The standard interpolation approach to image resizing is to fit the original picture with a continuous model and resample the function at the desired rate. However, one can obtain more accurate results if one applies a filter prior to sampling, a fact well known from sampling theory. The optimal solution corresponds to an orthogonal projection onto the underlying continuous signal space. Unfortunately, the optimal projection prefilter is difficult to implement when sine or high order spline functions are used. We propose to resize the image using an oblique rather than an orthogonal projection operator in order to make use of faster, simpler, and more general algorithms. We show that we can achieve almost the same result as with the orthogonal projection provided that we use the same approximation space. The main advantage is that it becomes perfectly feasible to use higher order models (e.g. splines of degree n=or>3). We develop the theoretical background and present a simple and practical implementation procedure using B-splines. Our experiments show that the proposed algorithm consistently outperforms the standard interpolation methods and that it provides essentially the same performance as the optimal procedure (least squares solution) with considerably fewer computations. The method works for arbitrary scaling factors and is applicable to both image enlargement and reduction.

We propose to re-size images using an oblique projection operator instead of the orthogonal one in order to obtain faster, simpler, and more general algorithms. The main advantage is that it becomes perfectly feasible to use higher order models (e.g., splines of degree n/spl ges/3). We develop the theoretical background and present a simple and practical implementation procedure that uses B-splines. Experiments show that the proposed algorithm consistently outperforms the standard interpolation method and that it essentially provides the same performance as the optimal procedure (least squares solution) with considerably less computations.

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 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.>

The least-squares polynomial spline approximation of a signal g(t) in L/sub 2/(R) is obtained by projecting g(t) on S/sup n/(R) (the space of polynomial splines of order n). It is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g(t) with a B-spline of order n. More specifically, the coefficients of the B-spline interpolation of order 2n+1 of the sampled filtered sequence are identical to the coefficients of the least-squares approximation of g(t) of order n. It is shown that this approximation can be obtained from a succession of three basic operations: prefiltering, sampling, and postfiltering, which confirms the parallel with the classical sampling/reconstruction procedure for bandlimited signals. The frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift-invariant basis functions of S/sup n/(R): the standard expansion in terms of B-spline coefficients, a representation in terms of sampled signal values, and a representation using orthogonal basis 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>

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.>

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.>

An unsupervised texture segmentation system using texture features obtained from a combination of spatial filters and nonlinear operators is described. Local texture features are evaluated in parallel by a succession of four basic operations: (1) a convolution for local structure detection (local linear transform); (2) a first nonlinearity of the form f(x)= mod x mod /sup alpha /; (3) an iterative smoothing operator; and (4) a second nonlinearity g(x). The Karhunen-Loeve transform is used to reduce the dimensionality of the resulting feature vector, and segmentation is achieved by thresholding or clustering in feature space. The combination of nonlinearities f(x)= mod x mod /sup alpha / (in particular, alpha =2) and g(x)=log x maximizes texture discrimination, and results in a description with variances approximately constant for all feature components and texture regions. This latter property improves the performance of both feature reduction and clustering algorithms significantly.>

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. Fisher's (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.< >

The authors present a linear feature-reduction technique for multicomponent or textured image segmentation. The transformation matrix is computed by simultaneously diagonalizing scatter matrices evaluated at two different spatial resolutions. Under reasonable conditions, this transform closely approximates the generalized Fisher linear discriminants which are optimal for region separability. Experimental examples suggest that this technique is superior to the Karhunen-Loeve transform for texture segmentation.< >

Handwriting can be characterized as a sequence of basic strokes connected according to rule. Handwriting so generated approximates that of humans very closely. Such a matching process can be used as the fundamental principle in a handwriting recognizer.