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Fast and exact visibility on digitized shapes and application to saliency-aware normal estimation

Romain Negro, Jacques-Olivier Lachaud

Abstract

Computing visibility on a geometric object requires heavy computations since it requires to identify pairs of points that are visible to each other, i.e. there is a straight segment joining them that stays in the close vicinity of the object boundary. We propose to exploit a specic representation of digital sets based on lists of integral intervals in order to compute eciently the complete visibility graph between lattice points of the digital shape. As a quite direct application, we show then how we can use visibility to estimate the normal vector eld of a digital shape in an accurate and convergent manner while staying aware of the salient and sharp features of the shape.

Fast and exact visibility on digitized shapes and application to saliency-aware normal estimation

Abstract

Computing visibility on a geometric object requires heavy computations since it requires to identify pairs of points that are visible to each other, i.e. there is a straight segment joining them that stays in the close vicinity of the object boundary. We propose to exploit a specic representation of digital sets based on lists of integral intervals in order to compute eciently the complete visibility graph between lattice points of the digital shape. As a quite direct application, we show then how we can use visibility to estimate the normal vector eld of a digital shape in an accurate and convergent manner while staying aware of the salient and sharp features of the shape.
Paper Structure (9 sections, 1 equation, 10 figures, 2 algorithms)

This paper contains 9 sections, 1 equation, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Visibility through tangency of chords
  3. Fast computation using integer intervals intersections
  4. Saliency-aware normal and curvature estimation
  5. Saliency-aware normal estimator.
  6. Experimental validation.
  7. Application to curvature estimation.
  8. Conclusion

Figures (10)

  • Figure 1: Examples of visibility and non visibility in 2D within the set $X$ (represented with black dots $\bullet$).
  • Figure 2: The set of visible points in $X$ from source $p$ is not $1$-connected.
  • Figure 3: Representation of a 2d cell complex (here the star of a given curve) as a lattice map. From 63 cells (with two coordinates) on the left, we get 11 intervals on the right.
  • Figure 4: Evolution of the visibility check algorithm for a $(2,1)$ vector. Green is the vector lattice map, black is the figure lattice map, red are the current intervals of positions where the visibility is still possible. The last red intervals are the visible positions. We travel the lattice maps from bottom-up and the found visibilities are drawn on the uppest figure.
  • Figure 5: Examples of visibilities from a source point: (left) the visibility is stopped at the sharp edge, (right) the visibility does not cross the gap between the two parts.
  • ...and 5 more figures

Theorems & Definitions (1)

  • definition 1: Visibility