In this paper, we present a novel method to estimate curvature of iso gray-level surfaces in gray-value images. Our method succeeds where standard isophote curvature estimation methods fail. There is neither a segmentation of the surface needed nor a parametric model assumed. Our estimator works on the orientation field of the surface. This orientation field and a description of local structure is obtained by the Gradient Structure Tensor. The estimated orientation field has discontinuities mod pi. It is mapped via the Knutsson mapping to a continuous representation. The principal curvatures of the surface, a coordinate invariant property, are computed in this mapped representation. From these curvatures, locally the bending energy is computed to describe the surface shape. An extensive evaluation shows that our curvature estimation is robust even in the presence of noise, independent of the scale of the object and furthermore the relative error stays small.

In this paper we present a new method for estimating confidence and curvature of 3-D curvilinear structures. The gradient structure tensor (GST) models shift-invariance. The eigenstructure of the tensor allows estimation of local dimensionality, orientation, and the corresponding confidence value. Local rotational invariance, which occurs often in images, causes a lower confidence estimate. This underestimation can be corrected for by a parabolic deformation of the data, in such a way that it becomes translational invariant. We show that the optimal deformation can be found analytically and yields a local curvature estimate as a valuable by-product. We tested our new method on synthetic images and applied it to the detection of channels in 3-D seismic delta.

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.

A method, based on the gradient square tensor (GST), to detect, classify and estimate the size of landmarks based on rods, plates and surfaces in 3D object-background images is described. Scale is automatically adapted to the local situation. Results show that the trace of the GST detects landmarks in composite objects and the determinant detects endpoints of rods. The relation between scale at maximum response and landmark size depends on landmark type. Landmarks can be classified by estimating cylindricality and planarity, derived from the GST-eigenvalues.

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.

The gradient-square tensor describes the orientation dependence of the squared directional derivative in images. The ratio of eigenvalues is a measure of local anisotropy. For an area showing shift invariance along some orientation (think of a piece of straight rail track) one of the tensor eigenvalues is zero. In practical situations (think of a piece of curved rail track) rotation invariance (perhaps around a remote center) occurs more often than shift invariance. Then curvature contributes to the smallest eigenvalue. In order to avoid this we deform a local area in such a way that the rotational symmetry becomes a translational one. Next the gradient square tensor defined on the transformed area, is expressed in derivatives of the original area. A curvature corrected anisotropy measure is defined. The correction turns out to be simple and straightforward. An average-curvature estimate for the area results as a valuable by product.

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.

We introduce a smart image sensor, the PSD-chip, designed for sheet-of-light range imaging. The sensor area consists of an array of position sensitive detector (PSD)-strips. The on-chip signal processing electronics is built up from both analog and digital circuitry. Our aim is to be able to record 1.5 million range values (rangels) per second at a 12-bits resolution.

The topography (not the field of radius values) of the medial axis of a convex d-dimensional polyhedron can be represented by: (1) The (2- to (d+1)-fold) intersections of internal Voronoi cell polyhedron hyperfaces. (2) The locus of first derivation discontinuities on the distance transform (DT). (3) A cross-linked tree of intersections, with the DT maximum as root, the polyhedron hyperfaces and edges as leaves. (4) The medial framework, the sub-tree of 1D intersections, with the edges of the DT maximum as root, the polyhedron edges as leaves. Based on representation 1 the method proposed locates ("constructs")-starting from the polyhedron edges, guided by representation 3, in order of increasing DT value-the intersections of representation 4. The 2D complexity is O(n/sub e/ log n/sub e/), with n/sub e/ the number of polygon edges. The 3D complexity is O(n/sub c/ log n/sub c/)

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.>

A metric defines the distance between any two points. The “natural” metrics of the digital world do not approximate the Euclidean metric of the continuous world well. Skeletonization (sometimes named topology preserving shrinking or homotopic thinning) is one example in which this leads to unacceptable results. In the present work we propose and demonstrate skeletonization using path-based metrics which are a better approximation of the Euclidean metric. Moreover, we achieve a good performance on sequential processors by processing each pixel only once in the calculations of binary (Hilditch) and grey-value (upper) skeletons.

Existing estimators for edge length in 2D and surface area in 3D are applied to a binary representation of the object. The authors estimate length and area through volume measurements. Volume is measured without thresholding and does not introduce a sampling error. Edges are transformed into volumes by giving them a constant height after which they can be shifted perpendicular to the edge over a small distance. Subtraction of two images shifted in opposite direction produces a volume that is proportional to the edge length. To guarantee a constant edge height along the edge or across the surface to be measured the authors apply a 'soft' clipping operation to the linear region of the edge. For accurate and isotropic displacement of edges the authors introduce a continuous space equivalent to the discrete local minimum and maximum filters. These filters are sampling invariant and allow shifts in the subpixel region.>

Sequential (search) methods for edge and line detection and parallel (filter) methods for edge detection are well-known. Filter methods for line detection are generally based on direction-by-direction filtering: each filter is parallel, but only special hardware architectures allow filtering in different directions simultaneously. The authors present a class of isotropic edge and line detection filters that work in arbitrary dimension: symmetry filters. The principle of the method is correlation of the filter window content with a mapped version. The mapping may be point inversion, blow or rotation, all relative to the window center. Covariance is used as correlation measure. In the paper only point inversion is considered. Moreover, the filter may contain weighting coefficients e.g. to emphasize contributions from the central part. For point inversion, lines give a positive correlation and edges a negative correlation (twofold symmetry and antisymmetry).>

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.>

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.