates of the nodes. Geometrical constraints that describe the propagation of the movement of the nodes are introduced, which are then efficiently utilized to reduce the number of independent structure parameters. A stochastic relaxation algorithm has been used to determine optimum global motion estimates and the parameters describing the structure of the wire-frame model. For the initialization of the motion and structure parameters, a modified feature based algorithm is used. Experimental results with simulated facial image sequences are given.>

Wavelet kernels in the form of complex exponentials with linear instantaneous frequencies are discussed. These wavelets do not satisfy the commonly used admissibility condition; furthermore, they are neither band-, nor space-limited functions and they are not absolutely integrable. It is shown, however, that the wavelet transforms using these kernels are still invertable and the inversion formulas are given. Such kernels are closely related to wave propagation related phenomena like the diffraction and the holography, and therefore, the obtained mathematical results coincide to well known optical applications. The kernels may be useful in space-depth analysis of data obtained from diffraction related imaging techniques.>

3-D motion estimation from a video sequence remains a challenging problem. Modelling the local interactions between the 3-D motion parameters is possible by using Gibbs random fields. An energy function which gives the joint probability distribution of the motion vectors, is constructed. The most probable motion vector set is found by maximizing the probability, represented by this distribution. Since the 3-D motion estimation problem is ill-posed, the regularization is achieved by an initial rigidity assumption. Afterwards, the rigidity is weakened hierarchically, until the finest level is reached. At the finest level, each point has its own motion vector and the "weak-connection" between these vectors are described by the energy function. The high computational cost Q decreased considerably by the multiprecision approach. The simulation results support all our discussions.>

This correspondence is about a Gibbs-Markov random field (GMRF) parameter estimation technique proposed by Derin and Elliott (1987). We will refer to this technique as the histogramming (H) method. First, the relation of the H method to the (conditional) maximum likelihood method is considered. Second, a bias-reduction based modification of the H method is proposed to improve its performance, especially in the case of small amounts of image data.>

A generalized filtering method based on the minimization of the energy of the Gibbs model is described. The well-known linear and median filters are all special cases of this method. It is shown that, with that selection of appropriate energy functions, the method can be successfully used to adapt the weights of the adaptive weighted median filter to preserve different textures within the image white eliminating the noise. The newly developed adaptive weighted median filter is based on a 3/spl times/3 square neighborhood structure. The weights of the pixels are adapted according to the clique energies within this neighborhood structure. The assigned energies to 2- or 3-pixel cliques are based on the local statistics within a larger estimation window. It is shown that the proposed filter performance is better compared to some well-known similar filters like the standard, separable, weighted and some adaptive weighted median filters.>

Proposes a novel formulation where 3D global and local motion estimation and the adaptation of a generic wireframe model to a particular speaker are considered simultaneously within an optical flow based framework including the photometric effects of the motion. We use a flexible wireframe model whose local structure is characterized by the normal vectors of the patches which are related to the coordinates of the nodes. Geometrical constraints that describe the propagation of the movement of the nodes are introduced, which are then efficiently utilized to reduce the number of independent structure parameters. A stochastic relaxation algorithm has been used to determine optimum global motion estimates and the parameters describing the structure of the wireframe model. Results with both simulated and real facial image sequences are provided.>

In model-based coding of facial images, the accuracy of motion and depth parameter estimates strongly affects the coding efficiency. MBASIC (model-based analysis-synthesis image coding) is a simple and effective iterative algorithm recently proposed by Aizawa et el. (see Signal Processing: Image Communication, no.1, p.139-52, 1989) for 3-D motion and depth estimation when the initial depth estimates are relatively accurate. In this correspondence, we analyze its performance in the presence of errors in the initial depth estimates and propose a modification to MBASIC algorithm that significantly improves its robustness to random errors with only a small increase in the computational load.

The authors present a new technique for the display end of a holographic three-dimensional television system and describe the computer simulations. The technique is based on the reproduction of the desired pattern, in this case the hologram, using traveling surface waves. The proposed method is simpler and more efficient than the methods available in the literature and it solves the display resolution and refreshing rate problems completely. Simulations show that the proposed system will work as desired when implemented in real time.>

The design and the implementation of an automated, design-rule-based, visual printed circuit board (PCB) inspection system are presented. The system employs mathematical-morphology-based image processing algorithms. This systems detects PCB defects related to the conducting structures on PCBs by checking a set of geometric design rules. For this purpose, an image segmentation algorithm and a defect detection algorithm are designed. The defect detection algorithm is capable of verifying the minimum conductor spacing, minimum conductor trace width, and the minimum land width requirements on digital binary PCB images. Also, an existing defect detection algorithm is modified for its implementation in the system.>

An algorithm that yields textured and connected binary fractals is presented. The texture is imposed by modelling the fractal as a Markov random field (MRF) at every resolution level. The model size and the parameters specify the texture. The generation starts at a coarser level and continues at finer levels. Connectivity, which is a global property, is maintained by restricting the flow of the sample generating Markov chain within a limited subset of all possible outcomes of the Markov random field. The texture is controlled by the parameters of the MRF model being used. Sample patterns are shown.>

A VLSI median filter unit has been designed and implemented in 3- mu m M/sup 2/CMOS, using full-custom VLSI design techniques. The unit consists of two single-chip median filters, one extensible and one real-time. The chips are bit-level pipelined systolic structures based on odd/even transposition sorting. The extensible chip is designed for applications requiring variable window sizes and variable word-lengths, whereas the other one is for real-time applications. Various median filtering techniques are easily realized by using the designed chips together with reasonable external hardware.>

A system of digital filters is presented, which is applied directly to the in-line hologram and maps the hologram into an image of the boundaries (edges) of the objects encoded in it provided that the object distribution function is real. The first stage is an approximate inverse filter in the form of a truncated series expansion in the Fourier transform domain, having basis functions coskb (u2+v2). The first term in the expansion corresponds to conventional optical reconstruction which is contaminated by the twin-image; and the twin-image is suppressed by the successive terms. The non-linear post filtering consists of a covariance filter which replaces each input pixel by the estimate of the local covariance. A thresholding may be added prior to the covariance filtering if the edges of the reconstructed objects are not sharp enough.