Robust Image Topological Feature Extraction

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Robust Image Topological Feature Extraction

• Karlis Freivalds, Paulis Kikusts

University of Latvia
Theory Days at Jõulumäe
October 2008
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Image Analysis

• Image analysis steps
• Pixels ? features
• Features ? objects
• Objects ? knowledge
• Knowledge ? decision
• Features
• Edge
• Ridge
• Valley (ravine)
• Corner

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Topological Features

• Image is treated as 3D surface
• Ridge
• Valley
• Junction
• Peak
• Pit
• Saddle
• (Edge)

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Ridges in Images
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Ridge Image Examples
Text
Line drawings
Natural objects
Industrial applications
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Mathematics

• Image smooth function f(x, y)
• Derivatives
• Directional derivative in direction p (px, py)
• Second derivative in two directions (unit
  vectors) p and q

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Principal Orthogonal Directions

• Two directions play the key role
• Gradient direction
• Tangent direction h (hx, xy) (gy, gx).
• fh 0
• Decision is based on fgg, fhh and fhg

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Ridge Definition

• Point (x, y) is called a ridge point if and only
  if

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Analytical example

• f(x, y) y2 x2

K3 xy 0 gives x 0 and y 0 K1 K2 always
true Ridge x 0
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Another example

• f(x, y) y2 10x2

K3 xy 0 gives x 0 and y 0 K1 always
true K2 Ridge x 0 Theorem Ridge of a
quadratic surface is always a line.
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Naive implementation

• fx (f(k 1, l) f(k 1, l)) / 2,
• fy (f(k, l 1) f(k, l 1)) / 2,
• g
• gx fx / g,
• gy fy / g,
• hx gy,
• hy gx,
• fxx f(k 1, l) 2f(k, l) f(k 1, l),
• fyy f(k, l 1) 2f(k, l 1) f(k, l 1),
• fxy (f(k 1, l 1) f(k 1, l 1) f(k
  1, l 1) f(k 1, l 1)) / 4,
• fgg fxxgxgx 2fxygxgy fyygygy,
• fhh fxxhxhx 2fxyhxhy fyyhyhy,
• fgh fxxgxhx fxy(gxhy gyhx) fyygyhy.
• Ridge conditions

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Naive implementation
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Equivalent Formulation

• Hessian matrix
• Eigenvalues of H
• Eigenvectors of H
• Ridge

R. Haralick, Ridges and Valleys on Digital
Images, Computer Vision, Graphics, and Image
Processing, vol. 22, no. 10, pp. 2838,Apr. 1983.
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Algorithm

• Calculate eigenvalues and eigenvectors
• See if
• See if there is a sign change of
• in the direction of
• Problems
• Unreliable
• Requires interpolation
• Relatively Slow

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Lee and Kim Algorithm

• 4 directions of discrete grid are analyzed
• Orthogonal directions of greatest curvature are
  selected
• Point is classified as ridge, valley, peak or pit
  based on sign changes of gradients in these
  directions.
• Problems
• Unreliable
• No clear distinction between ridges and peaks

Lee, S. and Kim, Y. J. 1995. Direct Extraction of
Topographic Features for Gray Scale Character
Recognition. IEEE Trans. Pattern Anal. Mach.
Intell. 17, 7 (Jul. 1995), 724-729.
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New Discrete Algorithm

• Consider 4 directions of discrete grid
• Find direction of minimum second derivative (that
  will be direction perpendicular to ridge), let
  fhh be that second derivative.
• Test if the current point is a local maximum in
  that direction
• Let fgg be the second derivative in the
  perpendicular direction.
• Test if abs(fhh) gt abs(fgg)
• If all tests are true return abs(fhh) as ridge
  strength
• Else return that this is not a ridge point

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Implementation

• // Calculation of ridge pixel strength at the
  given point, 0 if the point is not a ridge point
• // assumes that int image holds the image values
  in row by row.
• int RidgePixelStrengthOptimized(int x, int y)
• int p image (x ywidth)
• int f p
• int fpp4 int fhh,fhhI, fhh_tmp, fhhI_tmp
• fpp0 2(f - p1 - p-1) // calculate
  magnitudes of directional second derivatives
• fpp1 - pwidth1 - p-width-1
• fpp2 2(f - pwidth - p-width)
• fpp3 - pwidth-1 - p-width1
• if (fpp1gtfpp0)fhh fpp1fhhI 1 else
  fhh fpp0fhhI 0 // find the maximum one
• if (fpp2gtfpp3)fhh_tmp fpp2fhhI_tmp
  2 else fhh_tmp fpp3fhhI_tmp 3
• if(fhh_tmp gt fhh)fhh fhh_tmpfhhI
  fhhI_tmp
• fhh 2f // diagonal
• if(fhhltlowerThreshold) return 0 //
  optimization early exit

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Results
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Sensitivity to Noise
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Sensitivity to Noise (1)
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Sensitivity to Noise (2)
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Sensitivity to Noise (3)
Note All images were processed with the same
parameters
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Results

• Connected lines
• Thin lines
• No artifacts
• Low noise sensitivity
• High performance 45MPix/s 108 video frames(720
  576) per second (AMD Athlon 2.6GHz)

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Compared to Lee and Kim Algorithm

• Simpler
• Faster
• No spurious ridges

Original image Lee Kim Freivalds
Kikusts
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Scale Selection

• Gaussian blurring (Geusebroek et.al. 2002)
• Scale space and automatic scale selection
  (Lindeberg 1996)
• Adaptive blurring (Elder, Zucker 1998)

Geusebroek, J., Smeulders, A. W., and Weijer, J.
v. 2002. Fast Anisotropic Gauss Filtering. In
Proceedings of the 7th European Conference on
Computer Vision-Part I. LNCS, vol. 2350. pp.
99-112. T. Lindeberg "Edge detection and ridge
detection with automatic scale selection",
International Journal of Computer Vision, vol 30,
number 2, pp. 117--154, 1998. Earlier version
presented at IEEE Conference on Pattern
Recognition and Computer Vision, CVPR'96, San
Francisco, California, pages 465--470, june
1996 Elder, J. and Zucker, S. 1998. Local scale
control for edge detection and blur estimation.
IEEE Pattern Anal. Machine Intell., 20(7) 699716
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Topological Edge Detection

• Edge conditions
• fgg 0
• fggg lt 0
• Algorithm
• Find the discrete direction of maximum gradient
• See if the second derivative has sign change from
  positive to negative in that direction

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Comparison to Canny Detector

• Similar quality
• Simpler
• Faster
• Original Canny Proposed

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Results Shooting Target
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Conclusion

• New robust ridge, edge detection algorithms
• High quality
• High performance suitable for real-time
  applications
• Simple implementation
• Straightforward hardware implementation

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