on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

fast implementation of the continuous wavelet transform with integer scales we describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m in this approach the input signal and the analyzing wavelet are both represented by polynomial splines the algorithm uses a combination of moving sum and zero padded filters and its complexity per scale is o n where n is the signal length the computation is exact and the implementation is noniterative across scales we also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection first and second derivative operators or gabor like time frequency signal analysis

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

scale space derived from b splines this paper proposes a scale space theory based on b spline kernels our aim is twofold 1 present a general framework and show how b splines provide a flexible tool to design various scale space representations in particular we focus on the design of continuous scale space and dyadic scale space frame representations a general algorithm is presented for fast implementation of continuous scale space at rational scales in the dyadic case efficient frame algorithms are derived using b spline techniques to analyze the geometry of an image the relationship between several scale space approaches is explored the behavior of edge models the properties of completeness causality and other properties in such a scale space representation are examined in the framework of b splines it is shown that besides the good properties inherited from the gaussian kernel the b spline derived scale space exhibits many advantages for modeling visual mechanism including the efficiency compactness orientation feature and parallel structure

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

scale based detection of corners of planar curves a technique for detecting and localizing corners of planar curves is proposed the technique is based on gaussian scale space which consists of the maxima of absolute curvature of the boundary function presented at all scales the scale space of isolated simple and double corners is first analyzed to investigate the behavior of scale space due to smoothing and interactions between two adjacent corners the analysis shows that the resulting scale space contains line patterns that either persist terminate or merge with a neighboring line next the scale space is transformed into a tree that provides simple but concise representation of corners at multiple scales finally a multiple scale corner detection scheme is developed using a coarse to fine tree parsing technique the parsing scheme is based on a stability criterion that states that the presence of a corner must concur with a curvature maximum observable at a majority of scales experiments were performed to show that the scale space corner detector is reliable for objects with multiple size features and noisy boundaries and compares favorably with other corner detectors tested

multiscale curvature based shape representation using b spline wavelets this paper presents a new multiscale curvature based shape representation technique with application to curve data compression using b spline wavelets the evolution of the curve is implemented in the b spline scale space which enjoys a number of advantages over the classical gaussian scale space for instance the availability of fast algorithms the b spline wavelet transforms are used to efficiently estimate the multiscale curvature functions based on the curvature scale space image we introduce a coarse to fine matching algorithm which automatically detects the dominant points and uses them as knots for curve interpolation

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

efficient synthesis of gaussian filters by cascaded uniform filters gaussian filtering is an important tool in image processing and computer vision in this paper we discuss the background of gaussian filtering and look at some methods for implementing it consideration of the central limit theorem suggests using a cascade of simple filters as a means of computing gaussian filters among simple filters uniform coefficient finite impulse response digital filters are especially economical to implement the idea of cascaded uniform filters has been around for a while 13 16 we show that this method is economical to implement has good filtering characteristics and is appropriate for hardware implementation we point out an equivalence to one of burt s methods 1 3 under certain circumstances as an extension we describe an approach to implementing a gaussian pyramid which requires approximately two addition operations per pixel per level per dimension we examine tradeoffs in choosing an algorithm for gaussian filtering and finally discuss an implementation

multiscale corner detection by using wavelet transform a multiscale corner detection algorithm based on the wavelet transform of contour orientation is proposed it can utilize both the information of the local extrema and modulus of transform results to detect corners and arcs effectively the ramp width of contour orientation profile which can be computed using the transformed modulus of two scales reveals the difference between corner and arc and is utilized in the determination of corner points the experimental results have shown that the detector is more effective than both the single and multiple scale detectors they also demonstrate that the detector is insensitive to boundary noise in addition the proposed method is more efficient than the other multiscale corner detector because it operates on fewer number of scales which can be implemented by a fast transform algorithm

spl lambda spl tau space representation of images and generalized edge detector an image and surface representation based on regularization theory is introduced in this paper this representation is based on a hybrid model derived from the physical membrane and plate models the representation called the spl lambda spl tau representation has two dimensions one dimension represents smoothness or scale while the other represents the continuity of the image or surface it contains images surfaces sampled both in scale space and the weighted sobolev space of continuous functions thus this new representation can be viewed as an extension of the well known scale space representation we have experimentally shown that the proposed hybrid model results in improved results compared to the two extreme constituent models i e the membrane and the plate models based on this hybrid model a generalized edge detector ged which encompasses most of the well known edge detectors under a common framework is developed the existing edge detectors can be obtained from the generalized edge detector by simply specifying the values of two parameters one of which controls the shape of the filter spl tau and the other controls the scale of the filter spl lambda by sweeping the values of these two parameters continuously one can generate an edge representation in the spl lambda spl tau space which is very useful for developing a goal directed edge detection scheme for a specific task the proposed representation and the edge detector have been evaluated qualitatively and quantitatively on several different types of image data such as intensity range and stereo images

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

window functions represented by b spline functions characteristics of window functions provided by b spline functions are analyzed in comparison to other known windows for the same highest sidelobe level those windows have a mainlobe bandwidth between those of gaussian and cos sup a x windows are as good as those of cos sup a x ones in the sidelobe falloff

diffusion on the torus for hamiltonian maps for a mapping of the torust2 we propose a definition of the diffusion coefficientd suggested by the solution of the diffusion equation oft2 the definition ofd based on the limit of moments of the invariant measure depends on the set where an initial uniform distribution is assigned for the algebraic automorphism of the torus the limit is proved to exist and to have the same value for almost all initial sets in the subfamily of parallelograms numerical results show that it has the same value for arbitrary polygons and for arbitrary moments

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed