We lay a philosophical framework for the design of overcomplete multidimensional signal decompositions based
on the union of two or more orthonormal bases. By combining orthonormal bases in this way, tight (energy
preserving) frames are automatically produced. The advantage of an overcomplete (tight) frame over a single
orthonormal decomposition is that a signal is likely to have a more sparse representation among the overcomplete
set than by using any single orthonormal basis. We discuss the question of the relationship between pairs of bases
and the various criteria that can be used to measure the goodness of a particular pair of bases. A particular case
considered is the dual-tree Hilbert-pair of wavelet bases. Several definitions of optimality are presented along
with conjectures about the subjective characteristics of the ensembles where the optimality applies. We also
consider relationships between sparseness and approximate representations.
The quality of medical ultrasound images is limited by inherent poor resolution due to the finite temporal bandwidth of the acoustic pulse and the non-negligible width of the system point-spread function. One of the major difficulties in designing a practical and effective restoration algorithm is to develop a model for the tissue reflectivity
that can adequately capture significant image features without being
computationally prohibitive. The reflectivities of biological tissues
do not exhibit the piecewise smooth characteristics of natural images
considered in the standard image processing literature; while the
macroscopic variations in echogenicity are indeed piecewise smooth,
the presence of sub-wavelength scatterers adds a pseudo-random component
at the microscopic level. This observation leads us to propose modelling
the tissue reflectivity as the product of a piecewise smooth echogenicity
map and a unit-variance random field. The chief advantage of such
an explicit representation is that it allows us to exploit representations
for piecewise smooth functions (such as wavelet bases) in modelling
variations in echogenicity without neglecting the microscopic pseudo-random
detail. As an example of how this multiplicative model may be exploited,
we propose an expectation-maximisation (EM) restoration algorithm
that alternates between inverse filtering (to estimate the tissue
reflectivity) and logarithmic wavelet denoising (to estimate the echogenicity
map). We provide simulation and in vitro results to demonstrate
that our proposed algorithm yields solutions that enjoy higher resolution,
better contrast and greater fidelity to the tissue reflectivity compared
with the current state-of-the-art in ultrasound image restoration.
Marine seismic imaging involves reconstructing subsurface reflectivity from some scattered acoustic data generally observed near the ocean surface. The procedure can be framed as a linearized inverse scattering problem and is often called least-squares migration (LSM). LSM has been shown to be effective in optimizing the reconstruction of subsurface reflectivity, particularly in cases of missing or undersampled data or uneven subsurface illumination.
In standard LSM, the reflectivity model parameters are usually defined as a grid of point scatterers over the area or volume to be migrated. We propose an approach to pre-stack LSM using the Dual Tree Complex Wavelet Transform (DT-CWT) as a basis for the reflectivity.
Wavelet bases have a reputation for decorrelating or diagonalizing a range of non-stationary signals. In LSM, diagonalization of the model space affords a more accurate but practical representation of prior information about the subsurface reflectivity model parameters. The DT-CWT is chosen for its key advantages compared to other wavelet transforms. These include shift invariance, directional selectivity, perfect reconstruction, limited redundancy and efficient computation.
A complex wavelet based LSM algorithm, derived in a Bayesian framework, is presented. Minimization of the least-squares cost function is performed in the wavelet domain rather than the standard reflectivity model domain.
Overcomplete transforms, such as the Dual-Tree Complex Wavelet
Transform, can offer more flexible signal representations than
critically-sampled transforms such as the Discrete Wavelet
Transform. However the process of selecting the optimal set of
coefficients to code is much more difficult because many different
sets of transform coefficients can represent the same decoded
image. We show that large numbers of transform coefficients can
be set to zero without much reconstruction quality loss by forcing
compensatory changes in the remaining coefficients. We develop a
system for achieving these coding aims of coefficient elimination
and compensation, based on iterative projection of signals between
the image domain and transform domain with a non-linear process
(e.g.~centre-clipping or quantization) applied in the transform
domain. The convergence properties of such non-linear feedback
loops are discussed and several types of non-linearity are
proposed and analyzed. The compression performance of the
overcomplete scheme is compared with that of the standard Discrete
Wavelet Transform, both objectively and subjectively, and is found
to offer advantages of up to 0.65 dB in PSNR and significant
reduction in visibility of some types of coding artifacts.
Image registration is essential in spread spectrum based watermark detection of a distorted image. Traditional registration approaches include the use of templates and feature matching between the distorted image and a reference image. However, neither of these techniques adequately addresses localized transformation. We propose a new registration scheme based on motion estimation between the distorted image and a reference copy. Complex wavelets provide a hierarchical framework for our motion estimation algorithm and radial basis functions provide the means to correct erroneous motion vectors. Experimental results show that our approach can estimate the distortion quite accurately and allow correct watermark detection.
This paper presents an objective perceptual distortion measure quantifying the visibility of edge-like blocking artifacts in coded image sequences resulting from popular transform coding techniques. The prime motivation for this work is the awareness that properties of the human visual system should be central to the design and evaluation of image coding algorithms. The perceptual metric is the output of a visual model incorporating both the spatial and temporal characteristics of the visual system. Parameters of the model are based on results from a number of visual experiments in which sensitivities to simulated blocking artifacts were measured under various spatio-temporal background conditions. The visual model takes a pair of original and distorted sequences as inputs. Distortions are calculated along the vertical and horizontal directions. Visibility dependencies on spatial, temporal and motion activities of the background are incorporated using linear filtering and motion estimation. Pixel-based distortions are combined over local spatial and temporal regions to generate an overall distortion measure for each orientation. The final model output is the sum of the vertical and horizontal distortion measures. The model was applied to coded image sequences and the resulting distortion measures were compared to outcomes of subjective ranking tests. Results indicate that the perceptual distortion measure agrees well with human evaluation.
We describe techniques of encoding compressed images and video such that the encoded bit stream is resilient to transmission errors which occur on typical noisy channels, particularly to and from mobile users. These errors may occur in the form of random bit errors, bursty bit errors or packet losses, or combinations of these, and the proposed techniques are shown to suffer only gradual and graceful degradation as the error rate increases. This contrasts markedly with the severe error sensitivity of conventional image compression standards, such as JPEG and MPEG. A key tool that we employ is the error resilient entropy code (EREC)6 which largely overcomes the problems of loss of synchronization and severe error propagation, inherent in most entropy coded data streams. Unlike traditional forward error correction methods, the error-resilient techniques which we employ do not add significant redundancy to the compressed data and therefore do not waste capacity or degrade the compression performance under good channel conditions. They are shown to work particularly well with wavelet compression because the localized image defects, which errors do cause, are less visible with wavelets than with DCT-based compression, due to the smoother boundaries of the wavelet basis functions.
This paper describes a new wavelet-based approach to the motion estimation problem for digital video. A complex- valued discrete wavelet transform is used to decompose each frame into a subsampled directionally bandpass filtered hierarchy. The transform is defined so that at each level there is an approximate correspondence between local translation and coefficient phase shift. This relationship is used to estimate motion with each orientation subband. The estimates are combined over all orientations and scales using a coarse-to-fine refinement strategy to produce a fractional-pel accurate motion field with a directional confidence measure. The technique is suitable for video compression schemes and can also be used for stereo vision and image registration.
In this paper we are concerned with the design of 2D biorthogonal, 2-channel filter banks where the sampling is on the quincunx lattice. Such systems can be used to implement the nonseparable Discrete Wavelet Transform and also to construct the nonseparable scaling and wavelet functions. One important consideration of such systems (brought into attention by wavelet theory) is the regularity or smoothness of the scaling and wavelet functions. The regularity is related to the zero-property--the number of zeros of the filter transfer function at the aliasing frequency (((omega) 1,(omega) 2) equals ((pi) ,(pi) ) for the quincunx lattice). In general the greater the number of zeros, the greater the regularity. It has been shown previously by the authors that the transformation of variables is an effective and flexible way of designing multidimensional filter banks. However the wavelet aspects of the filter banks (i.e., regularity) were not considered. In this paper we shall show how the zero- property can be easily imposed through the transformation of variables technique. A large number of zeros can be imposed with ease. Arbitrarily smooth scaling and wavelet functions can be constructed. Several design examples will be given to illustrate this.
In this paper we present a technique to design 2-channel filter banks in 3 dimensions where the sampling is on the FCO (face centered orthorhombic) lattice. The ideal 3-D subband is of the truncated octahedron shape. It is based on the transformation of variable method and it is equivalent to the generalized McClellan transformation. The filters are FIR, have linear phase, and achieve perfect reconstruction. Although the subband shape is quite complicated the ideal frequency characteristics are well approximated. This is illustrated with an example. The technique provides the flexibility of controlling the frequency characteristics of the filters with ease. The filters can be implemented quite efficiently due to the highly symmetrical nature of the coefficients of the transformation.
A visual model which gives a distortion measure for Blocking Artifacts in images is presented. Given the original and the reproduced image as inputs, the model output is a single numerical value which quantifies the visibility of blocking error in the reproduced image. The model is derived based on the human visual sensitivity to horizontal and vertical edge artifacts which result from blocking. Psychovisual experiments have been carried out using a novel experimental technique to measure the sensitivity to edge artifacts with the variation of edge length, edge amplitude, background luminance and background activity. The model parameters are estimated based on these sensitivity measures. The final model has been tested on real images, and the results show that the error predicted by the model correlate well with the subjective ranking.
Many conventional video coding schemes, such as the CCITT H.261 recommendation, are based on the independent processing of nonoverlapping image blocks. An important disadvantage with this approach is that blocking artifacts may be visible in the decoded frames. We propose a coding scheme based entirely on the processing of overlapping, windowed data blocks, thus eliminating blocking eftects. Motion estimation and, in part, compensation are performed in the frequency domain using a complex lapped transform (CLT), which can be viewed as a complex extension of the lapped orthogonal transform (LOT). The motion compensation algorithm is equivalent to overlapped compensation in the spatial domain, but also allows image interpolation for subpixel displacements and sophisticated loop filters to be conveniently applied in the frequency domain. For inter- and intraframe coding, we define the modified fast lapped transform (MFLT). This is a modified form of the LOT that entirely eliminates blocking artifacts in the reconstructed data. The transform is applied in a hierarchical structure, and performs better than the discrete cosine transform (DCT) for both coding modes. The proposed coder is compared with the H.261 scheme and is found to have significantly improved performance.
In this paper we present an method for designing 2-D linear phase FIR diamond subband filters having the perfect reconstruction property. It is based on a transformation of variable technique and is equivalent to the generalized McClellan transformation. We present methods to design a whole class of transformations. The method provides the flexibility of controlling the frequency characteristics of the filters with ease. With this method the problem consists of two parts: design of the transformation and design of the 1-D filters. The filters designed with this method can be implemented efficiently by employing separable processing along the diagonal directions. Several numerical design examples are presented to illustrate the flexibility of the design method.
Many conventional video coding schemes, such as the CCITT H.261 recommendation, are based on the independent processing of non-overlapping image blocks. An important disadvantage with this approach is that blocking artifacts may be visible in the decoded frames. In this paper, we propose a coding scheme based entirely on the processing of overlapping, windowed data blocks, thus eliminating blocking effects. Motion estimation and compensation are both performed in the frequency domain using a complex lapped transform (CLT), which may be viewed as a complex extension of the lapped orthogonal transform (LOT). The motion compensation algorithm is equivalent to overlapped compensation in the spatial domain, but also allows image interpolation for sub-pel displacements and sophisticated loop filters to be conveniently applied in the frequency domain. For inter- and intra-frame coding, we define the modified fast lapped transform (MFLT). This is a modified form of the LOT, which entirely eliminates blocking artifacts in the reconstructed data. The transform is applied in a hierarchical structure, and performs better than the discrete cosine transform (DCT) for both coding modes. The proposed coder is compared with the H.261 scheme, and is found to have significantly improved performance.
In many source and data compression schemes, information relating to positions of high energy samples or areas of importance often needs to be relayed to the decoder. The error resilient positional code (ERPC) is an efficient fixed rate coding scheme for encoding such positional information, or equivalently, sparse binary data patterns. It has also been designed with good channel error robustness properties, whose performance degrades gracefully with worsening channel conditions, without the possibility of breakdown or loss of sync.
In this paper, the coding efficiency of the ERPC is compared to a few other standard schemes, and as well as being efficient, its error extension is shown to be low and non-catastrophic. The ERPC is then applied to an efficient error robust adaptive image coding example based on a SBC/VQ codec capable of operating in harsh channel conditions without the aid of channel coding.
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