P.W. Verbeek

Three-Dimensional Skeletonization: Principle and Algorithm

An algorithm is proposed for skeletonization of 3-D images. The criterion to preserve connectivity is given in two versions: global and local. The latter allows local decisions in the erosion process. A table of the decisions for all possible configurations is given in this paper. The algorithm using this table can be directly implemented both on general purpose computers and on dedicated machinery.

Recursive Gaussian derivative filters

We propose a strategy to design recursive implementations of the Gaussian filter and Gaussian regularized derivative filters. Each recursive filter consists of a cascade of two stable Nth-order subsystems (causal and anti-causal). The computational complexity is 2N multiplications per pixel per dimension independent of the size (/spl sigma/) of the Gaussian kernel. The filter coefficients have a closed-form solution as a function of scale (/spl sigma/) and recursion order N (N=3, 4, 5). The recursive filters yield a high accuracy and excellent isotropy in n-D space.

An efficient uniform cost algorithm applied to distance transforms

The uniform-cost algorithm is a special case of the A*-algorithm for finding the shortest paths in graphs. In the uniform-cost algorithm, nodes are expanded in order of increasing cost. An efficient version of this algorithm is developed for integer cost values. Nodes are sorted by storing them at predefined places (bucket sort), keeping the overhead low. The algorithm is applied to general distance transformation. A constrained distance transform is an operation which calculates at each pixel of an image the distance to the nearest pixel of a reference set, distance being defined as minimum path length. The uniform-cost algorithm, in the constrained case, proves to be the best solution for distance transformation. It is fast, the processing time is independent of the complexity of the image, and memory requirements are moderate. >

Theory of confocal fluorescence imaging in the Programmable Array Microscope (PAM).

The programmable array microscope (PAM) uses a spatial light modulator (SLM) to generate an arbitrary pattern of conjugate illumination and detection elements. The SLM dissects the fluorescent light imaged by the objective into a focal conjugate image, Ic, formed by the ‘in‐focus’ light, and a nonconjugate image, Inc, formed by the ‘out‐of‐focus’ light. We discuss two different schemes for confocal imaging using the PAM. In the first, a grid of points is shifted to scan the complete image. The second, faster approach, uses a short tiled pseudorandom sequence of two‐dimensional patterns. In the first case, Ic is analogous to a confocal image and Inc to a conventional image minus Ic. In the second case Ic and Inc are the sum and the difference, respectively, of a conventional and a confocal image. The pseudorandom sequence approach requires post‐processing to retrieve the confocal part, but generates significantly higher signal levels for an equivalent integration time.

Edge preserving orientation adaptive filtering

In this paper we describe a new stragegy for combining orientation adaptive filtering and edge preserving filtering. The filter adapts to the local orientation and avoids filtering across borders. The local orientation for steering the filter will be estimated in a fixed sized window which never contains two orientation fields. This can be achieved using generalized Kuwahara filtering. This filter selects from a set of fixed sized windows that contain the current pixel, the orientation of the window with the highest anisotropy. We compare out filter strategy with a multi-scale approach. We found that our filter strategy has a lower complexity and yields a constant improvement of the SNR.

Low-level image processing by max–min filters

Abstract A systematic framework is given that accommodates existing max-min filter methods and suggests new ones. Putting the upper and lower envelopes UPP = MIN (MAX) and LOW = MAX (MIN) in the roles that MAX, MIN or original play in existing filters we can distinguish edges in ramp edges and texture (or noise) edges; all methods presented come in three versions: for edges, ramp edges and non-ramp (“texture”) edges. The ramp versions of Philips dynamic thresholding and Lee edge detection are considerably less noise sensitive. For images with little noise the texture version of dynamic thresholding brings out fine textures while ignoring ramps. Lee edge-detection can in all versions be extended to a sharp “Laplacian” and an edge enhancer. Starting out from square-full several shapes of the maximum filter are tried out. The round-full filter gives least artefacts; when crescent updating is used it takes size-linear rather than size-quadratic time. The suboptimal round-sparse filter takes size-independent time.

Shading from shape, the eikonal equation solved by grey-weighted distance transform

Abstract For an optical or acoustical wavefront running through a medium of space variant refraction index the eikonal equation connects local front arrival time with local refraction index. So-called difference approximation methods are known for solving the spatial wavefront development with time and thus, indirectly, the eikonal equation. Here a novel fast method for the calculation of an approximative solution of the eikonal equation is proposed. From literature it is known that by solving an eikonal equation one can construct a line pattern rendition of a given image. We have generalized this method and made it fit for line engravings. We have found yet another kind of image display based on solving an eikonal equation: shading from shape. We propose to construct a matte 3-D surface (shape) that, when illuminated perpendicularly and imaged in eye or camera, yields a grey value (shading, luminance) field that renders the image. Both methods have been applied in a recent design for a Dutch coin.

On the location error of curved edges in low-pass filtered 2-D and 3-D images

The authors study the location error of curved edges in two- and three-dimensional images after analog and digital low-pass filtering. The zero crossing of a second derivative filter is a well-known edge localization criterion. The second derivative in gradient direction (SDGD) produces a predictable bias in edge location towards the centers of curvature while the linear Laplace filter produces a shift in the opposite direction. Their sum called PLUS (PLUS=Laplace+SDGD) leads to an edge detector that finds curved edges one order more accurately than its constituents. This argument holds irrespective of the dimension. The influence of commonly used low-pass filters (such as the PSF originating from diffraction limited optics using incoherent light (2-D), the Gaussian filter with variable cutoff point (D-D), and the isotropic uniform filter (D-D)) is studied. >

Curvature estimation in oriented patterns using curvilinear models applied to gradient vector fields

Curved oriented patterns are dominated by high frequencies and exhibit zero gradients on ridges and valleys. Existing curvature estimators fail here. The characterization of curved oriented patterns based on translation invariance lacks an estimation of local curvature and yields a biased curvature-dependent confidence measure. Using parameterized curvilinear models we measure the amount of local gradient energy along the model gradient as a function of model curvature. Minimizing the residual energy yields a closed-form solution for the local curvature estimate and the corresponding confidence measure. We show that simple curvilinear models are applicable in the analysis of a wide variety of curved oriented patterns.

Implementation of cellular-logic operators using 3∗3 convolution and table lookup hardware

Abstract Cellular-logic operations such as erosion, dilation, contour extraction, skeletonization, local majority voting, and pepper-and-salt noise removal are essential in processing binary images. It is shown that cellular-logic operations, like some homomorphic filters, can be constructed from a 3∗3 convolution and a nonlinear table lookup, features of many commercially available image-processing systems. The proposed method extends the field of application of such systems from enhancement and other preprocessing of gray-valued images to the processing and measurement of objects in the segmented image.