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A Proper Choice of Vertices for Triangulation Representation of Digital Images Ivana Kolingerova, Josef Kohout, Michal Rulf, Vaclav Uher, Proc. 2010 International Conference on Computer Vision and Graphics: Part II, pp. 41-48, 2010. Milestones

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A Proper Choice of Vertices for Triangulation Representation of Digital Images

Ivana Kolingerova, Josef Kohout, Michal Rulf, Vaclav Uher, Proc. 2010 International Conference on Computer Vision and Graphics: Part II, pp. 41-48, 2010. Milestones and Advances in Image Analysis

Stephanie Jennewein

  • 04. December 2012

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Contens

1 Motivation 2 Triangulation 3 A Proper Choice of Vertices 4 Summary Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Motivation

Triangulation representation of digital images enables geometric transformations very simple low compression in comparison with frequency-based methods Can we save disk space while preserving a good quality?

Strategy

choose the triangulation vertices randomly

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation

What to do:

1 assess proper number of vertices (computed from compression rate or

given by the user)

2 choose set of pixels 3 compute triangulation with Delaunay triangulation 4 decoding Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation

What to do:

1 assess proper number of vertices (computed from compression rate or

given by the user)

2 choose set of pixels 3 compute triangulation with Delaunay triangulation 4 decoding Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

2) choose set of pixels in general, choose edge points

Edge Point

A strong change in the grey values within a neighbourhood indicates an edge.

source: IPCV 2011-12

edge detecting operators:

◮ Roberts’s operator ◮ Laplace operator ◮ Gaussian operator Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

Roberts’s operator OpRoberts(i, j, fi,j) = | fi,j − fi+1,j+1

  • 1
  • 1

fi,j

| + | fi+1,j − fi,j+1

  • 1
  • 1

fi,j

|

[i,j] belongs to the set of vertices if

OpRoberts(i, j, fi,j) > T picture f pixel i,j threshold T

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

Laplace operator (Laplace4) OpLaplace4(i, j, fi,j) = | fi,j−1 + fi,j+1 + fi−1,j + fi+1,j − 4fi,j

  • 1

1

  • 4

1 1

fi,j

|

[i,j] belongs to the set of vertices if

OpLaplace4(i, j, fi,j) > T picture f pixel i,j threshold T

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

Laplace operator (Laplace8)

OpLaplace8(i, j, fi,j) = | fi−1,j−1 + fi−1,j + fi−1,j+1 + fi,j−1 + fi,j+1 + fi+1,j−1 + fi+1,j + fi+1,j+1 − 8fi,j

  • 1

1 1 1

  • 8

1 1 1 1

fi,j

|

[i,j] belongs to the set of vertices if

OpLaplace8(i, j, fi,j) > T picture f pixel i,j threshold T

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

Gaussian operator OpGauß(i, j, fi,j) =

r

  • k=−v

r

  • l=−v

|fi,j − fi+k,j+l · exp(−k2 + l2 2σ2 )|

[i,j] belongs to the set of vertices if

OpGauß(i, j, fi,j) > T picture f pixel i,j threshold T

  • v and r: influence factor of the point in this area

σ : vicinity area, width, standart deviation

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

Figure: The 9-10% pixels with the highest evaluation according to the presented

  • perators, a) Roberts, b) Laplace4, c) Laplace8, d) Gauß

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 2) choose set of pixels

Store coodrinates and the intensity of the chosen pixels

Strategy

choose the pixels randomly ⇒ don’t have to store coordinates Reason: coordinates can be recomputed from the seed of the random generator during decoding random point: chosen randomly edge point: chosen by an edge operator

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation

What to do:

1 assess proper number of vertices (computed from compression rate or

given by the user)

2 choose set of pixels 3 compute triangulation with Delaunay triangulation 4 decoding Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 4) compute triangulation with Delaunay triangulation

choose triangles in such a way, that the following property is fulfilled:

empty circumcircle criterion:

the circumcircle of any triangle does not contain any of the given vertices in its interior goal: maximize the minimum angle of all the angles of the triangles in the triangulation ambiguity: two neighbouring triangles have the same circumcircle remedy: choose diagonal with lower intensity gradient

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation

What to do:

1 assess proper number of vertices (computed from compression rate or

given by the user)

2 choose set of pixels 3 compute triangulation with Delaunay triangulation 4 decoding Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Triangulation - 5) decoding

5) decoding values of intensities inside triangles are interpolated from the known vertex intensity values coordinates of random points can be reconstructed with the seed of the random generator

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Contens

1 Motivation 2 Triangulation 3 A Proper Choice of Vertices 4 Summary Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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A Proper Choice of Vertices

Comparing edge detection operators using edge points and random points goal: highest fidelity and at least some compression Laplace:

◮ best for a low number of edge points ◮ and high number of random points

Roberts:

◮ best for a high number of edge points ◮ and low number of random points

Gauß:

◮ worst results ◮ slowest operator Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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A Proper Choice of Vertices

Figure: The image Fruits: Dependence of MSE on the total number of points of which 8-10% are edge points

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A Proper Choice of Vertices

Figure: The image Fruits: Dependence of MSE on the total number of points of which 8% are random points

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A Proper Choice of Vertices

Why not choose only random points?

Figure: The image Fruits Figure: 20% of points: only random points (MSE 99.11)

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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A Proper Choice of Vertices

Choosing only edge points

Figure: The image Fruits Figure: 20% of points: only edge points (MSE 131.94)

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A Proper Choice of Vertices

Figure: The image Fruits: Only random and only edge points

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A Proper Choice of Vertices

Proper choice with acceptable quality and some compression: for most common images:

◮ number random points: 10 − 15% of the image size ◮ number edge points: 5 − 10% of the image size

for images with many edges:

◮ number random points: 10 − 15% of the image size ◮ number edge points: 15 − 20% of the image size Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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A Proper Choice of Vertices

Figure: The image Fruits; a) input, b) result - 11% of edge points, 15% of random points, MSE=20.65

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A Proper Choice of Vertices

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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A Proper Choice of Vertices

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Contens

1 Motivation 2 Triangulation 3 A Proper Choice of Vertices 4 Summary Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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Summary

triangulation is simple and good for geometric transformations points for vertices of triangles can be chosen randomly, because the coordinates can be reconstructed during decoding to keep high quality of the image, one has to find a proper rate of random and edge points the Laplace-operator is best since we want to achieve a high number

  • f random points while preserving the quality

Milestones and Advances in Image Analysis (Stephanie Jennewein) A Proper Choice of Vertices for Triangulation Representation of Digital Images

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References

Ivana Kolingerova, Josef Kohout, Michal Rulf, Vaclav Uher: A proper choice

  • f vertices for triangulation representation of digital images. Proc. 2010

International Conference on Computer Vision and Graphics: Part II, pp. 41-48, 2010. Lecture-Notes: IPCV 2011/12 and 2012/13 Josef Kohout, On Digital Image Representation by the Delaunay Triangulation.Department of Computer Science and Engineering, University

  • f West Bohemia, Univerzitni 22, 306 14 Plze, Czech Republic

besoft@kiv.zcu.cz De Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational geometry. In: Algorithms and applications. Springer, Heidelberg (1997) Milan Sonka, Vaclav Hlavac, Roger Boyle: Image Processing, Analysis, and Machine Vision. ITP (1999) Galic, I., Weickert, J., Welk, M.: Towards PDE-based Image Compression. In: Para- gios, N., Faugeras, O., Chan, T., Schn orr, C. (eds.) VLSM 2005. LNCS, vol. 3752, pp. 3748. Springer, Heidelberg (2005)

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