Framing pyramids

Burt and Adelson (1983) introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous-domain frames associated with the LP.     

Centered pyramids

Quadtree-like pyramids have the advantage of re-suiting in a multiresolution representation where each pyramid node has four unambiguous parents. Such a centered topology guarantees a clearly defined up-projection of labels. This concept has been successfully and extensively used in applications of contour detection, object recognition and segmentation. Unfortunately, the quadtree-like type of pyramid has poor approximation powers because of the employed piecewise-constant image model. This paper deals with the construction of improved centered image pyramids in terms of general approximation functions. The advantages of the centered topology such a symmetry, consistent boundary conditions and accurate up-projection of labels are combined with a more faithful image representation at coarser pyramid levels. We start by introducing a general framework for the design of least squares pyramids using the standard filtering and decimation tools. We give the most general explicit formulas for the computation of the filter coefficients by any (well behaving) approximation function in both the continuous (L(2)) and the discrete (l(2)) norm. We then define centered pyramids and provide the filter coefficients for odd spline approximation functions. Finally, we compare the centered pyramid to the ordinary one and highlight some applications.     

Multiresolution maximum intensity volume rendering by morphological adjunction pyramids

We describe a multiresolution extension to maximum intensity projection (MIP) volume rendering, allowing progressive refinement and perfect reconstruction. The method makes use of morphological adjunction pyramids. The pyramidal analysis and synthesis operators are composed of morphological 3-D erosion and dilation, combined with dyadic downsampling for analysis and dyadic upsampling for synthesis. In this case the MIP operator can be interchanged with the synthesis operator. This fact is the key to an efficient multiresolution MIP algorithm, because it allows the computation of the maxima along the line of sight on a coarse level, before applying a two-dimensional synthesis operator to perform reconstruction of the projection image to a finer level. For interpolation and resampling of volume data, which is required to deal with arbitrary view directions, morphological sampling is used, an interpolation method well adapted to the nonlinear character of MIP. The structure of the resulting multiresolution rendering algorithm is very similar to wavelet splatting, the main differences being that (i) linear summation of voxel values is replaced by maximum computation, and (ii) linear wavelet filters are replaced by nonlinear morphological filters.     

Morphological pyramids with alternating sequential filters

The aim of this paper is to find a relationship between alternating sequential filters (ASF) and the morphological sampling theorem (MST) developed by Haralick et al. (1987). The motivation behind this approach is to take advantage of the computational efficiency offered by the MST to implement morphological operations. First, we show alternative proofs for opening and closing in the sampled and unsampled domain using the basis functions. These proofs are important because they show that it possible to obtain any level of a morphological pyramid in one step rather than the traditional two-step procedure. This decomposition is then used to show the relationship of the open-closing in the sampled and unsampled domain. An upper and a lower bound, for the above relationships, are presented. Under certain circumstances, an equivalence is shown for open-closing between the sampled and the unsampled domain. An extension to more complicated algorithms using a union of openings and an intersection of closings is also proposed. Using the Hausdorff metric, it is shown that a morphologically reconstructed image cannot have a better accuracy than twice the radius of the reconstruction structuring element. Binary and gray scale examples are presented.     

Motion compensated sequence coding using image pyramids

Image coding using image pyramids is a technique originally developed for still pictures. The authors report an image sequence coding technique with motion compensation using image pyramids, which gives greater data compression than conventional motion estimation algorithms.<>     

Spatially scalable video compression employing resolution pyramids

A spatially scalable video coding scheme for low bit rates is proposed. The codec is especially well suited for communications applications because it is based on motion-compensated predictive coding which provides the necessary low-delay property. The frames to be coded are decomposed into a Gaussian pyramid. Motion estimation and compensation are performed between corresponding pyramid levels of successive frames. We show that, to fulfill specific needs of spatial scalability, the motion compensation on each level must result in compatible prediction errors (displaced frame differences, DFD). Compatibility of the prediction errors means that the pyramid formed by independently obtained DFD's (the DFD pyramid) is close to a Gaussian pyramid decomposition of the DFD of the highest resolution level. From the DFD pyramid, a least squares Laplacian pyramid is derived, which is quantized and coded. The DFD encoder outputs an embedded bit stream. Thus, the coder control may truncate the bit stream at any point, and can keep a fixed rate. The motion vector fields obtained at the different resolution levels are also encoded by employing a pyramid approach. Simulation results show that the proposed coder achieves a coding gain compared to simulcast coding     

Low entropy image pyramids for efficient lossless coding

An efficient image source coding technique gives good compression performance at low computational complexity. This research introduces an efficient coding technique, based on pyramid coding, that involves transforming an image into an equivalent lower entropy form prior to lossless coding. The proposed method is also a multiresolution technique that facilitates progressive image transmission.     

Orientation-sensitive interpolative pyramids for lossless andprogressive image coding

This paper presents a modified JPEG coder that is applied to the compression of mixed documents (containing text, natural images, and graphics) for printing purposes. The modified JPEG coder proposed in this paper takes advantage of the distinct perceptually significant regions in these documents to achieve higher perceptual quality than the standard JPEG coder. The region-adaptivity is performed via classified thresholding being totally compliant with the baseline standard. A computationally efficient classification algorithm is presented, and the improved performance of the classified JPEG coder is verified.     

Rotation-invariant texture retrieval with Gaussianized steerable pyramids.

This paper presents a novel rotation-invariant image retrieval scheme based on a transformation of the texture information via a steerable pyramid. First, we fit the distribution of the subband coefficients using a joint alpha-stable sub-Gaussian model to capture their non-Gaussian behavior. Then, we apply a normalization process in order to Gaussianize the coefficients. As a result, the feature extraction step consists of estimating the covariances between the normalized pyramid coefficients. The similarity between two distinct texture images is measured by minimizing a rotation-invariant version of the Kullback-Leibler Divergence between their corresponding multivariate Gaussian distributions, where the minimization is performed over a set of rotation angles.     

Investigations on ‘doping stacking fault’ pyramids

So-called ‘doping stacking fault’ pyramids discovered in Sb-doped Czochralski pulled silicon single crystals have been shown to be invisible in the electron microscope and in X-ray topography. Several hypotheses were examined in order to find an explanation for the nature of the faults. The idea that planes of foreign atoms (Sb) might be built into the crystal, or that a single-plane stacking fault (with $\text{R}\text{= ?}\text{1}\text{4}\text{[111]}$) might be present had to be abandoned. The assumption of intrinsic/extrinsic stacking fault pairs composing the pyramids appears to be most likely. The possibility of the occurance of doping stacking faults together with ‘normal’ stacking fault pyramids in epitaxial wafers is discussed.     

Near-capacity irregular variable length coding and irregular unity rate coding

In this contribution we introduce an EXtrinsic Information Transfer (EXIT) chart matching technique for the design of two serially concatenated irregular codecs, each constituted by a variety of component codes. This approach facilitates a higher degree of design freedom than matching the EXIT function of an irregular codec to that of a regular codec, comprising only a single component code. As a result, a narrower EXIT chart tunnel can be created, facilitating operation at E_b/N₀ values that are closer to the channel?s capacity bound. This is demonstrated for a serial concatenation of iteratively decoded Irregular Variable Length Coding (IrVLC) and Irregular Unity Rate Coding (IrURC), which is favourably compared with an IrVLC and regular Unity Rate Coding (URC) based benchmarker. Finally, we show that the iterative decoding complexity of our IrVLCIrURC scheme can be reduced by about 25% upon employing a method of jointly performing EXIT chart matching, while seeking a reduced iterative decoding complexity.     

Nonlinear multiresolution signal decomposition schemes. I.Morphological pyramids

Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis. The proposed theory is based on the following ingredients: 1) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases toward higher levels and 2) each step toward a higher level is implemented by an (information-reducing) analysis operator, whereas each step toward a lower level is implemented by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. Several examples of pyramid decomposition schemes are shown to be instances of the proposed theory: a particular class of linear pyramids, morphological skeleton decompositions, the morphological Haar pyramid, median pyramids, etc. Furthermore, the paper makes a distinction between single-scale and multiscale decomposition schemes, i.e., schemes without or with sample reduction. Finally, the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks.     

Microphone array optimization by stochastic region contraction

The authors deal with optimal microphone placement and gain for a linear one-dimensional array often in a confined environment. A power spectral dispersion function (PSD) is used as a core element for a min-max objective function (PSDX). Derivation of the optimal spacings and gains of the microphones is a hard computational problem since the min-max objective function exhibits multiple local minima (hundreds or thousands). The authors address the computational problem of finding the global optimal solution of the PSDX function. A new method, stochastic region contraction (SRC), is proposed. It achieves a computational speedup of 30-50 when compared to the commonly used simulated-annealing method. SRC is ideally suited for coarse-gain parallel processing     

Compression of Laplacian pyramids through orthogonal transforms and improved prediction

Scalable representation of visual signals, such as image and video signals, has become a subject of active research since early 1980s. Scalability allows the adaptation of the bit rate and/or the resolution of the transmitted data to the network bandwidth and/or the rendering capability of the receiving device. For many years, spatial scalability has been achieved through wavelets, but recently the Laplacian pyramid (LP) has become an alternative choice because of reduced aliasing in the lower resolutions. In this paper, we focus on the coding efficiency of the LP with a view to transmitting it over a communication channel. In particular, we aim to improve the compression efficiency of the LP detail layers through improved interlayer prediction and orthogonal spatial transforms. First, we consider an LP in the open-loop configuration and propose to improve its rate-distortion performance by compressing it to a critically sampled representation. We derive four different orthogonal spatial transforms from the upsampling and downsampling filters that can achieve this representation, and apply them on the detail layers. The application of these transforms to the detail layers renders a fixed number of transform coefficients either zero or redundant, thus making their transmission unnecessary. Then we consider the compression of an LP in the closed-loop configuration through similar spatial transforms. Because of the introduction of quantization in the prediction loop, these spatial transforms applied on the detail layers do not produce the same number of zero or redundant transform coefficients as in the open-loop case. Nevertheless, the insight obtained from the open-loop coding leads us to enhance the interlayer prediction, and the subsequent application of the spatial transforms to the new detail layers aims to achieve better energy compaction.     

Shortening for irregular QC-LDPC codes

Shortening is a technique to obtain codes of shorter length and lower rate from a given LDPC code by putting infinite reliability on some variable nodes, whose positions are assumed to be available to both encoder and decoder. In this paper, we propose a shortening algorithm suitable for irregular QC-LDPC codes. The efficiency of the proposed algorithm is verified by both theoretical analysis and simulation.     

Optimal progressive lossless image coding using reduced pyramids with variable decimation ratios

This article introduces a novel method for the construction of extended reduced pyramids with rational decimation ratios from stage to stage. It is shown that this construction produces more accurate interpolation, and thus more efficient lossless compression, than conventional reduced pyramids.     

Fast codeword search algorithm for image coding based onmean-variance pyramids of codewords

The fast codeword search algorithm based on the mean pyramids of codewords is currently used in image coding applications. A more efficient algorithm is presented for fast codeword searching which is based on the use of mean-variance pyramids of codewords. Experimental results confirm the effectiveness of the proposed method     

Design of capacity-approaching irregular low-density parity-checkcodes

We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds     

Use of multiresolution wavelet feature pyramids for automatic registration of multisensor imagery.

The problem of image registration, or the alignment of two or more images representing the same scene or object, has to be addressed in various disciplines that employ digital imaging. In the area of remote sensing, just like in medical imaging or computer vision, it is necessary to design robust, fast, and widely applicable algorithms that would allow automatic registration of images generated by various imaging platforms at the same or different times and that would provide subpixel accuracy. One of the main issues that needs to be addressed when developing a registration algorithm is what type of information should be extracted from the images being registered, to be used in the search for the geometric transformation that best aligns them. The main objective of this paper is to evaluate several wavelet pyramids that may be used both for invariant feature extraction and for representing images at multiple spatial resolutions to accelerate registration. We find that the bandpass wavelets obtained from the steerable pyramid due to Simoncelli performs best in terms of accuracy and consistency, while the low-pass wavelets obtained from the same pyramid give the best results in terms of the radius of convergence. Based on these findings, we propose a modification of a gradient-based registration algorithm that has recently been developed for medical data. We test the modified algorithm on several sets of real and synthetic satellite imagery.