a steerable complex wavelet construction and its application to image denoising this work addresses the design of a novel complex steerable wavelet construction the generation of transform space feature measurements associated with corner and edge presence and orientation properties and the application of these measurements directly to image denoising the decomposition uses pairs of bandpass filters that display symmetry and antisymmetry about a steerable axis of orientation while the angular characterization of the bandpass filters is similar to those previously described the radial characteristic is new as is the manner of constructing the interpolation functions for steering the complex filters have been engineered into a multirate system providing a synthesis and analysis subband filtering system with good reconstruction properties although the performance of our proposed denoising strategy is currently below that of recently reported state of the art techniques in denoising it does compare favorably with wavelet coring approaches employing global thresholds and with an oracle shrinkage technique and presents a very promising avenue for exploring structure based denoising in the wavelet domain

a general weighted median filter structure admitting negative weights weighted median smoothers which were introduced by edgemore in the context of least absolute regression over 100 years ago have received considerable attention in signal processing during the past two decades although weighted median smoothers offer advantages over traditional linear finite impulse response fir filters it is shown in this paper that they lack the flexibility to adequately address a number of signal processing problems in fact weighted median smoothers are analogous to normalized fir linear filters constrained to have only positive weights it is also shown that much like the mean is generalized to the rich class of linear fir filters the median can be generalized to a richer class of filters admitting positive and negative weights the generalization follows naturally and is surprisingly simple in order to analyze and design this class of filters a new threshold decomposition theory admitting real valued input signals is developed the new threshold decomposition framework is then used to develop fast adaptive algorithms to optimally design the real valued filter coefficients the new weighted median filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior weighted median smoother structures

on modifications of newton s second law and linear continuum elastodynamics in this paper we suggest a new perspective where newton s second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid macroscopic bodies we confirm a finding of willis that the density of a body at a given frequency of oscillation can be anisotropic the relation between the force and the acceleration is non local but causal in time conversely for every response function satisfying these properties and having the appropriate high frequency limit there is a model which realizes that response function in many circumstances the differences between newton s second law and the new law are small but there are circumstances such as in specially designed composite materials where the difference is enormous for bodies which are not seemingly rigid the continuum equations of elastodynamics govern behaviour and also need to be modified the modified versions of these equations presented here are a generalization of the equations proposed by willis to describe elastodynamics in composite materials it is argued that these new sets of equations may apply to all physical materials not just composites the willis equations govern the behaviour of the average displacement field whereas one set of new equations governs the behaviour of the averageweighted displacement field where the weighted displacement field may attach zero weight to hidden areas in the material where the displacement may be unobservable or not defined from knowledge of the average weighted displacement field one obtains an approximate formula for the ensemble averaged energy density two other sets of new equations govern the behaviour when the microstructure has microinertia i e where there are internal spinning masses below the chosen scale of continuum modelling in the first set the average displacement field is assumed to be observable while in the second set an average weighted displacement field is assumed to be observable

design of multiparameter steerable functions using cascade basis reduction an efficient method of computing the least squares optimal basis functions to steer any function locally is presented the method combines the lie group theoretic and the singular value decomposition approaches its efficiency is demonstrated with the design of basis functions to steer a gabor function under the four parameter linear transformation group

optimal weighted median filtering under structural constraints a new expression for the output moments of weighted median filtered data is derived the noise attenuation capability of a weighted median filter can now be assessed using the l vector and m vector parameters in the new expression the second major contribution of the paper is the development of a new optimality theory for weighted median filters this theory is based on the new expression for the output moments and combines the noise attenuation and some structural constraints on the filter s behavior in certain special cases the optimal weighted median filter can be obtained by merely solving a set of linear inequalities this leads in some cases to closed form solutions for optimal weighted median filters some applications of the theory developed in this paper in 1 d signal processing and image processing are discussed throughout the analysis some striking similarities are pointed out between linear fir filters and weighted median filters

recursive weighted median filters admitting negative weights and their optimization a recursive weighted median rwm filter structure admitting negative weights is introduced much like the sample median is analogous to the sample mean the proposed class of rwm filters is analogous to the class of infinite impulse response iir linear filters rwm filters provide advantages over linear iir filters offering near perfect stopband characteristics and robustness against noise unlike linear iir filters rwm filters are always stable under the bounded input bounded output criterion regardless of the values taken by the feedback filter weights rwm filters also offer a number of advantages over their nonrecursive counterparts including a significant reduction in computational complexity increased robustness to noise and the ability to model resonant or vibratory behavior a novel recursive decoupling adaptive optimization algorithm for the design of this class of recursive wm filters is also introduced several properties of rwm filters are presented and a number of simulations are included to illustrate the advantages of rwm filters over their nonrecursive counterparts and iir linear filters

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

weighted median filters admitting complex valued weights and their optimization this paper introduces the concept of complex weighted median wm filtering admitting complex weighting unlike previous approaches in the literature that only allowed positive real valued weights the new wm structures exhibit improved performance as they exploit the richness of unrestricted complex weighting to this end the newly defined complex wm structures synthesize filtering operations whereas the prior structures could only attain smoothing properties due to the inherent constraints imposed on the weights in order to overcome the two dimensional 2 d search burden associated with the computation of the complex wm two fast robust and very efficient approximations are introduced adaptive optimization algorithms for their design are developed leading to simple lms type weight updates several simulations are shown illustrating the performance of the new complex wm filter structures

steerable weighted median filters a filter is steerable if transformed i e rotated scaled etc versions of its impulse response can be expressed as linear combinations of a fixed set of basis functions steerability is important for numerous image processing applications however it is a property presently shared only by a specific class of linear filters on the other hand several classes of nonlinear filters such as weighted median filters wmfs may offer certain advantages over linear filters such as robustness and edge preserving capabilities in this paper the concept of steerability is extended to encompass wmfs it will be shown that in general a steerable wmf design technique needs to be capable of handling negative weights although methods that allow the design of wmfs admitting negative weights have already been proposed such methods do not necessarily produce filters that are steerable as opposed to the approach presented in this work experimental results illustrate the applicability of steerable wmfs in two applications namely edge detection and orientation analysis

steerable wedge filters for local orientation analysis steerable filters have been used to analyze local orientation patterns in imagery such filters are typically based on directional derivatives whose symmetry produces orientation responses that are periodic with period spl pi independent of image structure we present a more general set of steerable filters that alleviate this problem

tuning the smoothness of the recursive median filter the median filter is a special case of nonlinear filters used for smoothing signals since the output of the median filter is always one of the input samples it is conceivable that certain signals could pass through the median filter unaltered these signals define the signature of a filter and are referred to as root signals median filters are known to possess the convergence property meaning that by repeating median filtering a root signal will be found starting from any input signal by associating the nonlinear operation of median filtering with a two terms cost function an optimization process that minimizes that function is obtained cost functions of the same type are associated with different recursive median filtering schemes by replacing the actual values inside the filter s window with the original signal the convergence behavior of these filters and their smoothness are studied by changing the positions of the replacements during filtering a tuning effect of the smoothness is obtained simulation results show that the proposed filtering schemes provide improved performance over the standard recursive median filter succeeding in preserving small details and fine textures

iterative truncated arithmetic mean filter and its properties the arithmetic mean and the order statistical median are two fundamental operations in signal and image processing they have their own merits and limitations in noise attenuation and image structure preservation this paper proposes an iterative algorithm that truncates the extreme values of samples in the filter window to a dynamic threshold the resulting nonlinear filter shows some merits of both the fundamental operations some dynamic truncation thresholds are proposed that guarantee the filter output starting from the mean to approach the median of the input samples as a by product this paper unveils some statistics of a finite data set as the upper bounds of the deviation of the median from the mean some stopping criteria are suggested to facilitate edge preservation and noise attenuation for both the long and short tailed distributions although the proposed iterative truncated mean itm algorithm is not aimed at the median it offers a way to estimate the median by simple arithmetic computing some properties of the itm filters are analyzed and experimentally verified on synthetic data and real images

weighted order statistics filters their classification some properties and conversion algorithm stack filters that are based on threshold logic with nonnegative weights and threshold value are called weighted order statistics wos filters and are classified into four different types type 0 trivial type 1 decreasing type 2 increasing and type 3 mixed a significant property of the threshold logic is that it only needs n 1 tuples to represent its function representation if the input of this function has it variables this associative representation is very useful in neural computing nonlinear filtering etc the authors propose a significant classification of threshold logic such that the behaviors of wos filters can be understood easily based on this classification in the paper all type 0 type 1 and type 2 wos filters are shown to possess the convergence property however type 3 wos filters do not necessarily possess the convergence property hence the authors investigate the convergence behaviors of some type 3 wos filters including symmetric weighted median filters and the type 3 filters proposed in wendt 1990 finally an efficient algorithm to determine whether a positive boolean function corresponds to a weighted median filter is proposed the authors use the property all threshold logics are regular to make the algorithm tractable

generalized multichannel image filtering structures recent works in multispectral image processing advocate the employment of vector approaches for this class of signals vector processing operators that involve the minimization of a suitable error criterion have been proposed and shown appropriate for this task in this framework two main classes of vector processing filters have been reported in the literature astola et al 1990 introduce the well known class of vector median filters vmf which are derived as maximum likelihood ml estimates from exponential distributions trahanias et al see ibid vol 2 no 4 p 528 34 1993 and vol 5 no 6 p 868 80 1996 study the processing of color image data using directional information considering the class of vector directional filters vdf we introduce a new filter structure the directional distance filters ddf which combine both vdf and vmf in a novel way we show that ddf are robust signal estimators under various noise distributions they have the property of chromaticity preservation and finally compare favorably to other multichannel image processing filters

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

deformable kernels for early vision early vision algorithms often have a first stage of linear filtering that extracts from the image information at multiple scales of resolution and multiple orientations a common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs a technique is presented that allows 1 computing the best approximation of a given family using linear combinations of a small number of basis functions and 2 describing all finite dimensional families i e the families of filters for which a finite dimensional representation is possible with no error the technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations the relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi scale 2d edge detection kernels the implementation issues are also discussed

design of linear combination of weighted medians this paper introduces a novel nonlinear filtering structure the linear combination of weighted medians lcwm the proposed filtering scheme is modeled on the structure and design procedure of the linear phase fir highpass hp filter in that the linear phase fir hp filter can be obtained by changing the sign of the filter coefficients of the fir lowpass lp filter in the odd positions the hp filter can be represented as the difference between two lp subfilters that have all positive coefficients this representation of the fir hp filter is analogous to the difference of estimates doe such as the difference of medians dom the dom is essentially a nonlinear hp filter that is commonly used in edge detection based on this observation we introduce a class of lcwm filters whose output is given by a linear combination of weighted medians of the input sequence we propose a method of designing the 1 d and 2 d lcwm filters satisfying required frequency specifications the proposed method adopts a transformation from the fir filter to the lcwm filter we show that the proposed lcwm filter can offer various frequency filtering characteristics including lp bandpass bp and hp responses

wavelet based rotational invariant roughness features for texture classification and segmentation we introduce a rotational invariant feature set for texture segmentation and classification based on an extension of fractal dimension fd features the fd extracts roughness information from images considering all available scales at once in this work a single scale is considered at a time so that textures with scale dependent properties are satisfactorily characterized single scale features are combined with multiple scale features for a more complete textural representation wavelets are employed for the computation of single and multiple scale roughness features because of their ability to extract information at different resolutions features are extracted in multiple directions using directional wavelets and the feature vector is finally transformed to a rotational invariant feature vector that retains the texture directional information an iterative k means scheme is used for segmentation and a simplified form of a bayesian classifier is used for classification the use of the roughness feature set results in high quality segmentation performance furthermore it is shown that the roughness feature set exhibits a higher classification rate than other feature vectors presented in this work the feature set retains the important properties of fd based features namely insensitivity to absolute illumination and contrast

combination median filter a detail and structure preserving smoothing filter is introduced it is called the combination median filter as it uses directional median multilevel median and median filters to smooth different regions of the image the decision about the region is made by robust dixon s r test which is well known in statistics for outlier detection the threshold value of dixon s test can be kept constant as a result the filtering algorithm operates like a quasi nonadaptive filter and no computation of local statistics is involved some properties of the filter as well as detailed experimental results that demonstrate its superior performance are presented

weighted median filters for multichannel signals weighted medians over multichannel signals are not uniquely defined due to its simplicity astola s vector median vm has received considerable attention particularly in image processing applications in this paper we show that the vm and its direct extension the weighted vm are limited as they do not fully utilize the cross channel correlation in fact vm treats all sub channel components independent of each other by revisiting the principles of maximum likelihood estimation of location in a multivariate signal space we propose two new and conceptually simple multichannel weighted median filters which can capture cross channel information effectively their optimal filter derivations are also presented followed by a series of simulations from color image denoising to array signal processing where the advantages of the new filtering structures are illustrated

representations of the schrodinger virasoro algebras in this paper it is proved that an irreducible weight module with finite dimensional weight spaces over the schrodinger virasoro algebras is a highest lowest weight module or a uniformly bounded module furthermore indecomposable modules of the intermediate series over these algebras are completely determined