Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces

Pottmann, Helmut; Opitz, Karsten

doi:10.1016/0167-8396(94)90057-4

Cited by 35 publications

(20 citation statements)

References 16 publications

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“…Introductions and summaries of the geometry relevant to the relatively basic aspects that we use in this paper (Sects. 2.1.2 and especially 2.2.2, with sufficient local explanations) are available (Pottmann and Opitz 1994;Koenderink and van Doorn 2002), but one should note that these papers use the geometry differently, as befits the very different applications they aim at.…”

Section: A1 Notation and Coordinate Transformations (Zeroth Order)mentioning

confidence: 97%

“…Most of the formal geometry we used in our present model was developed mathematically in the 1940s (e.g. Strubecker 1942) forms the basis for our application), and it has occasionally been used in computer-vision (Pottmann and Opitz 1994;Koenderink and van Doorn 2002) and in visual science (Koenderink 2003), but for different purposes and in different guises -always because it correctly respects the special role of one dimension (say, of visual rays) with respect to that of others (say, the two dimensions of the visual field). The fact that it allows metric shape (i.e.…”

Section: Completing and Extending The Modelmentioning

confidence: 99%

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Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields

Noest

Berg

2006

Biol Cybern

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Any computation of metric surface structure from horizontal disparities depends on the viewing geometry, and analysing this dependence allows us to narrow down the choice of viable schemes. For example, all depth-based or slant-based schemes (i.e. nearly all existing models) are found to be unrealistically sensitive to natural errors in vergence. Curvature-based schemes avoid these problems and require only moderate, more robust view-dependent corrections to yield local object shape, without any depth coding. This fits the fact that humans are strikingly insensitive to global depth but accurate in discriminating surface curvature. The latter also excludes coding only affine structure. In view of new adaptation results, our goal becomes to directly extract retinotopic fields of metric surface curvatures (i.e. avoiding intermediate disparity curvature). To find a robust neural realisation, we combine new exact analysis with basic neural and psychophysical constraints. Systematic, step-by-step 'design' leads to neural operators which employ a novel family of 'dynamic' receptive fields (RFs), tuned to specific (bi-)local disparity structure. The required RF family is dictated by the non-Euclidean geometry that we identify as inherent in cyclopean vision. The dynamic RF-subfield patterns are controlled via gain modulation by binocular vergence and version, and parameterised by a cell-specific tuning to slant. Our full characterisation of the neural operators invites a range of new neurophysiological tests. Regarding shape perception, the model inverts widely accepted interpretations: It predicts the various types of errors that have often been mistaken for evidence against metric shape extraction.

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Section: A1 Notation and Coordinate Transformations (Zeroth Order)mentioning

confidence: 97%

Section: Completing and Extending The Modelmentioning

confidence: 99%

Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields

Noest

Berg

2006

Biol Cybern

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“…It is a well-researched geometry [35,36,37,38,39,40]. English textbooks are available [31,30,46]. In the work of Pottmann and Liu [30] on architectural design, the isotropic direction is set by the direction of gravity, similar to the depth dimension in our application.…”

Section: Pictorial Space As a Homogeneous Spacementioning

confidence: 99%

Gauge Fields in Pictorial Space

Koenderink¹,

Doorn²

2012

SIAM J. Imaging Sci.

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Abstract. "Pictorial space" is the mental structure that appears to be the scaffold for the visual awareness when looking "into" (as opposed to "at") a picture. Its structure differs from the "visual space" that is the scaffold for the visual awareness when looking into the scene in front of the observer. The structure of pictorial space has been probed empirically and explored theoretically. Here we propose a framework that allows one to handle cases that have been encountered empirically, but thus far have not been explored in a formal, geometrical setting. The framework allows one to handle many idiosyncrasies of human visual observers, as well as to characterize the (frequent) individual differences in a principled manner. This opens the door to a principled formalism of the structure (e.g., quality) of the pictorial spaces evoked by various methods of presentation, as required for applications.

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“…Section 2.1. It naturally appears when properties of functions are to be geometrically visualized and interpreted at hand of their graph surfaces [17]. In particular, this holds for the visualization of stress properties in planar elastic systems at hand of their Airy surfaces [30].…”

Section: Previous Workmentioning

confidence: 99%

Discrete Surfaces in Isotropic Geometry

Pottmann

Liu

2007

Mathematics of Surfaces XII

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Abstract. Meshes with planar quadrilateral faces are desirable discrete surface representations for architecture. The present paper introduces new classes of planar quad meshes, which discretize principal curvature lines of surfaces in so-called isotropic 3-space. Like their Euclidean counterparts, these isotropic principal meshes meshes are visually expressing fundamental shape characteristics and they can satisfy the aesthetical requirements in architecture. The close relation between isotropic geometry and Euclidean Laguerre geometry provides a link between the new types of meshes and the known classes of conical meshes and edge offset meshes. The latter discretize Euclidean principal curvature lines and have recently been realized as particularly suited for freeform structures in architecture, since they allow for a supporting beam layout with optimal node properties. We also present a discrete isotropic curvature theory which applies to all types of meshes including triangle meshes. The results are illustrated by discrete isotropic minimal surfaces and meshes computed by a combination of optimization and subdivision.

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Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces

Cited by 35 publications

References 16 publications

Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields

Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields

Gauge Fields in Pictorial Space

Discrete Surfaces in Isotropic Geometry

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