This PDF file contains the editorial “Guest Editorial: Special Section on Advances in Mathematical Imaging” for JEI Vol. 6 Issue 04

We consider a cross-section topology that is defined on grayscale images. The main interest of this topology is that it keeps track of the grayscale information of an image. We define some basic notions relative to that topology. Furthermore, we indicate how to acquire a homotopic kernel and a leveling kernel. Such kernels can be seen as "ultimate" topological simplifications of an image. A kernel of a real image, though simplified, is still an intricated image from a topological point of view. We introduce the notion of an irregular region. The iterative removal of irregular regions in a kernel enables us to selectively simplify the topology of the image. Through an example, we show that this notion leads to a method for segmenting some grayscale images without the need to define and tune parameters.

Currently images are more often noisy than regular, and universal contour extraction operators are few. The purely discrete operators are rather crude and inaccurate, while the more precise operators, which relying on the continuous approach, are complicated and rather slow. Using the restriction of gray level images to 3 x 3 masks interpreted as 3D digital surfaces, we present a new way of computing exactly the normal vector, or gradient, at regular points. Despite the fact that regular points are seldom on a general image, we deduce from this theoretical result a new discrete contour extraction operator, based on max area triangles contained in 3 x 3 masks (the MAT operator), which yields high quality results,comparable with those of continuous operators.

An often reoccurring problem in digital image processing is the application of operators from differential geometry to discrete representations of curves and surfaces. We propose the use of feature detectors to improve the estimation of differentials of discrete functions. To this end we replace a differential operator by a bank of feature detectors and difference operators. The purpose of the feature detectors is first to examine the local behavior of the function. Next, depending on the outcome, the feature detectors select the most appropriate difference operator. For example, if the function behaves locally as a linear function, they select a difference operator that is well suited for linear functions. We show that this technique can be put on a firm mathematical basis. In particular, when designing a bank of feature detectors, we use Groebner bases for the functional decomposition and combination of the detectors. We illustrate the mathematical results with several practical examples.

Many imaging systems involve a loss of information that requires the incorporation of prior knowledge in the restoration/reconstruction process. We focus on the typical case of 3D reconstruction from an incomplete set of projections. An approach based on constrained optimization is introduced. This approach provides a powerful mathematical framework for selecting a specific solution from the set of feasible solutions; this is done by minimizing some criteria depending on prior densitometric information that can be interpreted through a generalized support constraint. We propose a global optimization scheme using a deterministic relaxation algorithm based on Bregman's algorithm associated with half-quadratic minimization techniques. When used for 3D vascular reconstruction from 2D digital subtracted angiography (DSA) data, such an approach enables the reconstruction of a well-contrasted 3D vascular network in comparison with results obtained using standard algorithms.

An adaptive approach to restoration of images corrupted by blurring, combined with additive, impulsive and multiplicative noise is proposed. It is based on the combination of nonlinear filters, iterative filtering procedures, and the principles of local adaptation. Numerical simulations and test images illustrating the efficiency of the approach are presented.

Battle-Lemarié's wavelet has a nice generalization in a bivariate setting. This generalization is called bivariate box spline wavelets. The magnitude of the filters associated with the bivariate box spline wavelets is shown to converge to an ideal high-pass filter when the degree of the bivariate box spline functions increases to ∞. The passing and stopping bands of the ideal filter are dependent on the structure of the box spline function. Several possible ideal filters are shown. While these filters work for rectangularly sampled images, hexagonal box spline wavelets and filters are constructed to process hexagonally sampled images. The magnitude of the hexagonal filters converges to an ideal filter. Both convergences are shown to be exponentially fast. Finally, the computation and approximation of these filters are discussed.

A novel and computationally efficient approach to an adaptive mammographic image feature enhancement using wavelet-based multiresolution analysis is presented. On wavelet decomposition applied to a given mammographic image, we integrate the information of the tree-structured zero crossings of wavelet coefficients and the information of the low-pass-filtered subimage to enhance the desired image features. A discrete wavelet transform with pyramidal structure is employed to speedup the computation for wavelet decomposition and reconstruction. The spatiofrequency localization property of the wavelet transform is exploited based on the spatial coherence of image and the principle of human psychovisual mechanism. Preliminary results show that the proposed approach is able to adaptively enhance local edge features, suppress noise, and improve global visualization of mammographic image features. This wavelet-based multiresolution analysis is therefore promising for computerized mass screening of mammograms.

A frame is a family {f_i}^∞_i=1 of elements in a Hilbert space H with the property that every element in H can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications, it is a problem that the calculation requires inversion of an operator on H. The projection method is introduced to avoid this problem. The basic idea is to consider finite subfamilies {f_i}ⁿ_i=1 of the frame and the orthogonal projection P_n onto span {f_i}ⁿ_i=1. For f ⊂ H, P_nf has a representationas a linear combination of f_i, i = 1,2,...,n, and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case, we have avoided theinversion problem. In the same spirit, we approximate the solution to a moment problem. It turns out that the class of "well-behavingframes" are identical for the two problems we consider.

To characterize color values measured by color devices (e.g., scanners, color copiers, and color cameras) in a device-independent fashion these values must be transformed to colorimetric tristimulus values. The measured RGB 3-vectors are not a linear transformation away from such colorimetric vectors, however, but still the best transformation between these two data sets, or between RGB values measured under different illuminants, can easily be determined. Two well-known methods for determining this transformation are the simple least-squares fit (LS) procedure and Vrhel’s principal component method. The former approach makes no a priori statement about which colors will be mapped well and which will be mapped poorly. Depending on the data set a white reflectance may be mapped accurately or inaccurately. In contrast, the principal component method solves for the transform that exactly maps a particular set of basis surfaces between illuminants (where the basis is usually designed to capture the statistics of a set of spectral reflectance data) and hence some statement can be made about which colors will be mapped without error. Unfortunately, even if the basis set fits real reflectances well this does not guarantee good color correction. Here we propose a new, compromise, constrained regression method based on finding the mapping which maps a single (or possibly two) basis surface(s) without error and, subject to this constraint, also minimizes the sum of squared differences between the mapped RGB data and corresponding XYZ tristimuli values. The constrained regression is particularly useful either when it is crucial to map a particular color with great accuracy or when there is incomplete calibration data. For example, it is generally desirable that the device coordinates for a white reflectance should always map exactly to the XYZ tristimulus white. Surprisingly, we show that when no statistics about reflectances are known then a white-point preserving mapping affords much better correction performance compared with the naive least-squares method. Colorimetric results are improved further by guiding the regression using a training set of measured reflectances; a standard data set can be used to fix a white-point-preserving regression that does remarkably well on other data sets. Even when the reflectance statistics are known, we show that correctly mapping white does notincur a large colorimetric overhead; the errors resulting from white-point preserving least-squares fitting and straightforward least-squares are similar.

For signal representation, it is always preferred that a signal be represented using a minimum number of parameters. In any transform coding scheme, the central operation is the reduction of correlation and thereby, with appropriate coding of the transform coefficients, allows data compression to be achieved. The objective of data encoding is to transform a data array into a statistically uncorrelated set. This step is typically considered a "decorrelation" step, because in the case of unitary transformations, the resulting transform coefficients are relatively uncorrelated. Most unitary transforms have the tendency to compact the signal energy into relatively few coefficients. The compaction of energy thus achieved permits a prioritization of the spectral coefficients, with the most energetic ones receiving a greater allocation of encoding bits. The transform efficiency and ease of implementation are to a large extent mutually incompatible. There are various transforms such as Karhunen-Loeve, discrete cosine transforms, etc., but the choice depends on the amount of reconstruction error that can be tolerated and the computational resources available. We apply an approximate Fourier series expansion (AFE) to sampled one-dimensional signals and images, and investigate some mathematical properties. Additionally, we extend the expansion to an approximate cosine expansion (ACE) and show that, for the purpose of data compression with minimum error reconstruction of images, the performance of ACE is better than AFE. For comparison purposes, the results are also compared with a discrete cosine transform (DCT).

Mathematical morphology has produced an important class of nonlinear filters. Unfortunately, design methods existing for these types of filter tend to be computationally intractable or require some expert knowledge of mathematical morphology. Genetic algorithms (GAs) provide useful tools for optimization problems which are made difficult by substantial complexity and uncertainty. Although genetic algorithms are easy to understand and simple to implement in comparison with deterministic design methods, they tend to require long computation times. But the structure of a genetic algorithm lends itself well to parallel implementation and, by parallelization of the GA, major improvements in computation time can be achieved. A method of morphological filter design using GAs is described, together with an efficient parallelization implementation, which allows the use of massively parallel computers or inhomogeneous clusters of workstations.

A two-stage high-precision algorithm for detecting the orientation and position of the surface mount device (SMD) is described. In the preprocessing step, a coarse orientation of the SMD is obtained by line fitting. A high-precision fuzzy Hough transform (FHT) is applied to the corner points to estimate precisely the orientation of the device, with its position determined by using four detected corner points. The FHT employed has a real-valued accumulator over the limited range of angles that is determined in the preprocessing step. Computer simulation with a number of test images shows that the parameters obtained by the presented algorithm are more accurate than those by conventional methods such as the moment method, projection method, and Hough transform methods. It can be applied to fast and accurate automatic inspection and placement systems.

This PDF file contains the editorial “Book Rvw: Computer visualization," by Gallagher for JEI Vol. 6 Issue 04