The need for more realistic 3D scene representations has fomented the development of models for a wide range of applications. In this context, solutions that attempt to model the light's behavior through the plenoptic function have provided considerable advancements using neural-based approaches, often presenting a trade-off between rendering time and model sizes. In this work, we propose a pruning framework to reduce the sizes of these models by computing the visibility over the training data, applicable to different 3D scene representations. In particular, we implement first a solution suitable for the 3D Gaussian Splatting, and then we exemplify the solution for the Neural Radiance Fields (NeRF)-style of rendering using PlenOctrees. We show that our pruning solution produces smaller models in terms of the number of elements – be they voxels, points, or Gaussians – with minimal losses in terms of rendering novel views. We further assess our solution by combining it with state-of-the-art (SOTA) compression solutions for both rendering schemes. Results over the NeRF-Synthetic dataset show comparable metrics to the SOTA for PlenOctrees, achieving marginal gains for lower bitrates. For 3DGS, the combination of our pruning method and compression solutions achieves a compression ratio of up to 37.5 times over the uncompressed 3DGS models, with only a 0.5 dB decrease in rendering quality. When compared against other SOTA compression methods, our solution produces models 1.4 times smaller, with less than a 0.1 dB loss over novel views for synthetic data, and models 1.9 times smaller with less than 0.2 dB loss when synthesizing novel views on real-world, outdoor content.

3D scene representation has been a central theme of study for a wide range of applications, and the representation of light behavior is one of the relevant topics when producing realistic models. In this work, we create a framework to assess the representation of non-Lambertian scenes by generating a pipeline to create plenoptic point clouds (PPCs) systematically and evaluating them against implicit solutions, such as Neural Radiance Fields (NeRF)-like models. We compare such approaches according to rendering quality and compression efficiency. On the compression side, we propose an encoding scheme for PPC, leveraging the occlusion masks of the points and the Moving Picture Expert Group's (MPEG) Geometry-Based Solid Content Test Model (GeS-TM). Rendering results over the training views show that the uncompressed PPC outperforms 3D Gaussian Splatting (3DGS) by 1.51 dB, on average, for the 8 scenes of the NeRF Synthetic 360 dataset. In compression efficiency, 3DGS outperforms the compressed PPCs by 0.7 dB in BD-PSNR on average. Our occlusion-aware encoding scheme reduces the size of uncompressed PPCs up to 800 times, outperforming current encoding schemes for PPC by 1.9 dB in BD-PSNR.

This paper describes a novel lossless compression method for point cloud geometry, building on a recent lossy compression method that aimed at reconstructing only the bounding volume of a point cloud. The proposed scheme starts by partially reconstructing the geometry from the two depthmaps associated to a single projection direction. The partial reconstruction obtained from the depthmaps is completed to a full reconstruction of the point cloud by sweeping section by section along one direction and encoding the points which were not contained in the two depthmaps. The main ingredient is a list-based encoding of the inner points (situated inside the feasible regions) by a novel arithmetic three dimensional context coding procedure that efficiently utilizes rotational invariances present in the input data. State-of-the-art bits-per-voxel results are obtained on benchmark datasets.

This paper describes a novel lossless point cloud compression algorithm that uses a neural network for estimating the coding probabilities for the occupancy status of voxels, depending on wide three dimensional contexts around the voxel to be encoded. The point cloud is represented as an octree, with each resolution layer being sequentially encoded and decoded using arithmetic coding, starting from the lowest resolution, until the final resolution is reached. The occupancy probability of each voxel of the splitting pattern at each node of the octree is modeled by a neural network, having at its input the already encoded occupancy status of several octree nodes (belonging to the past and current resolutions), corresponding to a 3D context surrounding the node to be encoded. The algorithm has a fast and a slow version, the fast version selecting differently several voxels of the context, which allows an increased parallelization by sending larger batches of templates to be estimated by the neural network, at both encoder and decoder. The proposed algorithms yield state-of-the-art results on benchmark datasets. The implementation will be available at https://github.com/marmus12/nnctx

The paper proposes a new lossy way of encoding the geometry of point clouds. The proposed scheme reconstructs the geometry from only the two depth maps associated to a single projection direction and then proposes a progressive reconstruction process using suitably defined anchor points. The reconstruction from the two depth images follows several primitives for analyzing and encoding, several of which are only optional. The resulting bitstream is embedded and can be truncated at various levels of reconstruction of the bounding volume. The encoding tools for encoding the needed entities are extremely simple and can be combined flexibly. The scheme can also be combined with the G-PCC coding, for reconstructing in a lossless way the sparse point clouds. The experiments show improvement of the rate-distortion performance of the proposed method when combined with the G-PCC codec as compared to G-PCC codec alone.

This paper proposes a lenslet image compression method scalable from low bitrates to fully lossless. The subaperture images are split into two sets: a set of reference views, encoded by a standard lossy or lossless compressor, and the set of dependent views, which are reconstructed by sparse prediction from the reference set using the geometrical information from the depth map. The set of reference views may contain all views and all views may also be dependent views, in which case the sparse predictive stage does not reconstruct from scratch the views, but it refines in a sequential order all views by combining in an optimal way the information about the same region existing in neighbor views. The encoder transmits to the decoder a segmented version of the scene depthmap, the encoded versions of the reference views, displacements for each region from the central view to each of the dependent views, and finally the sparse predictors for each region and each dependent view. The scheme can be configured to ensure random access to the dependent views, while the reference views are compressed in a backward compatible way, e.g., using JPEG 2000. The experimental results show performance better than that of the baseline standard compressor used, JPEG 2000.

We propose a low complexity method for estimating direction of arrival (DOA) when the positions of the array sensors are affected by errors with known magnitude bound. This robust DOA method is based on solving an optimization problem whose solution is obtained in two stages. First, the problem is relaxed and the corresponding power estimation has an expression similar to that of standard beamforming. If the relaxed solution does not satisfy the magnitude bound, an approximation is made by projection. Unlike other robust DOA methods, no eigenvalue decomposition is necessary and the complexity is similar to that of MVDR. For low and medium SNR, the proposed method competes well with more complex methods and is clearly better than MVDR.

The paper deals with encoding the contours of given regions in an image. All contours are represented as a sequence of contour segments, each such segment being defined by an anchor (starting) point and a string of contour edges, equivalent to a string of chain-code symbols. We propose efficient ways for anchor points selection and contour segments generation by analyzing contour crossing points and imposing rules that help in minimizing the number of anchor points and in obtaining chain-code contour sequences with skewed symbol distribution. When possible, part of the anchor points are efficiently encoded relative to the currently available contour segments at the decoder. The remaining anchor points are represented as ones in a sparse binary matrix. Context tree coding is used for all entities to be encoded. The results for depth map compression are similar (in lossless case) or better (in lossy case) than the existing results.

This letter proposes a method for lossless coding the left disparity image, L, from a stereo disparity image pair (L,R), conditional on the right disparity image, R, by keeping track of the transformation of the constant patches from R to L. The disparities in R are used for predicting the disparities in L, and the locations of the pixels where the prediction is erroneous are encoded in a first stage, conditional on the patch-labels of R image, allowing the decoder to already reconstruct with certainty some elements of the L image, e.g., the disparity values at certain pixels and parts of the contours of left image patches. Second, the contours of the patches in L image that are still unknown after first stage are conditionally encoded using a mixed conditioning context: the usual causal current context from the contours of L and a noncausal context extracted from the contours in the correctly estimated part of L obtained in the first stage. The depth values in the patches of L image are finally encoded, if they are not already known from the prediction stage. The new algorithm, dubbed conditional crack-edge region value (C-CERV), is shown to perform significantly better than the non-conditional coding method CERV and than another existing conditional coding method, over the Middlebury corpus. C-CERV is shown to reach lossless compression ratios of 100-250 times for those images that have a high precision of the disparity map.

We introduce a method to create lossy versions of one image, either by successively merging the constant regions of the original image, or by iteratively splitting the regions from a created lossy image using horizontal or vertical line segments. Merging and split decisions are greedily taken, according to the best slope towards next point in the rate-distortion curve. For each created lossy image, the region contours and the optimal depth values can be entropy coded in three ways: with a new algorithm, or with two existing lossless coding algorithms. The obtained results compare favorably with the existing lossy methods.

We investigate the problem of sparse modeling for predictive coding and introduce an efficient algorithm for computing sparse stereo linear predictors for lossless audio compression. Sparse linear predictive coding offers both improved compression and reduction of decoding complexity compared with non-sparse linear predictive coding. The modeling part amounts to finding the optimal structure of a sparse linear predictor using a fully implementable minimum description length (MDL) approach. The MDL criterion, simplified conveniently under realistic assumptions, is approximately minimized by a greedy algorithm which solves sequentially least squares partial problems, where theLDLTfactorization ensures numerically stable solutions and facilitates a quasi-optimal quantization of the parameter vector. The overall compression system built around this modeling tool is shown to achieve the main goals: improved compression and, even more importantly, faster decoding speeds than the state of the art lossless audio compression methods. The optimal MDL sparse predictors are shown to provide parametric spectra that constitute new alternative spectral descriptors, capturing important regularities missed by the optimal MDL non-sparse predictors.

This paper introduces an efficient method for lossless compression of depth map images, using the representation of a depth image in terms of three entities: 1) the crack-edges; 2) the constant depth regions enclosed by them; and 3) the depth value over each region. The starting representation is identical with that used in a very efficient coder for palette images, the piecewise-constant image model coding, but the techniques used for coding the elements of the representation are more advanced and especially suitable for the type of redundancy present in depth images. Initially, the vertical and horizontal crack-edges separating the constant depth regions are transmitted by 2D context coding using optimally pruned context trees. Both the encoder and decoder can reconstruct the regions of constant depth from the transmitted crack-edge image. The depth value in a given region is encoded using the depth values of the neighboring regions already encoded, exploiting the natural smoothness of the depth variation, and the mutual exclusiveness of the values in neighboring regions. The encoding method is suitable for lossless compression of depth images, obtaining compression of about 10-65 times, and additionally can be used as the entropy coding stage for lossy depth compression.

We present a sliding window RLS for sparse filters, based on the greedy least squares algorithm. The algorithm adapts a partial QR factorization with pivoting, using a simplified search of the filter support that relies on a neighbor permutation technique. For relatively small window size, the proposed algorithm has a lower complexity than recent exponential window RLS algorithms. Time-varying FIR channel identification simulations show that the proposed algorithm can also give better mean squared coefficient errors.

In this paper we study various chain codes, which are representations of binary image contours, in terms of their ability to compress in the best way the contour information using memory models. We consider five chain codes, including the widely used AF8 and 30T codes, and note that they correspond to memory models of first and second order for contour representation. In order to provide predictive distributions for the arithmetic coding, memory distribution models such as Markov models and context trees utilized in adaptive configurations are used on top of the chain codes. By additionally accounting for all side costs we obtain losslessly decodable files and find the best performer to be the context tree modeling applied to the sequence of 30T chain codes, surpassing all results recently reported in the literature for the same data set of bilevel images.

We propose a context conditioning scheme for encoding the prediction residuals when compressing files containing companded signals. Our scheme encompasses decompanding of the signals, performing linear prediction in the decompanded domain, and then companding back the predicted value into a companded prediction (CP) value, which will differ from the true companded value by an amount called companded prediction residual (CPR). The proposed context conditioning scheme for encoding the CPR, uses a probability distribution conditional on a context made up of two quantities: (1) the predicted value and (2) a scale parameter of the background probability distribution function assumed for the decompanded domain residuals. Various context building schemes and various storing strategies can be used to obtain the necessary conditional coding distribution of the CPR, to be used with an arithmetic coder or range coder. The implementation in fixed point precision can be done very efficiently and with very low memory requirements.

In this paper we study how to encode W-long vectors, with N in the range of hundreds, at low bit rates of 0.5 bit per sample or lower. We adopt a vector quantization structure, where an overall gain is encoded with a scalar quantizer and the remaining scaled vector is encoded using a vector quantizer built out by combining smaller (length L) binary codes known to be efficient in filling the space, the important examples discussed here being the Golay codes. Due to the typical nonstationary distribution of the long vectors, a piecewise stationary plus contamination model is assumed. The generic solution is to encode the outliers using Golomb-Rice codes, and for each L-long subvector to encode the vector of absolute values using the nearest neighbor in a certain shell of a chosen binary {0,1} code, the sign information being transmitted separately. The rate-distortion optimization problem can be very efficiently organized and solved for the unknowns, which include the Hamming weights of the chosen shells for each of the nb= [N/L] subvectors, and the overall gain g. The essential properties which influence the selection of a certain binary code as a building block are its space filling properties, the number of shells of various Hamming weights (allowing more or less flexibility in the rate-distortion optimization), the closeness of N to a multiple of L, and the existence of fast search of nearest neighbor on a shell. We show results when using the Golay codes for vector quantization on audio coding applications.

This paper presents a new algorithm for identifying the language of words in a multilingual automatic speech recognition system. The new algorithm uses as input written words and it is composed of a method for language modelling and a method to select the language of a given word based on the available models. We also present two selection rules for the model's parameters. One of the rules uses a free parameter that controls the accuracy of the resulted model, as well as its size. On average, the classification accuracy of the new algorithm is above 70% for the first best and above 80% for the first two best.

In this paper we study the problem of optimally quantizing the least square estimates and we introduce a method where the quantized estimate vector is obtained by a sequence of interleaved optimizationquantization scalar stages. We show how the general approach can be reduced to a simple and efficient algorithm when connecting it to the LDLTsolver for general LS problems. The application to quantization of the linear prediction coefficients for audio lossless coding reveals the high performance of the approach, leading to topmost performance in the class of frame based audio coders, surpassing significantly the performance of the current MPEG4-ALS standard.

The paper presents exact and implementable solutions to the problem of universal coding of approximate repeats by using the normalized maximum likelihood model for the class of Markov sources of first order, incorporating constraints which are standard in the context of fast searching similarities over full genomes. A coding scheme combining universal codes for memoryless sources and for sources with memory is then presented. The results when compressing the full human genome show that the combined scheme is able to provide slight improvements over the existing state of the art. As a side result, interesting pairs of sequences may be found, which are highly similar by the new NML model for Markov sources, but have a lower similarity score when evaluated with the NML for memoryless sources.

We present the NML model for classes of models with memory described by first order dependencies. The model is used for efficiently locating and encoding the best regressor present in a dictionary. By combining the order-1 NML with the order-0 NML model the resulting algorithm achieves a consistent improvement over the earlier order-0 NML algorithm, and it is demonstrated to have superior performance on the practical compression of the human genome

This paper introduces a novel prediction structure for improving the lossless compression ratio, by accounting for companding nonlinearities of different sample-based audio formats. This applies to a wide class of formats including a-law, μ-law, DAT-LP (digital audio tape recorders), DV-LP (digital video camcorders), and HDCD (high definition compatible digital). The proposed prediction structure obtains significant compression improvements (8-12%) over traditional linear prediction for a-law, μ-law, DAT-LP, DV-LP and also small compression improvements for HDCD. The improvement in compression can also be used for the detection of nonlinearities in HDCD format, making possible to play HDCD CDs at improved audio resolution in ordinary public domain players.

This article introduces an algorithm for the lossless compression of DNA files, which contain annotation text besides the nucleotide sequence. First a grammar is specifically designed to capture the regularities of the annotation text. A revertible transformation uses the grammar rules in order to equivalently represent the original file as a collection of parsed segments and a sequence of decisions made by the grammar parser. This decomposition enables the efficient use of state-of-the-art encoders for processing the parsed segments. The output size of the decision-making process of the grammar is optimized by extending the states to account for high-order Markovian dependencies. The practical implementation of the algorithm achieves a significant improvement when compared to the general-purpose methods currently used for DNA files.

We discuss a lossless data compression system that uses fixed tree machines to encode data. The idea is to create a sequence of tree machines and a robust escape method aimed at preventing expansion of the encoded string for data whose statistics deviate from those represented by the machines. The resulting algorithm is shown to have superior compression of short files compared to other methods.

MOTIVATION: The protein lysate microarray is a developing proteomic technology for measuring protein expression levels in a large number of biological samples simultaneously. A challenge for accurate quantification is the relatively narrow dynamic range associated with the commonly used chromogenic signal detection system. To facilitate accurate measurement of the relative expression levels, each sample is serially diluted and each diluted version is spotted on a nitrocellulose-coated slide in triplicate. Thus, each sample yields multiple measurements in different dynamic ranges of the detection system. This study aims to develop suitable algorithms that yield accurate representations of the relative expression levels in different samples from multiple data points. RESULTS: We evaluated two algorithms for estimating relative protein expression in different samples on the lysate microarray by means of a cross-validation procedure. For this purpose as well as for quality control we designed a 1440-spot lysate microarray containing 80 identical samples of purified bovine serum albumin, printed in triplicate with six 2-fold dilutions. Our analysis showed that the algorithm based on a robust least squares estimator provided the most accurate quantification of the protein lysate microarray data. We also demonstrated our methods by estimating relative expression levels of p53 and p21 in either p53(+/+) or p53(-/-) HCT116 colon cancer cells after two drug treatments and their combinations on another lysate microarray. AVAILABILITY: http://www.cs.tut.fi/~mirceanc/lysate_array_bioinformatics.htm

This article presents an efficient algorithm for DNA sequence compression, which achieves the best compression ratios reported over a test set commonly used for evaluating DNA compression programs. The algorithm introduces many refinements to a compression method that combines: (1) encoding by a simple normalized maximum likelihood (NML) model for discrete regression, through reference to preceding approximate matching blocks, (2) encoding by a first order context coding and (3) representing strings in clear, to make efficient use of the redundancy sources in DNA data, under fast execution times. One of the main algorithmic features is the constraint on the matching blocks to include reasonably long contiguous matches, which not only reduces significantly the search time, but also can be used to modify the NML model to exploit the constraint for getting smaller code lengths. The algorithm handles the changing statistics of DNA data in an adaptive way and by predictively encoding the matching pointers it is successful in compressing long approximate matches. Apart from comparison with previous DNA encoding methods, we present compression results for the recently published human genome data.

The use of normalized maximum likelihood (NML) model for encoding sequences known to have regularities in the form of approximate repetitions was discussed. A particular version of the NML model was presented for discrete regression, which was shown to provide a very powerful yet simple model for encoding the approximate repeats in DNA sequences. Combining the model of repeats with a simple first order Markov model, a fast lossless compression method was obtained that compares favorably with the existing DNA compression programs. It is remarkable that a simple model, which recursively updates a small number of parameters, is able to reach the state of the art compression ratio for DNA sequences with much more complex models. Being a minimum description length (MDL) model, the NML model may later prove to be useful in studying global and local features of DNA or possibly of other biological sequences.

In this paper, alphabet extension for Markov sources is studied such that each extension tree is grown by splitting the node with the maximum value for a weight as a generalization of the leaf probability in Tunstall's (1967) algorithm. We show that the optimal asymptotic rate of convergence of the per-symbol code length to the entropy does not depend on an a priori selected proportional allocation of the sizes of the extension trees at the states. We show this without imposing restrictive conditions on the weight by which the trees are extended. Further, we prove the asymptotic optimality of an algorithm that allocates an increasing total number of leaves among the states. Finally, we give exact formulas for all the relevant quantities of the trees grown.

We propose the application of a new transform-based coding method in conjunction with Golomb-Rice (G-R) codes to lower significantly the complexity, which can be used in various applications, e.g. the multiple description coding. The theoretical evaluations predict no important loss in compression performance, while the complexity is considerably reduced. Since GR codes are very fast and well suited for exponentially decaying distributions, they were implemented during the last decade in image and audio compressors. In all these schemes, the selection of the code parameter is performed presuming Laplacian distribution of prediction errors. We derive the selection method for the GR code parameter also for the case of Gaussian inputs.

We present a design method for optimal energy compaction IIR filters, where the numerator and denominator may have different degrees. The design is performed via iterative relaxations, where the numerator is optimized given the denominator, followed by optimization of denominator given the numerator. The two optimization problems involved are solved using semidefinite programming (SDP) techniques, where the real positiveness of the causal part of the product filter is formulated in two alternative ways: first using the Kalman-Yakubovich-Popov (KYP) lemma, and second, by a less-known parameterization (Genin et al., 2000; Stoica et al., 2000), which we show to be more convenient numerically. Numerical results show the effectiveness of the proposed method and the improvements when compared with optimal FIR compaction filters or constrained IIR compaction filters (restricted to have allpass polyphase components).

We present two methods of entropy coding for the lattice codevectors. We compare our entropy coding methods with one method previously presented in the literature from the point of view of rate-distortion as well as of the computation complexity and memory requirements. The results are presented for artificial Laplacian and Gaussian data, as well as for LSF parameters of speech signals. In the latter case, the multiple scale lattice VQ (MSLVQ) is used for quantization, which reduces the rate gain of the entropy coding method when compared with the fixed rate case, but allows a dynamic allocation of the bits in the whole speech coding scheme.

An n-state Markov model for symbol occurrences is extended to an equivalent source for variable length strings of symbols in a dictionary at every state i, which are to be encoded with the string index in the dictionary. The algorithm for building the n dictionaries optimizes the rate subject to a given total number of entries in the dictionaries, and it is practical even for Markov sources with thousands of states. The speed of the algorithm stems from encoding by table look-ups of the strings instead of single symbols. For this the n dictionaries need be known both to the encoder and the decoder. A static version of the algorithm is very well suited for creation of compressed files with random access. An adaptive version is shown to be faster than the methods in the PPM class, while providing only slightly lower compression ratios.

This paper proposes a vector quantization (VQ) scheme that improves the quality of the reconstructed image by correcting quantization artifacts. The basic idea of the coding scheme is to treat separately the blocks containing edges (edge blocks), as they contain important perceptual information. The edge blocks are identified using an edge detection algorithm and are labeled by an escape index, and their reconstruction is done via an interpolation procedure that exploits the spatial correlation in the image. The remaining blocks (smooth blocks) are coded using standard VQ.

This paper introduces a new lattice quantization scheme, the multiple-scale lattice vector quantization (MSLVQ), based on the truncation of the D/sub 10//sup +/ lattice. The codebook is composed of several copies of the truncated lattice scaled with different scaling factors. A fast nearest neighbor search is introduced. We compare the performance of predictive MSLVQ for quantization of line spectrum frequency (LSF) coefficients with the quantization technique used in the G.729 codec and show the better performance of our method in terms of spectral distortion. The MSLVQ scheme achieves the transparent quality at 21 bits/frame.

A new filtering architecture is proposed, generalizing some previously introduced multilevel median filters. An efficient design procedure for the new filtering architecture is demonstrated for image restoration application. Simulation results show a good noise rejection performance, combined with a fine detail preservation capability.

This paper generalizes trellis based near-lossless image compression schemes, in order to satisfy two constraints: a local one, which enforces an upper bound /spl epsiv/ for the reconstruction error at every pixel, and a global one, which penalizes the quantity R+/spl lambda/D (the rate and distortion over each line of the image). By changing /spl lambda/ we obtain gradual changes from lossless coding (/spl lambda//spl infin/) to DPCM near-lossless coding (/spl lambda/=0), resulting in a distinct rate-distortion curve for each tolerance /spl epsiv/. We propose here a trellis-based fixed-slope near-lossless compression approach, that fits the adaptive context coding scheme ASBOSC introduced in 1997. The experimental results over a set of test images in near-lossless mode are better than the results obtained with other near-lossless techniques.

This paper proposes the use of mean absolute error (MAE) optimal Boolean and stack filters for sequential prediction in lossless grey-level image coding. FIR-Boolean hybrid filters are introduced as variations of Boolean filter structure and shown to be very effective for the prediction task. Different instances of optimal filtering are considered for realizing the prediction stage. First, the use of global-optimal predictors is analyzed, when the global MAE-optimal filter is used as a predictor. Then more refined structures, block-optimal and adaptive-size-block-optimal are considered, where predictors are adapted to local characteristics. These structures prove most suitable when small prediction masks are used. Extensive simulations are carried out for analyzing and comparing the performance of the newly introduced predictors and various other sequential predictors.

This paper introduces a filter class which generalizes the extended threshold Boolean filter (ETBF) to encompass many other classes of linear and nonlinear filters. The proposed filter uses two observation windows and it can be described as a locally adaptive linear filter. One window (usually the largest) collects the inputs of the linear filter while the threshold decomposition of the other one is used to compute the weights. As in the case of ETBF, the new filter possesses an MSE-optimal design similar to linear filters. An efficient iterative design algorithm is developed for the proposed filter which also applies for ETBF. Experimental results show that the new method can provide better filtering performance than ETBF.

In this paper we introduce a new class of adaptive nonlinear predictors by allowing the parameters of the L-predictor to be selected according to the transitions in a finite state machine (FSM) context modeller. A procedure for the adaptive design of the general unconstrained FSM-context L-predictor is proposed and compared with the classical design techniques for some particular FSM-L predictors. The application of the new predictor for lossless compression of gray level images is examined for different FSM structures.

We develop a new procedure for the optimal stack filter design under structural constraints, e.g. for minimizing an error criterion and simultaneously preserving the shape of some signals. The training framework for optimal stack filter design developed by Tabus, Petrescu and Gabbouj (see IEEE Transactions on Image Processing, Special Issue on Nonlinear Image Processing, IP-5, p.1-18, June 1996) provides us with a proper setting for imposing structural constraints to the optimal filter, in order to preserve some desired details of the image. The target application is optimal stack filter design for image restoration, the goal being the "most efficient" noise attenuation.

A training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks. The targeted families of nonlinear filters are the Boolean filters and stack filters. The main merit of this framework at the implementation level is perhaps the absence of constraining models, making it nearly universal in terms of application areas. We develop fast procedures to design optimal or close to optimal filters, based on some representative training set. Furthermore, the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution. Symmetry constraints are imposed on the data and, consequently, on the resulting optimal solutions for improved performance and ease of implementation. The case study is dedicated to natural images. The properties of optimal Boolean and stack filters, when the desired signal in the training set is the image of a natural scene, are analyzed. Specifically, the effect of changing the desired signal (using various natural images) and the characteristics of the noise (the probability distribution function, the mean, and the variance) is analyzed. Elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets. A remarkably low sensitivity and, consequently, a good generalization power of Boolean and stack filters are revealed. Boolean-based filters are thus shown to be not only suitable for image restoration but also robust, making it possible to build libraries of "optimal" filters, which are suitable for a set of applications.

For the nonlinear classes of filters based on the median archetype, e.g. stack, weighted order statistics (WOS), morphological filters the techniques exploiting recursiveness of embedded structures were not used yet. We investigate in this paper the possibilities of improving the speed of the optimal design techniques restating the optimal design problem as a sequence of optimal problems for embedded structures. The speed up effect of recursive-in-order design techniques is very significant but it is not the only benefit of this principle: it also allows the evaluation of the optimal structure of the filter, comparing the performances of the optimal filters having various structural parameters and observing where the performance index curve starts to flatten with the increase in the structure "size".