J.J. Koenderink and A.J. van Doorn, "Two-plus-one-dimensional differential geometry" Pattern Recognition Letters , Vol. 15, pp. 439-443, May 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....nearby scene and an underestimation for the far scene or vice versa. All the distortion, however, takes place only in the Z dimension. Thus the resulting depth function involves an affine transformation. The invariants of these shape maps have been studied in the work of Koenderink and van Doorn [22, 23]. 6 Conclusions An algorithm independent stability analysis of structure from motion has been presented. The analysis did not make any assumptions about the scene, and was based solely on the fact that the depth of the scene in order for the scene to be visible has to be positive. As input ....

J. Koenderink and A. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439--443, 1994.

....and perceptual psychology for a long time. With the advent of digital computers and the possibility of constructing anthropomorphic robotic devices that perceive the world in a way similar to the way humans and animals perceive it, computational studies are beginning to be devoted to this problem [15]. Many synthetic models have been proposed over the years in an attempt to account for the systematic distortion between physical and perceptual space. These range from Euclidean geometry [10] to hyperbolic [18] and affine [25] geometry. Many other interesting approaches have also been proposed, ....

J. Koenderink and A. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439--443, 1994. 25

....A geomorphological ridge is thus the path of steepest ascent leading from a pass to a peak. A geomorphological dale is the path of steepest descent leading from a pass to a pit. This ridge definition has been proposed as a useful feature for image analysis [Koenderink and van Doorn, 1993] [Koenderink and van Doorn, 1994] and has been successfully applied to image segmentation [Griffin et al. 1992] Notably the geomorphological ridge definition makes no mention of ridge shape. Such ridges may be long and sharply peaked in the direction perpendicular to the steepest ascent. They may also be short and show almost ....

....whether a point lies on a geomorphological ridge or not based only on the local shape of the landscape around it. The global character of geomorphological ridges has created some problems and confusion concerning a precise mathematical definition. The articles [Koenderink and van Doorn, 1993] and [Koenderink and van Doorn, 1994] clarify the matter from a modern differential geometric point of view. In particular they discuss some historical attempts to define geomorphological ridges from local properties alone and conclude that this is doomed to fail from the very start . Despite the lacking relation to geomorphological ....

Koenderink, J. and van Doorn, A. (1994). Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15(5):439--444.

....as the height of the terrain, and many interesting features, e.g. cell walls in microscopic images, can be described by watersheds. One of the major categories of approaches to extract watersheds are the ridge detectors, which were first proposed in the early part of the 19th century (see [2, 8] for a historical overview) 1] gives a good overview over the existing classes of ridge detectors. However, it is well known that they do not model the way water runs downhill [2] and can therefore not be used to extract watersheds. Another theory was proposed in the second half of the 19th ....

....extract watersheds are the ridge detectors, which were first proposed in the early part of the 19th century (see [2, 8] for a historical overview) 1] gives a good overview over the existing classes of ridge detectors. However, it is well known that they do not model the way water runs downhill [2], and can therefore not be used to extract watersheds. Another theory was proposed in the second half of the 19th century by Maxwell, Jordan, and Cayley (see [4, 6] and references therein) It is based on the observation that for generic surfaces there is a unique slope line through every ....

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J. J. Koenderink and A. J. van Doorn. Two-plus-onedimensional differential geometry. Pattern Recognition Letters, 15(5):439--443, May 1994.

....a primary cue. Since the agglomerations mainly form blob or ridge like structures (see Figure 2 for two examples) we have decided to use a ridge detector , which builds upon the earlier methods for ridge detection described in (Haralick 1983, Eberly, Gardner, Morse, Pizer Scharlach 1994, Koenderink van Doorn 1994, Pizer, Burbeck, Coggins, Fritsch Morse 1994) and is defined as follows (Lindeberg 1996a) Introduce a local (p; q) coordinate system at each image point, defined by the mixed second order derivative being zero (i.e. L pq = 0) Then, we can detect (possibly elongated, bright) blob features ....

Koenderink, J. J. & van Doorn, A. J. (1994), `Two-plus-one-dimensional differential geometry', Pattern Recognition Letters 15(5), 439--444.

....x0 ffl y0 ffl = Gammafi ffl ff ffl . The exact relationship of fi 0 ffl to x 0 0 ffl is characterized by the locations of local minima of the function A , which in the image processing literature are often referred to as courses. These are not the courses in the topographical sense [30]. To be more precise, we are interested in the local minima of A in the direction corresponding to the largest second derivative. We compute the largest eigenvalue, 1 , of the Hessian, H, of A , that is the matrix of the second derivatives of A with respect to x 0 0 ffl and fi 0 ffl , ....

J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439--443, 1994.

....to accept these parameters as thresholds, and to compute appropriate thresholds for the second derivative from them internally. Finally, it should be stressed that the lines extracted are not ridges in the topographic sense, i.e. they do not define the way water runs downhill or accumulates [7]. In fact, they are much more than a ridge in the sense that a ridge can be regarded in isolation, while a line needs to model its surroundings. If a ridge detection algorithm is used to ex6 (a) Lines detected with bias removal (b) Detail of (a) c) Detail of (a) without bias removal Figure 8. ....

J. J. Koenderink and A. J. van Doorn. Two-plus-onedimensional differential geometry. Pattern Recognition Letters, 15(5):439--443, May 1994.

....will be isolated, they occur quite often in real images and lead to fragmented lines without a semantic reason. Furthermore, the ridge positions found by this operator will often be in wrong positions due to the nature of the differential geometric property used, even for images without noise [4, 17]. In the second sub category, ridges are found at points where one of the principal curvatures of the image assumes a local maximum [18, 15] which is analogous to the approach taken to define ridges in advanced differential geometry [19] For lines with a flat profile it has the problem that two ....

....(12) to obtain the desired thresholds. Again, this is not a severe restriction of the algorithm, but only a matter of convenience. Finally, it should be stressed that the lines extracted are not ridges in the topographic sense, i.e. they do not define the way water runs downhill or accumulates [17, 37]. In fact, they are much more than a ridge in the sense that a ridge can be regarded in isolation, while a line needs to model its surroundings. If a ridge detection algorithm is used to extract lines, the asymmetry of the lines will invariably cause it to return biased results. ....

Jan J. Koenderink and Andrea J. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15(5):439--443, May 1994.

....nearby scene and an underestimation for the far scene or vice versa. All the distortion, however, takes place only in the Z dimension. Thus the resulting depth function involves an affine transformation. The invariants of these shape maps have been studied in the work of Koenderink and van Doorn [22, 23]. 6 Conclusions An algorithm independent stability analysis of structure from motion has been presented. The analysis did not make any assumptions about the scene, and was based solely on the fact that the depth of the scene in order for the scene to be visible has to be positive. As input ....

J. Koenderink and A. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439--443, 1994.

.... of Computer Science, University of British Columbia 5 Department of Computer Science, ETH Zurich The study of the drainage network on smooth mathematical surfaces has a long history, including mathematicians such as Cayley in 1859 [1] and Maxwell in 1870 [12] Koenderink and van Doorn [8, 9] credit Rothe [20] with publishing the first solution in 1915; Koenderink and van Doorn s work expresses this solution in the terminology of modern differential geometry and includes examples of why other attempts at a solution (before and since) are inadequate. The fact that TINs are not smooth ....

J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. Pat. Recog. Letters, 15:439--443, May 1994.

....TINs (triangulated irregular networks) Drainage has also been defined in terms of other surface networks. Smooth mathematical surfaces The study of drainage on smooth mathematical surfaces has a long history, including mathematicians such as Cayley [3] and Maxwell [16] Koenderink and van Doorn [11, 12] credit Rothe [27] with the first characterization of ridges and channels in 1915; Koenderink and van Doorn s work expresses this solution in the terminology of modern differential geometry and includes examples of why other attempts at characterizing drainage on smooth surfaces (especially using ....

....see next, these definitions capture the local definitions in a TIN by Frank et al. 6, 33] and extend them to satisfy our assumptions on global behavior of a drainage network. In smooth terrains these definitions omit some of the special slope lines that satisfy Rothe s differential equations [11, 12] because trickle paths in smooth terrains may merge only in the limit. The expected watercourse at the bottom of a smooth gutter, for example, does not materialize. Rothe s equations do not apply in non differentiable terrains such as TINs, however. 10 30 20 35 10 30 25 15 10 30 20 15 p ....

J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. Pat. Recog. Letters, 15:439--443, May 1994.

....x 0 ffl y 0 ffl = Gammafi ffl ff ffl . The exact relationship of fi 0 ffl to x 0 0 ffl is characterized by the locations of local minima of the function A , which in the image processing literature are often referred to as courses. These are not courses in the topographical sense [26]. To be more precise, we are interested in the local minima of A in the direction corresponding to the largest second derivative. We compute the largest eigenvalue, 1 , of the Hessian, H , of A , that is the matrix of the second derivatives of A with respect to x 0 0 ffl and fi 0 ffl , ....

J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439--443, 1994.

.... Hence, the computation of a drainage network is strongly related to the computation of point and lineal features of the underlying terrain (see [Yu96] for a survey) If we consider a terrain as a continuous surface, a drainage network can be characterized in terms of differential geometry (see [Koe93, Koe94]) There is no accepted definition of the drainage network on an RSG. As noted by Yu et al. Yu96] the network is defined as the result of an algorithmic process. As for the case of feature extraction, local methods based on image processing techniques are applied on RSGs [Dou73, Vin91] Mark ....

Koenderik, J.J., van Doorn, A.J., Two-plus-one-dimensional differential geometry, Pattern Recognition Letters, 15, 1994, pp.439-443.

....distributions of high speed steels (white dots represent carbide particles) The row index (1. 4) denotes the type of the distribution, which corresponds to the shape of the agglomerations. The column index (1. 7) describes the degree, basically reflecting the size of the agglomerations. in [5, 2, 7] is defined as follows [11] Introduce a local (p; q) coordinate system at each image point, defined by the mixed second order derivative being zero (i.e. L pq = 0) Then, we can detect (possibly elongated, bright) blob features from points which are simultaneously maximal with respect to space ....

Koenderink, van Doorn. Two-plus-one dimensional differential geometry, PRL 15(5):439--444, 1994.

.... psychology for a long time [2, 23, 24, 29] With the advent of digital computers and the possibility of constructing anthropomorphic robotic devices that perceive the world in a way similar to the way humans and animals perceive it, computational studies are beginning to be devoted to this problem [20]. Many synthetic models have been proposed over the years in an attempt to account for the systematic distortion between physical and perceptual space. These range from Euclidean geometry [14] to hyperbolic [24] and affine [32, 35] geometry. Many other interesting approaches have also been ....

J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439--443, 1994.

....ridge detection algorithms can be formulated in a similar way. If we follow a differential geometric approach, and define a bright (dark) ridge point as a point for which the brightness assumes a maximum (minimum) in the main principal curvature direction (Haralick 1983; Eberly et al. 1994; Koenderink and van Doorn 1994), then in a local (p; q) system characterized by the mixed second order derivative being zero, this definition can be written 8 : L p = 0; L pp 0; jL pp j jL qq j; or 8 : L q = 0; L qq 0; jL qq j jL pp j: In analogy with section 3, let us first sweep out a ridge surface in ....

J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. PRL, 15(5):439--444, 1994.

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J.J. Koenderink and A.J. van Doorn, "Two-plus-one-dimensional differential geometry" Pattern Recognition Letters , Vol. 15, pp. 439-443, May 1994.

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