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Neural Correlates of Boundary and Medial Axis Representations in Primate Visual Cortex
... Symmetry sets and medial axes found their way beyond computer (or "artificial") vision. Symmetry is in general an attractive feature for human vision [22]. 2 Recent results show that symmetry sets may playa role in biological vision as well [18,19,20]. Both psychophysical and physiological studies show that biological visual systems might have better performance when observing points in certain positions on the medial axis. ...
... The line M joins the intersection of the tangent at ,(t) and the x-axis to the midpoint of the segment from (0,0) to ,(t). The function 0 will now be given by eliminating a, b, h, f and will be the 4 X 4 determinant O(t,p,q):= C(t) Ct(t) Cx(p, q) Cy(p, q) (19) In (p, q) coordinates 0 = 0 is the equation of the locus of centers of the conics tangent to , at ,(t) and tangent to the x-axis at the origin. Writing ,(t) = (X(t), Y(t)) we have ...
Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.
... Such representations have found wide use in computer vision, image analysis, graphics, and computer aided design (Bloomenthal and Shoemake (1991); Storti et al. (1997)). Psychophysical and neurophysiological studies have shown evidence that medial relationships play an important role in the human visual system (Leyton (1992); Lee et al. (1995)). The medial locus was first proposed by Blum (1967), and its properties were later studied in 2D by Blum and Nagel (1978) and in 3D by Nackman and Pizer (1985). ...
... Many authors, in image analysis, geometry, human vision, computer graphics, and mechanical modeling, have come to the understanding, originally promulgated by Blum (1967), that the medial relationship 1 between points on opposite sides of a figure (Fig. 1) is an important factor in the object's geometric descrip- tion. Biederman (1987), Marr (1978), Burbeck (1996, Leyton (1992), Lee (1995, and others have produced psychophysical and neurophysiological evidence for the importance of medial relationships (in 2D projection ) in human vision. The relation has also been explored in 3D by Nackman (1985), Vermeer (1994, and Siddiqi (1999) , and medial axis modeling techniques have been applied by many researchers, includ- ing Bloomenthal (1991), Wyvill (1986), Singh (1998), Amenta (1998), Bittar (1995, Igarashi (1999) and Markosian (1999). ...
M-reps (formerly called DSLs) are a multiscale medial means for modeling and rendering 3D solid geometry. They are particularly well suited to model anatomic objects and in particular to capture prior geometric information effectively in deformable models segmentation approaches. The representation is based on figural models, which define objects at coarse scale by a hierarchy of figures - each figure generally a slab representing a solid region and its boundary simultaneously. This paper focuses on the use of single figure models to segment objects of relatively simple structure. A single figure is a sheet of medial atoms, which is interpolated from the model formed by a net, i.e., a mesh or chain, of medial atoms (hence the name m-reps), each atom modeling a solid region via not only a position and a width but also a local figural frame giving figural directions and an object angle between opposing, corresponding positions on the boundary implied by the m-rep. The special capability of an m-rep is to provide spatial and orientational correspondence between an object in two different states of deformation. This ability is central to effective measurement of both geometric typicality and geometry to image match, the two terms of the objective function optimized in segmentation by deformable models. The other ability of m-reps central to effective segmentation is their ability to support segmentation at multiple levels of scale, with successively finer precision. Objects modeled by single figures are segmented first by a similarity transform augmented by object elongation, then by adjustment of each medial atom, and finally by displacing a dense sampling of the m-rep implied boundary. While these models and approaches also exist in 2D, we focus on 3D objects. The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper. The accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.
... Object recognition is an essential task in image processing , and the skeleton or medial axis is a shape descriptor that is often used for this task. Thus, the computation of skeletons and symmetry sets of planar shapes is a subject that received a great deal of attention from the mathematical (see Bruce et al., 1985; Bruce and Giblin, 1992 and references therein), computational geometry (Preparata and Shamos, 1990), biological vision (Kovács and Julesz, 1994; Lee et al., 1995; Leyton, 1992 ), and computer vision communities (see for example Ogniewicz, 1993; Serra, 1982 and references therein) since the original work by Blum (1967 Blum ( , 1973). In the classical Euclidean case, the symmetry set of a planar curve (or of the boundary of a planar shape) is defined as the set of points equidistant from at leastFigure 1. Sketch of a Euclidean symmetry set. ...
A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.
... The distinction between boundary and area contrast has been around for some time (Hering, 1878;Mach, 1865) and has been attributed to "Mach-type" and "Hering-type" lateral inhibition by von Bekesy (1969). More recently, it has been correlated with cortical neural responses to the boundary of a stimulus that precede, by about 100 msec, the neural responses to the interior ofthe stimulus (T. Lee, Mumford, & Schiller, 1995). Paradiso and Nakayama (1991) investigated the spatial and temporal parameters affecting brightness perception and filling-in, using a forward I backward masking paradigm. ...
Visual backward masking not only is an empirically rich and theoretically interesting phenomenon but also has found increasing
application as a powerful methodological tool in studies of visual information processing and as a useful instrument for investigating
visual function in a variety of specific subject populations. Since the dual-channel, sustained-transient approach to visual
masking was introduced about two decades ago, several new models of backward masking and metacontrast have been proposed as
alternative approaches to visual masking. In this article, we outline, review, and evaluate three such approaches: an extension
of the dual-channel approach as realized in the neural network model of retino-cortical dynamics (Ogmen, 1993), the perceptual
retouch theory (Bachmann, 1984, 1994), and the boundary contour system (Francis, 1997; Grossberg & Mingolla, 1985b). Recent
psychophysical and electrophysiological findings relevant to backward masking are reviewed and, whenever possible, are related
to the aforementioned models. Besides noting the positive aspects of these models, we also list their problems and suggest
changes that may improve them and experiments that can empirically test them.
... This process could operate on visual information at a different spatial scales (Burbeck & Pizer, 1995;Burbeck & Zauberman, 1997). Also, there is some psychophysical and neurophysiological evidence that the visual system computes implicit or "medial" axes running through the center of shapes (Kovács & Julesz, 1994;Lee, Mumford, & Schiller, 1995). However, at present, it is unclear whether the detection of axes of elongation in multi-item displays (even on the basis of coarse "blob-like" information) can occur without selective attention. ...
Three experiments investigated the role of the global spatial structure of two-dimensional (2-D) shapes in terms of symmetry
and elongation on visual search for shape orientation. Experiment 1 demonstrated the often reported orientation search asymmetry
(i.e., a faster detection of a tilted target among vertical distractors than the reverse) for the global orientation of 2-D
polygons that possess a salient, “principal” axis of symmetry or elongation. Moreover, the search asymmetry depended on the
orientation of the principal axis, rather than on the orientation of local contours. Further exploration of this effect with
polygons (Experiment 2) showed that the search asymmetry for global orientation occurred for shapes containing an axis of
symmetry; elongation, on the other hand, did not seem to be crucial. Finally, Experiment 3 demonstrated orientation search
asymmetries with shapes composed of curved rather than straight contours: Here, the search asymmetry occurred as a function
of the orientation of both axes of symmetry and elongation. Overall, search for global orientation was less efficient than
search for local orientation. The results suggest that the perception of the global orientation of shapes is mediated by axis-based
descriptions in terms of perceptually salient axes of symmetry and elongation.
Many animals are challenged with the task of reorientation. Considerable research over the years has shown a diversity of species extract geometric information (e.g., distance and direction) from continuous surfaces or boundaries to reorient. How this information is extracted from the environment is less understood. Three encoding strategies that have received the most study are the use of principal axes, medial axis or local geometric cues. We used a modeling approach to investigate which of these three general strategies best fit the spatial search data of a highly-spatial corvid, the Clark's nutcracker (Nucifraga columbiana). Individual nutcrackers were trained in a rectangular-shaped arena, and once accurately locating a hidden goal, received non-reinforced tests in an L-shaped arena. The specific shape of this arena allowed us to dissociate among the three general encoding strategies. Furthermore, we reanalyzed existing data from chicks, pigeons and humans using our modeling approach. Overall, we found the most support for the use of the medial axis, although we additionally found that pigeons and humans may have engaged in random guessing. As with our previous studies, we find no support for the use of principal axes.
Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) three-point contact with two or more distinct points on the curve. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case.
The analysis of higher level global visual attributes such as geometric shapes is thought to be a function of specialist modules in the extrastriate cortices. However, V1 is unique because only in this area can one find cells tuned to orientation and other local features with the spatial precision needed for representing high resolution and geometric aspects of an image. Geometric computation that demands such resolution may need to involve V1 via feedback. V1 neurons are known to exhibit very different types of responses in the short latency (40–80 milliseconds post-stimulus time window) versus longer latencies (80–200 milliseconds)1–5 Here we report that the later part of V1 neurons’ responses is sensitive to geometric attributes of globally defined shapes. We propose a computational interpretation of these results.
In this paper we discuss a new approach to compute discrete skeletons of planar shapes which is based on affine distances, being therefore affine invariant. The method works with generic curves that may contain concave sections. A dynamical interpretation of the affine skeleton construction, based on curve evolution, is discussed as well. We propose an efficient implementation of the method and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.
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