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Model Fitting

Hao Jiang

Computer Science Department

Oct 8, 2009

Hough transform for circles

For a fixed radius r, unknown gradient direction

• Circle: center (a,b) and radius r222 )()( rbyax ii

Image space Hough space a

b

Source: K. Grauman

Hough transform for circles

For a fixed radius r, unknown gradient direction

• Circle: center (a,b) and radius r222 )()( rbyax ii

Image space Hough space

Intersection: most votes for center occur here.

Source: K. Grauman

Hough transform for circles

For an unknown radius r, unknown gradient direction

• Circle: center (a,b) and radius r222 )()( rbyax ii

Hough spaceImage space

b

a

r

Source: K. Grauman

Hough transform for circles

For an unknown radius r, unknown gradient direction

• Circle: center (a,b) and radius r222 )()( rbyax ii

Hough spaceImage space

b

a

r

Source: K. Grauman

Hough transform for circles

For an unknown radius r, known gradient direction

• Circle: center (a,b) and radius r222 )()( rbyax ii

Hough spaceImage space

θ

x

Source: K. Grauman

Hough transform for circles

For every edge pixel (x,y) :

For each possible radius value r:

For each possible gradient direction θ: // or use estimated gradient

a = x – r cos(θ)

b = y + r sin(θ)

H[a,b,r] += 1

end

end

Source: K. Grauman

Example: detecting circles with Hough

Crosshair indicates results of Hough transform,bounding box found via motion differencing.

Source: K. Grauman

Original Edges

Example: detecting circles with Hough

Votes: Penny

Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penny).

Source: K. Grauman

Original Edges

Example: detecting circles with Hough

Votes: Quarter

Coin finding sample images from: Vivek KwatraSource: K. Grauman

General Hough Transform

Hough transform can also be used to detection arbitrary shaped object.

11Template Target

General Hough Transform

Hough transform can also be used to detection arbitrary shaped object.

12Template Target

General Hough Transform

Hough transform can also be used to detection arbitrary shaped object.

13Template Target

General Hough Transform

Hough transform can also be used to detection arbitrary shaped object.

14Template Target

General Hough Transform Rotation Invariance

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Example

model shape Source: L. Lazebnik

Say we’ve already stored a table of displacement vectors as a function of edge orientation for this model shape.

Example

displacement vectors for model points

Now we want to look at some edge points detected in a new image, and vote on the position of that shape.

Example

range of voting locations for test point

Example

range of voting locations for test point

Example

votes for points with θ =

Example

displacement vectors for model points

Example

range of voting locations for test point

votes for points with θ =

Example

Application in recognition

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

B. Leibe, A. Leonardis, and B. Schiele,Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004

training image

visual codeword withdisplacement vectors

Source: L. Lazebnik

Application in recognition

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

test image

Source: L. Lazebnik

RANSAC

RANSAC – Random Sampling Consensus

Procedure: Randomly generate a hypothesis Test the hypothesis Repeat for large enough times and choose the best one

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Line Detection

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Line Detection

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Line Detection

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Line Detection

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Count “inliers”: 7

Line Detection

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Inlier number: 3

Line Detection

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After many trails, we decide this is the best line.

Probability of Success

What’s the success rate of RANSAC given a fixed number of validations?

Given a success rate, how to choose the minimum number of validations?

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Object Detection Using RANSAC

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Template Target

Scale and Rotation Invariant Feature

SIFT (D. Lowe, UBC)

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Stable Feature

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Stable Feature

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Local max/min point’s valuesare stable when the scale changes

SIFT

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Filteringthe imageusing filters at differentscales.(for example using Gaussian filter)

Difference of Gaussian

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SIFT Feature Points

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(b) Shows the points at the local max/min of DOG scale space forthe image in (a).