Delineation - Moodle 2020-2021 · Interactive Delineation •The user provides the start and end points (red x). •The algorithm does the rest (yellow line). Image. Gradient. 10

1

Delineation

•Dynamic Programming

•Deformable Models

•Hough Transform

•Graph Based Approaches

2

From Gradients to Outlines

3

From Deep-Nets to Outlines

—> Still work to do!

4

Mapping and Overlays

Connectivity matters!

5

Connectomics

Courtesy of C. Petersen

—> Topology needed

6

Analogy

• Uses Deep Nets to find the most promis ing locations to focus on.

• Performs tree-based search when possible.

• Relies on reinforcement learning and other ML techniques to train.

High level processing

Low level processing

7

Techniques

Semi-Automated Techniques:

• Dynamic Programming • Deformable Models

Fully Automated Techniques:

• Hough Transform • Graph Based Approaches

8

Reminder: Canny Limitations

There is no ideal value of s!

9

Interactive Delineation

• The user provides the start and end points (red x). • The algorithm does the rest (yellow line).

Image Gradient

10

Live Wire in Action

Mortensen and Barrett, SIGGRAPH’95

11

1D Dynamic Programming

where f denotes the gradient orientation.

k

xk

h(x1, x2, …, xn) = −n

∑k=1

g(xk) +n−1

∑k=1

r(xk, xk+1)

r(xk, xk+1) = diff(ϕ(xk), ϕ(xk+1))

12


s g

• N Locations • Q Quantized values

! Global optimum O(NQ2)

1 2 3 N…

13


r r r

r r

r

r

r

14

Notations:s Start point

L List of active nodes

c(u,v) Local costs for link u -> v

d(v) Total cost from s to v

s

vu

d(v)

c(u,v)

g


15

Dijkstra Path Expansion

Open nodes represent "unvisited" nodes. Filled nodes are visited ones, with color representing the distance: The greener, the shorter the path. Nodes in all the different directions are explored uniformly, appearing more-or-less as a circular wavefront.

https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm

16

Dijkstra’s Algorithm

Maintain a sorted list of paths

17

Live Wire

• Sorting is the expensive operation. Normally nlog(n), but can be reduced to log(n) if all costs are integer costs

• Local costs computed using gradient:

c(u,v) = 255 – ½ (g(u)+ g(v))

• Diagonal penalized by multiplying cost of non diagonal edges by:

• Add a constant cost for each edge.

18

Magnetic Lasso in Photoshop

Integrating the LiveWire into a powerful interface that allows a user to correct mistakes yields a useful too.

19

Tracing Neurons

▪ In the biomedical world, images are 3D cubes of data. ▪ The approach extends naturally to tracking of 3D

structures such as dendritic trees in the brain, blood vessels, etc …

20

Limitations

• The “optimal” path is not always the “best” one. • Difficult to impose global constraints. • The cost grows exponentially with the dimension

of the space in which we work.

--> Must often look for local, as opposed to global, optimum using gradient descent techniques.

21

Techniques


• Dynamic programming • Deformable Models


• Hough transform • Graph Based Approaches

22

Snakes

23

2—D Snake

Deformable contours that • Maximize the gradient along the curve; • Minimize their deformation energy. --> Interactive tools for contour detection that can be

generalized to handle sophisticated models

24

Energy Landscape

E = 1/L

Z 1

s=0(��G(x(s), y(s)) + �2(s))ds

= 1/L

Z 1

s=0(��G(x(s), y(s)) + �

2x

�s2

2

+�2y

�s2

2

)ds

= 1/L(��NX

i=0

G(xi, yi) +N�1X

i=1

((2xi � xi�1 � xi+1)2 + (2yi � yi�1 � yi+1)2)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

25

Matrix Notation

26

Local Optimum

But K is not invertible!

27

Dynamics

Embed curve in a viscous medium and solve at each step:

0 =∂E∂X

+ αdXdt

=∂EG∂X

+ KX + αdXdt

0 =∂E∂Y

+ αdYdt

=∂EG∂Y

+ KY + αdYdt

28

Iterating

At every step:

! Solve two linear equations at each iteration.

0 =�EG�X

+KXt + ↵(Xt �Xt�1) ) (K + ↵I)Xt = ↵Xt�1 ��EG�X

0 =�EG�Y

+KYt + ↵(Yt � Yt�1) ) (K + ↵I)Yt = ↵Yt�1 ��EG�Y

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

29

Derivatives of the Image Gradient

… …

30

Bilinear Interpolation

31

Open and Closed Snakes

32

Cysts Tumors in Ultrasound Images

Luo et al., 2017

Drawn by the physician. Refined by the Computer.

33--> Updated field boundaries.

Network Snakes

Butenuth, PhD 2008

34

Ribbon Snakes

+1/2WtKWW

35

DYNAMICS EQUATIONS

! Solve three linear equations at each iteration.

(K + ↵I)Xt = ↵Xt�1 ��EG�X

(K + ↵I)Yt = ↵Yt�1 ��EG�Y

(K + ↵I)Wt = ↵Wt�1 ��EG�W

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

36

Delineating Roads

37

Delineating Roads

38

Evaluation

It takes far fewer clicks to trace the roads using semi-automated tools than doing entirely by hand.

39

Ridge Lines

40

Building

41

3—D Snakes

Smooth 3—D snake Rectilinear 3—D snake

42

Dynamics Equations

(K + ↵I)Xt = ↵Xt�1 ��EG�X

(K + ↵I)Yt = ↵Yt�1 ��EG�Y

(K + ↵I)Zt = ↵Zt�1 ��EG�Z

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

! Solve three linear equations at each iteration.

43

Constrained Optimization

• Minimize F(S) subject to C(S) = 0

44

Site Modeling (1996)

45

Site Modeling (2019)

46

Real-Time 3D Shape

Ostlund et al., ECCV’12.

47

Sails

! Real-time performance evaluation.

48

Wings

49

Level Sets

50

Curve Evolution

Problems: 1. Involves curve sampling and resampling. 2. Curvature estimates are numerically unstable.

Generic formulation:@C

@t= ↵(s, t)~T + �(s, t) ~N

Reparameterization:@C

@t= �(s, t) ~N

Special cases:

Prairie fire � = ±1Di↵usion � = �0 � �1

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

51

Level Sets

Consider the curve as the zero level set of the surface.

! Much better numerical stability.

z = �(x, y, t)AAAB+XicbVBNT8JAEJ36ifhV9OhlI5hgQkjLRT2YEL14xMQKCTRkuyywYfuR3a1aKz/Fiwc1Xv0n3vw3LtCDgi+Z5OW9mczM8yLOpLKsb2NpeWV1bT23kd/c2t7ZNQt7tzKMBaEOCXkoWh6WlLOAOoopTluRoNj3OG16o8uJ37yjQrIwuFFJRF0fDwLWZwQrLXXNQukRnaNOY8jKD5Wkoo5LXbNoVa0p0CKxM1KEDI2u+dXphST2aaAIx1K2bStSboqFYoTTcb4TSxphMsID2tY0wD6Vbjo9fYyOtNJD/VDoChSaqr8nUuxLmfie7vSxGsp5byL+57Vj1T91UxZEsaIBmS3qxxypEE1yQD0mKFE80QQTwfStiAyxwETptPI6BHv+5UXi1KpnVfu6VqxfZGnk4AAOoQw2nEAdrqABDhC4h2d4hTfjyXgx3o2PWeuSkc3swx8Ynz/JcJHxAAAB+XicbVBNT8JAEJ36ifhV9OhlI5hgQkjLRT2YEL14xMQKCTRkuyywYfuR3a1aKz/Fiwc1Xv0n3vw3LtCDgi+Z5OW9mczM8yLOpLKsb2NpeWV1bT23kd/c2t7ZNQt7tzKMBaEOCXkoWh6WlLOAOoopTluRoNj3OG16o8uJ37yjQrIwuFFJRF0fDwLWZwQrLXXNQukRnaNOY8jKD5Wkoo5LXbNoVa0p0CKxM1KEDI2u+dXphST2aaAIx1K2bStSboqFYoTTcb4TSxphMsID2tY0wD6Vbjo9fYyOtNJD/VDoChSaqr8nUuxLmfie7vSxGsp5byL+57Vj1T91UxZEsaIBmS3qxxypEE1yQD0mKFE80QQTwfStiAyxwETptPI6BHv+5UXi1KpnVfu6VqxfZGnk4AAOoQw2nEAdrqABDhC4h2d4hTfjyXgx3o2PWeuSkc3swx8Ynz/JcJHxAAAB+XicbVBNT8JAEJ36ifhV9OhlI5hgQkjLRT2YEL14xMQKCTRkuyywYfuR3a1aKz/Fiwc1Xv0n3vw3LtCDgi+Z5OW9mczM8yLOpLKsb2NpeWV1bT23kd/c2t7ZNQt7tzKMBaEOCXkoWh6WlLOAOoopTluRoNj3OG16o8uJ37yjQrIwuFFJRF0fDwLWZwQrLXXNQukRnaNOY8jKD5Wkoo5LXbNoVa0p0CKxM1KEDI2u+dXphST2aaAIx1K2bStSboqFYoTTcb4TSxphMsID2tY0wD6Vbjo9fYyOtNJD/VDoChSaqr8nUuxLmfie7vSxGsp5byL+57Vj1T91UxZEsaIBmS3qxxypEE1yQD0mKFE80QQTwfStiAyxwETptPI6BHv+5UXi1KpnVfu6VqxfZGnk4AAOoQw2nEAdrqABDhC4h2d4hTfjyXgx3o2PWeuSkc3swx8Ynz/JcJHxAAAB+XicbVBNT8JAEJ36ifhV9OhlI5hgQkjLRT2YEL14xMQKCTRkuyywYfuR3a1aKz/Fiwc1Xv0n3vw3LtCDgi+Z5OW9mczM8yLOpLKsb2NpeWV1bT23kd/c2t7ZNQt7tzKMBaEOCXkoWh6WlLOAOoopTluRoNj3OG16o8uJ37yjQrIwuFFJRF0fDwLWZwQrLXXNQukRnaNOY8jKD5Wkoo5LXbNoVa0p0CKxM1KEDI2u+dXphST2aaAIx1K2bStSboqFYoTTcb4TSxphMsID2tY0wD6Vbjo9fYyOtNJD/VDoChSaqr8nUuxLmfie7vSxGsp5byL+57Vj1T91UxZEsaIBmS3qxxypEE1yQD0mKFE80QQTwfStiAyxwETptPI6BHv+5UXi1KpnVfu6VqxfZGnk4AAOoQw2nEAdrqABDhC4h2d4hTfjyXgx3o2PWeuSkc3swx8Ynz/JcJHx

0 = �t + �() |r�|

where =�xx�2y � 2�xy�x�y + �yy�2x

�2x + �2y

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

Corresponding evolution equation.

52

Topology Change

53

Level Set Smoothing

Smoothing occurs when

Desirable properties:

• Converges towards circles. • Total curvature decreases. • Number of curvature extrema and zeros of curvature decreases.

Relationship with Gaussian smoothing:

• Analogous to Gaussian smoothing of boundary over the short run, but does not cause self-intersections or overemphasize elongated parts.

• Can be implemented by Gaussian smoothing the characteristic function of a region.

�() = �AAACA3icbZC7SgNBFIZn4y3G26plmsFEiIVhN41aCEEbywiuCWSXcHYymwyZvTAzK4QlhY2vYmOhYutL2Pk2TpItNPGHgY//nMOZ8/sJZ1JZ1rdRWFldW98obpa2tnd298z9g3sZp4JQh8Q8Fh0fJOUsoo5iitNOIiiEPqdtf3Q9rbcfqJAsju7UOKFeCIOIBYyA0lbPLFddnyrANXcESQIn+BKf4jlXe2bFqlsz4WWwc6igXK2e+eX2Y5KGNFKEg5Rd20qUl4FQjHA6KbmppAmQEQxoV2MEIZVeNjtigo+108dBLPSLFJ65vycyCKUch77uDEEN5WJtav5X66YqOPcyFiWpohGZLwpSjlWMp4ngPhOUKD7WAEQw/VdMhiCAKJ1bSYdgL568DE6jflG3bxuV5lWeRhGV0RGqIRudoSa6QS3kIIIe0TN6RW/Gk/FivBsf89aCkc8coj8yPn8A04aV2A==AAACA3icbZC7SgNBFIZn4y3G26plmsFEiIVhN41aCEEbywiuCWSXcHYymwyZvTAzK4QlhY2vYmOhYutL2Pk2TpItNPGHgY//nMOZ8/sJZ1JZ1rdRWFldW98obpa2tnd298z9g3sZp4JQh8Q8Fh0fJOUsoo5iitNOIiiEPqdtf3Q9rbcfqJAsju7UOKFeCIOIBYyA0lbPLFddnyrANXcESQIn+BKf4jlXe2bFqlsz4WWwc6igXK2e+eX2Y5KGNFKEg5Rd20qUl4FQjHA6KbmppAmQEQxoV2MEIZVeNjtigo+108dBLPSLFJ65vycyCKUch77uDEEN5WJtav5X66YqOPcyFiWpohGZLwpSjlWMp4ngPhOUKD7WAEQw/VdMhiCAKJ1bSYdgL568DE6jflG3bxuV5lWeRhGV0RGqIRudoSa6QS3kIIIe0TN6RW/Gk/FivBsf89aCkc8coj8yPn8A04aV2A==AAACA3icbZC7SgNBFIZn4y3G26plmsFEiIVhN41aCEEbywiuCWSXcHYymwyZvTAzK4QlhY2vYmOhYutL2Pk2TpItNPGHgY//nMOZ8/sJZ1JZ1rdRWFldW98obpa2tnd298z9g3sZp4JQh8Q8Fh0fJOUsoo5iitNOIiiEPqdtf3Q9rbcfqJAsju7UOKFeCIOIBYyA0lbPLFddnyrANXcESQIn+BKf4jlXe2bFqlsz4WWwc6igXK2e+eX2Y5KGNFKEg5Rd20qUl4FQjHA6KbmppAmQEQxoV2MEIZVeNjtigo+108dBLPSLFJ65vycyCKUch77uDEEN5WJtav5X66YqOPcyFiWpohGZLwpSjlWMp4ngPhOUKD7WAEQw/VdMhiCAKJ1bSYdgL568DE6jflG3bxuV5lWeRhGV0RGqIRudoSa6QS3kIIIe0TN6RW/Gk/FivBsf89aCkc8coj8yPn8A04aV2A==AAACA3icbZC7SgNBFIZn4y3G26plmsFEiIVhN41aCEEbywiuCWSXcHYymwyZvTAzK4QlhY2vYmOhYutL2Pk2TpItNPGHgY//nMOZ8/sJZ1JZ1rdRWFldW98obpa2tnd298z9g3sZp4JQh8Q8Fh0fJOUsoo5iitNOIiiEPqdtf3Q9rbcfqJAsju7UOKFeCIOIBYyA0lbPLFddnyrANXcESQIn+BKf4jlXe2bFqlsz4WWwc6igXK2e+eX2Y5KGNFKEg5Rd20qUl4FQjHA6KbmppAmQEQxoV2MEIZVeNjtigo+108dBLPSLFJ65vycyCKUch77uDEEN5WJtav5X66YqOPcyFiWpohGZLwpSjlWMp4ngPhOUKD7WAEQw/VdMhiCAKJ1bSYdgL568DE6jflG3bxuV5lWeRhGV0RGqIRudoSa6QS3kIIIe0TN6RW/Gk/FivBsf89aCkc8coj8yPn8A04aV2A==

54

Shape Recovery

! Expansion stops at the boundaries.

Evolution equation: 0 = �t + �() |r�|

where: �() = kI(1� ✏)

kI =1

1 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

55

Level Sets

56

Level Sets

57

Techniques





58

Finding Lines

Input: • Canny edge points. • Gradient magnitude and orientation.

Output: • All straight lines in image.

59

Hough Transform

Given a parametric model of a curve:

• Map each contour point onto the set of parameter values for which the curves passes through it.

• Find the intersection for all parameter sets thus mapped.

60

Voting Scheme

x cos(✓) + y sin(✓) = r , 0 ✓ ⇡AAACKnicbVDLSsNAFJ3Ud31VXboZLEJFKYkKCiL42LhUsA9oQplMb9qhk0mcuRFL8Xvc+CtuXCji1g8xaSP4OjDDmXPu5c49fiyFQdt+swoTk1PTM7NzxfmFxaXl0spq3USJ5lDjkYx002cGpFBQQ4ESmrEGFvoSGn7/PPMbt6CNiNQ1DmLwQtZVIhCcYSq1S6d31OWRqbjYA2Rb2wPqGqG+nvSYauoe0Z3ssqkr4YaOrZzHotgule2qPQL9S5yclEmOy3bp2e1EPAlBIZfMmJZjx+gNmUbBJdwX3cRAzHifdaGVUsVCMN5wtOo93UyVDg0inR6FdKR+7xiy0JhB6KeVIcOe+e1l4n9eK8Hg0BsKFScIio8HBYmkGNEsN9oRGjjKQUoY1yL9K+U9phnHNN0sBOf3yn9Jfbfq7FV3r/bLJ2d5HLNknWyQCnHIATkhF+SS1AgnD+SJvJBX69F6tt6s93Fpwcp71sgPWB+fxBijyA==

x

y

r

q

61

Voting Scheme


x

y

r

q

62

Voting Scheme


x

y

r

q

63

Voting Scheme


x

y

r

q

64

Iris Detection

65

Occlusions

In theory:

In practice:

66

Circle Detection

Circle of equation:

Therefore:

67

Gradient Orientation

Can vote either along the entire circle or only at two points per value of the radius.

68

Simple Image

Voting scheme:

Result:

69

Eye Image

Image and accumulator:

Best four candidates:

70

Ellipses

71

Ellipse Detection

Ellipse of equation:

Therefore:

72

dx

d✓= �a sin(✓)

dy

d✓= b cos(✓)

� = atan(�dxd✓

,dy

d✓)

= atan(a sin(✓), b cos(✓))

= atan(a

btan(✓))

tan(�) =a

btan(✓)

) ✓ = atan( batan(�))

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

dx

d✓= �a sin(✓)

dy

d✓= b cos(✓)


,dy

d✓)


= atan(a

btan(✓))

tan(�) =a

btan(✓)



dx

d✓= �a sin(✓)

dy

d✓= b cos(✓)


,dy

d✓)


= atan(a

btan(✓))

tan(�) =a

btan(✓)



Gradient Orientation

For each ellipse point:

The accumulator need only be incremented for this q.

[dx

d✓,dy

d✓]

AAACEHicbZDLSsNAFIYnXmu9RV26GSyiCymJCLosunFZwV4gDWUymbRDJxdmTsQQ8ghufBU3LhRx69Kdb+O0DaitPwz8fOcczpzfSwRXYFlfxsLi0vLKamWtur6xubVt7uy2VZxKylo0FrHsekQxwSPWAg6CdRPJSOgJ1vFGV+N6545JxePoFrKEuSEZRDzglIBGffPI6QWS0Ny/L3K/B0MGpDgpUfaD3L5Zs+rWRHje2KWpoVLNvvnZ82OahiwCKohSjm0l4OZEAqeCFdVeqlhC6IgMmKNtREKm3HxyUIEPNfFxEEv9IsAT+nsiJ6FSWejpzpDAUM3WxvC/mpNCcOHmPEpSYBGdLgpSgSHG43SwzyWjIDJtCJVc/xXTIdFpgM6wqkOwZ0+eN+3Tuq39zVmtcVnGUUH76AAdIxudowa6Rk3UQhQ9oCf0gl6NR+PZeDPep60LRjmzh/7I+PgGwe6eSg==AAACEHicbZDLSsNAFIYnXmu9RV26GSyiCymJCLosunFZwV4gDWUymbRDJxdmTsQQ8ghufBU3LhRx69Kdb+O0DaitPwz8fOcczpzfSwRXYFlfxsLi0vLKamWtur6xubVt7uy2VZxKylo0FrHsekQxwSPWAg6CdRPJSOgJ1vFGV+N6545JxePoFrKEuSEZRDzglIBGffPI6QWS0Ny/L3K/B0MGpDgpUfaD3L5Zs+rWRHje2KWpoVLNvvnZ82OahiwCKohSjm0l4OZEAqeCFdVeqlhC6IgMmKNtREKm3HxyUIEPNfFxEEv9IsAT+nsiJ6FSWejpzpDAUM3WxvC/mpNCcOHmPEpSYBGdLgpSgSHG43SwzyWjIDJtCJVc/xXTIdFpgM6wqkOwZ0+eN+3Tuq39zVmtcVnGUUH76AAdIxudowa6Rk3UQhQ9oCf0gl6NR+PZeDPep60LRjmzh/7I+PgGwe6eSg==AAACEHicbZDLSsNAFIYnXmu9RV26GSyiCymJCLosunFZwV4gDWUymbRDJxdmTsQQ8ghufBU3LhRx69Kdb+O0DaitPwz8fOcczpzfSwRXYFlfxsLi0vLKamWtur6xubVt7uy2VZxKylo0FrHsekQxwSPWAg6CdRPJSOgJ1vFGV+N6545JxePoFrKEuSEZRDzglIBGffPI6QWS0Ny/L3K/B0MGpDgpUfaD3L5Zs+rWRHje2KWpoVLNvvnZ82OahiwCKohSjm0l4OZEAqeCFdVeqlhC6IgMmKNtREKm3HxyUIEPNfFxEEv9IsAT+nsiJ6FSWejpzpDAUM3WxvC/mpNCcOHmPEpSYBGdLgpSgSHG43SwzyWjIDJtCJVc/xXTIdFpgM6wqkOwZ0+eN+3Tuq39zVmtcVnGUUH76AAdIxudowa6Rk3UQhQ9oCf0gl6NR+PZeDPep60LRjmzh/7I+PgGwe6eSg==AAACEHicbZDLSsNAFIYnXmu9RV26GSyiCymJCLosunFZwV4gDWUymbRDJxdmTsQQ8ghufBU3LhRx69Kdb+O0DaitPwz8fOcczpzfSwRXYFlfxsLi0vLKamWtur6xubVt7uy2VZxKylo0FrHsekQxwSPWAg6CdRPJSOgJ1vFGV+N6545JxePoFrKEuSEZRDzglIBGffPI6QWS0Ny/L3K/B0MGpDgpUfaD3L5Zs+rWRHje2KWpoVLNvvnZ82OahiwCKohSjm0l4OZEAqeCFdVeqlhC6IgMmKNtREKm3HxyUIEPNfFxEEv9IsAT+nsiJ6FSWejpzpDAUM3WxvC/mpNCcOHmPEpSYBGdLgpSgSHG43SwzyWjIDJtCJVc/xXTIdFpgM6wqkOwZ0+eN+3Tuq39zVmtcVnGUUH76AAdIxudowa6Rk3UQhQ9oCf0gl6NR+PZeDPep60LRjmzh/7I+PgGwe6eSg==

[dy

d✓,�dx

d✓]

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✓AAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8oAAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8oAAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8oAAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8o

73

Generalized Hough

Finding a becher …

… or a lake.

74

Generalized Hough

• We want to find a shape defined by its boundary points in terms of the location of a reference point [xc,yc]. • For every boundary point p, we can compute the displacement vector r = [xc,yc] – p as a function of local gradient orientation f.

pr(f)

[xc,yc]

f

Ballard. , PR’81

75

R-Table

• Set of potential displacement vectors r,a given the boundary orientation f.

--> Generalized template matching.

76

Algorithm

1. Make an R-table for the shape to be located.

2. Form an accumulator array of possible reference points initialized to zero.

3. For each edge point,

• Compute the possible centers, that is, for each table entry, compute

• Increment the accumulator array

77

Real-Time Hough

78

Accumulator

79

From Delineation To Detection

Leibe & al. , ECCV’04

Training image

Visual codeword with displacement vectors

Instead of indexing displacements by gradient orientation, index by “visual codeword”.

K.Grauman

80

From Delineation To Detection

Leibe & al. , ECCV’04

Instead of indexing displacements by gradient orientation, index by “visual codeword”.

Test image

K.Grauman

81

Training

1. Build codebook of patches around extracted interest points using clustering.

2. Map the patch around each interest point to closest codebook entry. 3. For each codebook entry, store all positions it was found, relative to

object center.

--> Build an R table.

82

Testing

1. Given test image, extract patches, match to codebook entry. 2. Cast votes for possible positions of object center. 3. Search for maxima in voting space. 4. Extract weighted segmentation mask based on stored

masks for the codebook occurrences.

83

Pedestrian Detection

Gall & al. , PAMI’11

84

Occlusion Handling

85

Limitations

Computational cost grows exponentially with the number of model parameters:

!Only works for objects whose shape can be defined by a small number of parameters.

!Approach is robust but lacks flexibility.

86

Techniques





87

Magnitude and Orientation

88

Minimum Spanning Tree

Image modeled as a graph:

-> Generate minimal distance graph O(N log(N) algorithm)

89

Delineation 1998

Fischler & Heller, 1998.

Detect road centerlines

Find generic paths

Apply semantic filter

Find road widths

90

From Image To Roads

91

Road Editing

92

Dendrites And Axons

Fluorescent neurons in the adult mouse brain imaged imaged in vivo through a cranial window using a 2-photon microscope.

93

Delineation 2012: Neurites …

94

.. and Roads

Turetken et al, CVPR’12.

—> Machine plays a crucial role to ensure that the same algorithm works in different situations.

95

Histogram of Gradient Deviations

—> One histogram per radius interval plus four geometric features (curvature, tortuosity, .…).

96

Embedding

—> Same length feature vectors whatever the actual length of the path.

97

Path Classification

98

Finding the Best Tree

99

QMIP Formulation

100

Roads

101

Brainbow Images

102

Blood Vessels

103

Deep Tsunami

104

Reminder: AlexNet (2012)

Task: Image classification Training images: Large Scale Visual Recognition Challenge 2010 Training time: 2 weeks on 2 GPUs

Major Breakthrough: Training large networks has now been shown to be practical!!

105

Delineation 2011

Turetken et al, CVPR’12.

These two steps are closely related!

106

Reminder: U-NetImage Tubularity Map

Downsampling Upsampling

—> Train a U-Net to output a tubularity map.

Skip connection

Skip connection

107

Training U-Net

Mosinska et al, CVPR’18.

Train Encoder-decoder U-Net architecture using binary cross-entropy

Minimize

Lbce(x,y;w) = �1

i

PX

1

[yn log(ŷi) + (1� yi) log(1� ŷi)]

where

• ŷ = fw(x),

• x in an input image,

• y the corresponding ground truth.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

i

108

Network Output

BCELoss GroundtruthImage

109

Accounting For Topology

!ℒ#$%

ℒ&'(Minimize

L(x,y;w) = Lbce(x,y;w) + Ltop(x,y;w) ,

Ltop(x,y;w) /NX

n=1

MnX

m=1

klmn (y)� lmn (f(x,w))k22 ,

Ltop: Sum of square di↵erences between feature maps in a pretrained VGG.AAADgXicfVJNb9NAEHXiAk34SuHIZURalJQS4lz4qCpVcCiHFhVB2krZxFpvxsmq3rW7u6YNxv+D38WNXwPrxIiSVh1p5bdv3uyMZyZIIq5Nt/urUnVXbt2+s1qr3713/8HDxtqjIx2nimGfxVGsTgKqMeIS+4abCE8ShVQEER4Hp+8L//FXVJrH8ouZJTgUdCJ5yBk1lvLXKj+AyJjLMUoDB1xywb9hHUiAEy4zPJNUKTrbzOtQ22+RIISLLciK7yyH7QU6z9vwDHbs2fezgGHemtMX+bXK54XKxMnNKiDbsAWE1Gt/1TflBpKoODGxhUSnws/kjpePPpYXUVyyA1/mQCIMDfkO0SgTuS+hVT7Xhhf/uHCpMpujDUTxydSGjnp+b16c7RHK8eUOrZelrr+Fz6mAOAR9llKFMOZhiAolQw0BmnNECSFSk1qfoIkGLoGCHZtR1I5xDEd7e52632h2O925wVXglaDplHboN36SccxSYSfJIqr1wOsmZphRZTiLMK+TVGNC2Smd4MBCSQXqYTZfoRw2LDOGMFb22E2Ys5cjMiq0nonAKgU1U73sK8jrfIPUhK+HGZdJamwHFonCNAI7rWIfbXMUMhPNLKBMcVsrsClVlBm7tUUTvOVfvgr6vc6bjvep19x9V3Zj1XniPHVajue8cnadD86h03dY5Xd1o9qpvnRX3E236/YW0mqljHns/Gfu9h/7URLPAAADgXicfVJNb9NAEHXiAk34SuHIZURalJQS4lz4qCpVcCiHFhVB2krZxFpvxsmq3rW7u6YNxv+D38WNXwPrxIiSVh1p5bdv3uyMZyZIIq5Nt/urUnVXbt2+s1qr3713/8HDxtqjIx2nimGfxVGsTgKqMeIS+4abCE8ShVQEER4Hp+8L//FXVJrH8ouZJTgUdCJ5yBk1lvLXKj+AyJjLMUoDB1xywb9hHUiAEy4zPJNUKTrbzOtQ22+RIISLLciK7yyH7QU6z9vwDHbs2fezgGHemtMX+bXK54XKxMnNKiDbsAWE1Gt/1TflBpKoODGxhUSnws/kjpePPpYXUVyyA1/mQCIMDfkO0SgTuS+hVT7Xhhf/uHCpMpujDUTxydSGjnp+b16c7RHK8eUOrZelrr+Fz6mAOAR9llKFMOZhiAolQw0BmnNECSFSk1qfoIkGLoGCHZtR1I5xDEd7e52632h2O925wVXglaDplHboN36SccxSYSfJIqr1wOsmZphRZTiLMK+TVGNC2Smd4MBCSQXqYTZfoRw2LDOGMFb22E2Ys5cjMiq0nonAKgU1U73sK8jrfIPUhK+HGZdJamwHFonCNAI7rWIfbXMUMhPNLKBMcVsrsClVlBm7tUUTvOVfvgr6vc6bjvep19x9V3Zj1XniPHVajue8cnadD86h03dY5Xd1o9qpvnRX3E236/YW0mqljHns/Gfu9h/7URLPAAADgXicfVJNb9NAEHXiAk34SuHIZURalJQS4lz4qCpVcCiHFhVB2krZxFpvxsmq3rW7u6YNxv+D38WNXwPrxIiSVh1p5bdv3uyMZyZIIq5Nt/urUnVXbt2+s1qr3713/8HDxtqjIx2nimGfxVGsTgKqMeIS+4abCE8ShVQEER4Hp+8L//FXVJrH8ouZJTgUdCJ5yBk1lvLXKj+AyJjLMUoDB1xywb9hHUiAEy4zPJNUKTrbzOtQ22+RIISLLciK7yyH7QU6z9vwDHbs2fezgGHemtMX+bXK54XKxMnNKiDbsAWE1Gt/1TflBpKoODGxhUSnws/kjpePPpYXUVyyA1/mQCIMDfkO0SgTuS+hVT7Xhhf/uHCpMpujDUTxydSGjnp+b16c7RHK8eUOrZelrr+Fz6mAOAR9llKFMOZhiAolQw0BmnNECSFSk1qfoIkGLoGCHZtR1I5xDEd7e52632h2O925wVXglaDplHboN36SccxSYSfJIqr1wOsmZphRZTiLMK+TVGNC2Smd4MBCSQXqYTZfoRw2LDOGMFb22E2Ys5cjMiq0nonAKgU1U73sK8jrfIPUhK+HGZdJamwHFonCNAI7rWIfbXMUMhPNLKBMcVsrsClVlBm7tUUTvOVfvgr6vc6bjvep19x9V3Zj1XniPHVajue8cnadD86h03dY5Xd1o9qpvnRX3E236/YW0mqljHns/Gfu9h/7URLPAAADgXicfVJNb9NAEHXiAk34SuHIZURalJQS4lz4qCpVcCiHFhVB2krZxFpvxsmq3rW7u6YNxv+D38WNXwPrxIiSVh1p5bdv3uyMZyZIIq5Nt/urUnVXbt2+s1qr3713/8HDxtqjIx2nimGfxVGsTgKqMeIS+4abCE8ShVQEER4Hp+8L//FXVJrH8ouZJTgUdCJ5yBk1lvLXKj+AyJjLMUoDB1xywb9hHUiAEy4zPJNUKTrbzOtQ22+RIISLLciK7yyH7QU6z9vwDHbs2fezgGHemtMX+bXK54XKxMnNKiDbsAWE1Gt/1TflBpKoODGxhUSnws/kjpePPpYXUVyyA1/mQCIMDfkO0SgTuS+hVT7Xhhf/uHCpMpujDUTxydSGjnp+b16c7RHK8eUOrZelrr+Fz6mAOAR9llKFMOZhiAolQw0BmnNECSFSk1qfoIkGLoGCHZtR1I5xDEd7e52632h2O925wVXglaDplHboN36SccxSYSfJIqr1wOsmZphRZTiLMK+TVGNC2Smd4MBCSQXqYTZfoRw2LDOGMFb22E2Ys5cjMiq0nonAKgU1U73sK8jrfIPUhK+HGZdJamwHFonCNAI7rWIfbXMUMhPNLKBMcVsrsClVlBm7tUUTvOVfvgr6vc6bjvep19x9V3Zj1XniPHVajue8cnadD86h03dY5Xd1o9qpvnRX3E236/YW0mqljHns/Gfu9h/7URLP

110

BCELoss BCEandTopoLoss GroundtruthImage

Accounting for Topology

111

Iterative Refinement

Use the same network to progressively refine the results keeping the number of parameters constant

1 2 3

Image Iter1 Iter2 Iter3 Groundtruth

112

Delineation Steps

1.Compute a probability map.

2. Sample and connect the samples.

3. Assign a weight to the paths.

4. Retain the best paths.

113

Delineation 2019These two steps are performed by the same network

Mosinska et al, PAMI’19.

114

Dual Use U-Net

Tubularity Map

Path score

Image and

Binary Mask

115

Streets Of Toronto

— False negatives — False positives

116

Dendrites And Axons

117

On the Job

The new job is good, we do of course a lot of Deep Learning, but also some good old-school computer vision e.g. registration 😉 So the material from Computer Vision class is definitely helpful and I wouldn't change it to another all-Deep Learning class (even in the light of today's Turing Award).

Best,

Agata

118

1998 - 2038

?It is difficult to make predictions, especially about the future.

Sometimes attributed to Niels Bohr.

1998

!ℒ#$%

ℒ&'(

2018 2038

119

In Short

• Edge and image information is noisy.

• Models are required to make sense of it.

! An appropriate combination of graph-based techniques, machine learning, and semi-automated tools is required.

120

Alpha Go Analogy

• Uses Deep Nets to find the most promising locations to focus on.

• Performs Tree based search when possible.

• Relies on reinforcement learning and other ML techniques to train.

—> Beat the world champion in 2017.

Low-level processing

High-level processing

Documents

Delineation - Moodle 2020-2021 · Interactive Delineation •The user provides the start and end points (red x). •The algorithm does the rest (yellow line). Image. Gradient. 10