Nick G. Kingsbury

The dual-tree complex wavelet transform

The paper discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing. The authors use the complex number symbol C in CWT to avoid confusion with the often-used acronym CWT for the (different) continuous wavelet transform. The four fundamentals, intertwined shortcomings of wavelet transform and some solutions are also discussed. Several methods for filter design are described for dual-tree CWT that demonstrates with relatively short filters, an effective invertible approximately analytic wavelet transform can indeed be implemented using the dual-tree approach.

Image processing with complex wavelets

We first review how wavelets may be used for multi–resolution image processing, describing the filter–bank implementation of the discrete wavelet transform (DWT) and how it may be extended via separable filtering for processing images and other multi–dimensional signals. We then show that the condition for inversion of the DWT (perfect reconstruction) forces many commonly used wavelets to be similar in shape, and that this shape produces severe shift dependence (variation of DWT coefficient energy at any given scale with shift of the input signal). It is also shown that separable filtering with the DWT prevents the transform from providing directionally selective filters for diagonal image features. Complex wavelets can provide both shift invariance and good directional selectivity, with only modest increases in signal redundancy and computation load. However, development of a complex wavelet transform (CWT) with perfect reconstruction and good filter characteristics has proved difficult until recently. We now propose the dual–tree CWT as a solution to this problem, yielding a transform with attractive properties for a range of signal and image processing applications, including motion estimation, denoising, texture analysis and synthesis, and object segmentation.

A dual-tree complex wavelet transform with improved orthogonality and symmetry properties

We present a new form of the dual-tree complex wavelet transform (DT CWT) with improved orthogonality and symmetry properties. Beyond level 1, the previous form used alternate odd-length and even-length bi-orthogonal filter pairs in the two halves of the dual-tree, whereas the new form employs a single design of even-length filter with asymmetric coefficients. These are similar to the Daubechies orthonormal filters, but designed with the additional constraint that the filter group delay should be approximately one quarter of the sample period. The filters in the two trees are just the time-reverse of each other, as are the analysis and reconstruction filters. This leads to a transform, which can use shorter filters, which is orthonormal beyond level 1, and in which the two trees are very closely matched and have a more symmetric sub-sampling structure, but which preserves the key DT CWT advantages of approximate shift-invariance and good directional selectivity in multiple dimensions.

A distortion measure for blocking artifacts in images based on human visual sensitivity

A visual model that gives a distortion measure for blocking artifacts in images is presented. Given the original and reproduced image as inputs, the model output is a numerical value that 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 that result from blocking. Psychovisual experiments have been carried out to measure the visual sensitivity to these artifacts. In the experiments, typical edge artifacts are shown to subjects and the sensitivity to them is measured with the variation of background luminance, background activity, edge length, and edge amplitude. Synthetic test patterns are used as background images in the experiments. The sensitivity measures thus obtained are used to estimate the model parameters. The final model is tested on real images, and the results show that the error visibility predicted by the model correlates well with the subjective ranking.

The EREC: an error-resilient technique for coding variable-length blocks of data

Many source and data compression schemes work by splitting the input signal into blocks and producing variable-length coded data for each block. If these variable-length blocks are transmitted consecutively, then the resulting coder is highly sensitive to channel errors. Synchronization code words are often used to provide occasional resynchronization at the expense of some added redundant information. This paper introduces the error-resilient entropy code (EREC) as a method for adapting existing schemes to give increased resilience to random and burst errors while maintaining high compression. The EREC has been designed to exhibit graceful degradation with worsening channel conditions. The EREC is applicable to many problems and is particularly effective when the more important information is transmitted near the start of each variable-length block and is not dependent on following data. The EREC has been applied to both still image and video compression schemes, using the discrete cosine transform (DCT) and variable-length coding. The results have been compared to schemes using synchronization code words, and a large improvement in performance for noisy channels has been observed.

Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables

An approach to designing multidimensional linear-phase FIR diamond subband filters having the perfect reconstruction property is presented. It is based on a transformation of variables technique and is equivalent to the generalized McClellan transformation. Methods for designing a whole class of transformation are given. The approach consists of two parts; design of the transformation and design of the 1-D filters. The use of Lagrange halfband filters to design the 1-D filters is discussed. The modification of a particular Lagrange halfband filter which gives a pair of simple 1-D filters that are almost similar to each other in their frequency characteristics but still form a perfect reconstruction pair is presented. The design technique is extended to other types of two-channel sampling lattice and subband shapes, in particular, the parallelogram and the diagonally quadrant subband cases. Several numerical design examples are presented to illustrate the flexibility of the design method.

Motion estimation using a complex-valued wavelet transform

This paper describes a new motion estimation algorithm that is potentially useful for both computer vision and video compression applications. It is hierarchical in structure, using a separable two-dimensional (2-D) discrete wavelet transform (DWT) on each frame to efficiently construct a multiresolution pyramid of subimages. The DWT is based on a complex-valued pair of four-tap FIR filters with Gabor-like characteristics. The resulting complex DWT (CDWT) effectively implements an analysis by an ensemble of Gabor-like filters with a variety of orientations and scales. The phase difference between the subband coefficients of each frame at a given subpel bears a predictable relation to a local translation in the region of the reference frame subtended by that subpel. That relation is used to estimate the displacement field at the coarsest scale of the multiresolution pyramid. Each estimate is accompanied by a directional confidence measure in the form of the parameters of a quadratic matching surface. The initial estimate field is progressively refined by a coarse-to fine strategy in which finer scale information is appropriately incorporated at each stage. The accuracy, efficiency, and robustness of the new algorithm are demonstrated in comparison testing against hierarchical implementations of intensity gradient-based and fractional-precision block matching motion estimators.

Shift invariant properties of the dual-tree complex wavelet transform

We discuss the shift invariant properties of a new implementation of the discrete wavelet transform, which employs a dual-tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (2/sup m/:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses.

Design of Q-shift complex wavelets for image processing using frequency domain energy minimization

This paper proposes a new method of designing finite-support wavelet filters, based on minimization of energy in key parts of the frequency domain. In particular this technique is shown to be very effective for designing families of filters that are suitable for use in the shift-invariant dual-tree complex wavelet structure that has been developed by the author recently, and has been shown to be important for a range of image processing applications. The dual-tree structure requires most of the wavelet filters to have a well-controlled group delay, equivalent to one quarter of a sample period, in order to achieve optimal shift invariance. The proposed new design technique allows this requirement to be included along with the usual smoothness and perfect reconstruction properties to yield wavelet filters with a unique combination of features: linear phase, tight frame, compact spatial support, good frequency domain selectivity with low sidelobe levels, approximate shift invariance, and good directional selectivity in two or more dimensions.

Hidden Markov tree modeling of complex wavelet transforms

Multiresolution signal and image models such as the hidden Markov tree aim to capture the statistical structure of smooth and singular (edgy) regions. Unfortunately, models based on the orthogonal wavelet transform suffer from shift-variance, making them less accurate and realistic. We extend the HMT modeling framework to the complex wavelet transform, which features near shift-invariance and improved angular resolution compared to the standard wavelet transform. The model is computationally efficient (with linear-time computation and processing algorithms) and applicable to general Bayesian inference problems as a prior density for the data. In a simple estimation experiment, the complex wavelet HMT model outperforms a number of high-performance denoising algorithms, including redundant wavelet thresholding (cycle spinning) and the redundant HMT.