Murray Eden

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. >

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. >

Fast B-spline transforms for continuous image representation and interpolation

Efficient algorithms for the continuous representation of a discrete signal in terms of B-splines (direct B-spline transform) and for interpolative signal reconstruction (indirect B-spline transform) with an expansion factor m are described. Expressions for the z-transforms of the sampled B-spline functions are determined and a convolution property of these kernels is established. It is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering. Fast computational algorithms based on the recursive implementations of these filters are proposed. A B-spline interpolator can also be characterized in terms of its transfer function and its global impulse response (cardinal spline of order n). The case of the cubic spline is treated in greater detail. The present approach is compared with previous methods that are reexamined from a critical point of view. It is concluded that B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner. >

Analysis of emission tomographic scan data: limitations imposed by resolution and background

The proper analysis of positron emission tomographic scan data requires a careful knowledge of the limitations of the tomographic system used so that scan data can be collected and sampled in a manner consistent with those limitations. The present investigation was undertaken to clarify some of the limitations imposed by resolution. The usual imaging situation, e.g., 218FDG , C15O2, or 15O2 , involves imaging structures of limited size in all three dimensions which may appear either warm or cool in relation to some background level of activity. In emission tomography the importance of adequate data sampling within a given plane has been frequently emphasized. Little attention, however, has been given to proper z axis sampling for clinical scanning. The actual selection of regions of interest from scans can have a significant impact on the subsequent statistical analysis. Previous work on this subject has experimentally examined the relationship of object size to quantitative estimation in the hot spot-cold background situation for the one- and two-dimensional cases. Approximate three-dimensional recovery coefficients for the hot spot-cold background situation have been calculated. An examination of the factors discussed above, three-dimensional objects with varying contrast, z axis sampling, and selection of regions of interest, has not yet been addressed in the literature. The purpose of the present investigation is to examine these factors.

Multiresolution feature extraction and selection for texture segmentation

An approach is described for unsupervised segmentation of textured images. Local texture properties are extracted using local linear transforms that have been optimized for maximal texture discrimination. Local statistics (texture energy measures) are estimated at the output of an equivalent filter bank by means of a nonlinear transformation (absolute value) followed by an iterative Gaussian smoothing algorithm. This procedure generates a multiresolution sequence of feature planes with a half-octave scale progression. A feature reduction technique is then applied to the data and is determined by simultaneously diagonalizing scatter matrices evaluated at two different spatial resolutions. This approach provides a good approximation of R.A. Fishers (1950) multiple linear discriminants and has the advantage of requiring no a priori knowledge. This feature reduction methods appears to be an improvement on the commonly used Karhunen-Loeve transform and allows efficient texture segmentation based on simple thresholding. >

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 >

A family of polynomial spline wavelet transforms

Abstract This paper presents an extension of the family of orthogonal Battle/Lemarie spline wavelet transforms with emphasis on filter bank implementation. Spline wavelets that are not necessarily orthogonal within the same resoluton level, are constructed by linear combination of polynomial spline wavelets of compact support, the natural counterpart of classical B-spline functions. Mallats fast wavelet transform algorithm is extended to deal with these non-orthogonal basis functions. The impulse and frequency responses of the corresponding analysis and synthesis filters are derived explicitly for polynomial splines of any order n (n odd). The link with the general framework of biorthogonal wavelet transforms is also made explicit. The special cases of orthogonal, B-spline, cardinal and dual wavelets are considered in greater detail. The B-spline (respectively dual) representation is associated with simple FIR binomial synthesis (respectively analysis) filters and recursive analysis (respectively synthesis) filters. The cardinal representation provides a sampled representation of the underlying continuous functions (interpolation property). The distinction between cardinal and orthogonal representation vanishes as the order of the spline is increased; both wavelets tend asymptotically to the bandlimited sinc-wavelet. The distinctive features of these various representations are discussed and illustrated with a texture analysis example.

Recursive regularization filters: design, properties, and applications

Least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed. For each problem, there is a family of discrete regularization filters (R-filters) which allow an efficient determination of the solutions. These operators are stable symmetric lowpass filters with an adjustable scale factor. Two decomposition theorems for the z-transform of such systems are presented. One facilitates the determination of their impulse response, while the other allows an efficient implementation through successive causal and anticausal recursive filtering. A case of special interest is the design of R-filters for the first- and second-order difference operators. These results are extended for two-dimensional signals and, for illustration purposes, are applied to the problem of edge detection. This leads to a very efficient implementation (8 multiplies+10 adds per pixel) of the optimal Canny edge detector based on the use of a separable second-order R-filter. >

Enlargement or reduction of digital images with minimum loss of information

The purpose of this paper is to derive optimal spline algorithms for the enlargement or reduction of digital images by arbitrary (noninteger) scaling factors. In our formulation, the original and rescaled signals are each represented by an interpolating polynomial spline of degree n with step size one and Delta, respectively. The change of scale is achieved by determining the spline with step size Delta that provides the closest approximation of the original signal in the L(2)-norm. We show that this approximation can be computed in three steps: (i) a digital prefilter that provides the B-spline coefficients of the input signal, (ii) a resampling using an expansion formula with a modified sampling kernel that depends explicitly on Delta, and (iii) a digital postfilter that maps the result back into the signal domain. We provide explicit formulas for n=0, 1, and 3 and propose solutions for the efficient implementation of these algorithms. We consider image processing examples and show that the present method compares favorably with standard interpolation techniques. Finally, we discuss some properties of this approach and its connection with the classical technique of bandlimiting a signal, which provides the asymptotic limit of our algorithm as the order of the spline tends to infinity.

The L/sub 2/-polynomial spline pyramid

The authors are concerned with the derivation of general methods for the L/sub 2/ approximation of signals by polynomial splines. The main result is that the expansion coefficients of the approximation are obtained by linear filtering and sampling. The authors apply those results to construct a L/sub 2/ polynomial spline pyramid that is a parametric multiresolution representation of a signal. This hierarchical data structure is generated by repeated application of a REDUCE function (prefilter and down-sampler). A complementary EXPAND function (up-sampler and post-filter) allows a finer resolution mapping of any coarser level of the pyramid. Four equivalent representations of this pyramid are considered, and the corresponding REDUCE and EXPAND filters are determined explicitly for polynomial splines of any order n (odd). Some image processing examples are presented. It is demonstrated that the performance of the Laplacian pyramid can be improved significantly by using a modified EXPAND function associated with the dual representation of a cubic spline pyramid. >