The gradient of the intensity level is zero at the ridge points
f(x)
x
f ’(x)
f ’(x)
x
Gradient of the intensity level is maximal at the edge points
f ’’(x)
x
edges = {P = ( x, y ) / arg(max(| ∇I ( P ) |))}
Ù Local extremum along the gradient direction
∇(|| ∇I ||).∇( I ) = 0
∂ ∇I ∂x Ix + ∂ ∇I ∂y Iy = 0
CAS – Course Computer Vision : Edge detection - Part 1
z z z z
General recall Marr Theory Edge detection : what for ? Edge detection & Segmentation
CAS – Course Computer Vision : Edge detection - Part 1
1
General recalls
Physical meaning z Edge & ridge representations z Principal edge detection methods
z
CAS – Course Computer Vision : Edge detection - Part 1
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Illustration (2)
The second derivative is zero at the edge points
1
CAS – Course Computer Vision : Edge detection - Part 1
2D case : Edge as maximal gradient
• Maximal gradient :
1
CAS – Course Computer Vision : Edge detection - Part 1
1D case : Mathematical formulation
Two equivalent formulations 1) signal S(x) first derivative Sx is locally maximum. {edges} = { x | arg max(Sx(x)) locally)} 2) second derivative is equal to zero. {edges} = { x | Sxx(x)=0 }
●
Discrete space :
●
Ii,j at point (i,j)
CAS – Course Computer Vision : Edge detection - Part 1
1
Convention
I J
I0
J0
I(i0,j0)
i , j N2 0„ i „ n 1 0„ j „ m 1
Y X – Course Computer Vision : Edge detection - Part 1 CAS
CAS – Course Computer Vision : Edge detection - Part 1
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Physical meaning
Edges correspond to many different physical
z
properties from the “real” world :
contours of objects (1), z borders (2), z shadow (3), z change of intensity level, color, texture (4) z …
z
CAS – Course Computer Vision : Edge detection - Part 1
z z
z
Give a simplified representation of the image Provide information for higher level processing
z
Key difficulties
z
To extract physical information (and overcome the probleme of noise, shadow, lightening, ...) No unique approach
CAS – Course Computer Vision : Edge detection - Part 1
1
Ridge detection
f(x)
Edge detection
Edges are points were the intensity level changes sharply
Ridges are points were the intensity level is maximal
CAS – Course Computer Vision : Edge detection - Part 1
z
1
Edge detection and Segmentation
• Segmentation
∀ i Ri ≠ φ ∀ i, j i ≠ j A = U Ri
i
Ri ∩ R j = φ
• Two major “families” of segmentation
Differential approach Variational approach Mathematical Morphology Surface Model Markovian approach ….
CAS – Course Computer Vision : Edge detection - Part 1
High level processing 1
CAS – Course Computer Vision : Edge detection - Part 1
Edge detection : what for ?
z
Edge detection is a low level processing. Edge detection
Zero laplacien are given by zero-crossing points along the gradient direction => sub-pixel accuracy.
CAS – Course Computer Vision : Edge detection - Part 1
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III – Edge detection : Derivative approaches
z z z z z z z z
Mathematical definition Convolution masks An ill posed problem Gaussian filter & Optimal filters Crest/ridges points detection in 2D Extension to 3D images Multi-scale approach Performance analysis
CAS – Course Computer Vision : Edge detection - Part 1
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Definition
Edge points = points were the intensity level changes sharply.
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CAS – Course Computer Vision : Edge detection - Part 1
Chaining, thinning / Active contours / Derivative vs variational approaches
5. Conclusion
CAS – Course Computer Vision : Edge detection - Part 1
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I -Introduction
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Contour based segmentation
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Edge detection + Contour closing
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Region based segmentation
CAS – Course Computer Vision : Edge detection - Part 1
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II- First notions of Edge & Ridge Detection
Edge Detection Basis of Theory and Practice (1)
CAS – Computer Vision Course (2005.03.11) Veronique PRINET prinet@
CAS – Course Computer Vision : Edge detection - Part 1
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Contents
1. Introduction 2. First notions of Edge detection
Physical meaning / Mathematical representation / An ill posed problem / Overview of edge detection methods
3. Derivative approaches
Computation of the partial derivatives / Gaussian & Optimal filters / Multi-scale approach / Assessment / …