DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 1 Outline The problem - - PowerPoint PPT Presentation

DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 1

DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 2

Outline

The problem The objectives The solution Example: circle testing Experiment: circle testing Concluding remarks

DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 3

Curve Detection Problem

Hypothesis generation

• Binary image,
• target curve type,
• prior information

Hypothesis Testing

• Hypotheses
• Detected curves

Supported mainly by number of pixels additional supports

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Objectives

We are looking for a measure that can be

used independently of the process to generate hypotheses.

We are interested in adding useful statistic

supports as a result of testing. Ideally, the statistics should allow true positives to be separated from false positives.

The implementation should have reasonable

computational complexities.

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Hypothesis generation and hypothesis testing

In light of these objectives, note that

some of the most popular post- processing strategies either

do not provide the statistical support that

we are after, or

are closely coupled with the hypothesis

generation process, e.g., post-processing for Hough transform in the parameter space.

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The solution

Construct a system of curves, which the

hypothesis is a member of.

Derive a function that distributes pixels

in the image into members of the system.

Compute statistics of the hypothesis

from the distributions.

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The system of curves

Let h be an instance of a curve found by a

hypothesis generator such as Hough transform or RANSAC. The instance h is a member of the system: s1 = 0 and s2 = 0 are two distinct curves of the same type as h and intersect h at two points (respectively, four points) if h is a circle (respectively, an ellipse.)

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The transform

From (1), the transform

is used to map edge pixels in the binary image into member of the system.

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Mapping pixels (I)

How edge pixels are mapped to

members of the system:

Edge pixels of the hypothesis are mapped

into a particular member;

Edge pixels from a curve j different from

the hypothesis are mapped into members, with each one receiving O(deg(j) * deg(h)) edge pixels; and.

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Mapping pixels (II)

Edge pixels that are from uniformly

distributed noise are mapped so that a member with B edge pixels receives µB edge pixels, where µ is the level of noise

Result of the the mapping is recorded

as a histogram of distribution of edge pixels against a partitioned range of λ.

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An example: circle testing

Let h be a hypothesis with center (x0,y0)

and radius r0. Choose s1=0 and s2 =0 to be circles with radius sqrt(2) r0, with λ(s1)=0, λ(s2)=1. and λ(h) = ¼.

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DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 13

The system and the function

The system: The function:

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Implementation issues

In computing the statistics, it usually

suffices to look at a neighborhood around the the hypothesis.

In the example, since hypothesis is located

at λ = ¼, we may compute the statistics of the histogram in the λ range [0,1/2].

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Experiment in circle testing

The proposed method is compared with

the global threshold.

Hypotheses are generated by the

standard Hough transform for circle.

To compensate for size, both the global

threshold and the λ-Histogram are scaled by radius.

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The input image

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Comparison: definitions (I)

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Comparison: definitions (II)

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Result (I): the proposed

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Result (II): global threshold

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At 5% noise

true(1)/fasle(0) GT proposed C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 ) 1 . 7 6 6 7 7 . 6 8 3 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 5 ) 1 . 6 5 5 6 7 . 9 4 8 C i r c l e P a r a m e t e r ( 2 8 , 2 5 , 1 2 ) . 6 3 8 9 2 . 9 2 2 C i r c l e P a r a m e t e r ( 2 5 , 2 6 , 1 5 ) . 5 8 8 9 3 . 5 1 C i r c l e P a r a m e t e r ( 2 7 , 2 4 , 1 2 ) . 5 5 5 6 2 . 4 1 2 C i r c l e P a r a m e t e r ( 2 4 , 2 8 , 1 7 ) . 5 9 8 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 2 ) 1 . 5 5 . 3 2 6 C i r c l e P a r a m e t e r ( 2 3 , 2 8 , 1 8 ) . 4 9 7 true(1)/fasle(0) GT proposed C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 5 ) 1 . 6 5 5 6 7 . 9 4 8 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 ) 1 . 7 6 6 7 7 . 6 8 3 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 2 ) 1 . 5 5 . 3 2 6 C i r c l e P a r a m e t e r ( 2 5 , 2 6 , 1 5 ) . 5 8 8 9 3 . 5 1 C i r c l e P a r a m e t e r ( 3 , 2 3 , 1 5 ) . 4 4 4 4 3 . 3 7 4 C i r c l e P a r a m e t e r ( 2 5 , 2 4 , 2 1 ) . 4 4 8 3 . 3 6 7 C i r c l e P a r a m e t e r ( 2 7 , 2 , 1 5 ) . 4 3 3 3 3 . 3 3 C i r c l e P a r a m e t e r ( 2 7 , 3 , 1 5 ) . 3 8 8 9 3 . 2 4 9

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At 7% noise: global threshold

true(1)/false(0) GT proposed C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 ) 1 . 8 6 . 5 4 7 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 5 ) 1 . 7 2 2 2 7 . 3 5 5 C i r c l e P a r a m e t e r ( 2 8 , 2 5 , 1 2 ) . 6 3 8 9 2 . 4 9 4 C i r c l e P a r a m e t e r ( 2 5 , 2 6 , 1 5 ) . 6 2 2 2 3 . 3 6 8 C i r c l e P a r a m e t e r ( 2 7 , 2 4 , 1 2 ) . 5 8 3 3 C i r c l e P a r a m e t e r ( 2 4 , 2 8 , 1 7 ) . 5 5 8 8 C i r c l e P a r a m e t e r ( 2 3 , 2 7 , 1 8 ) . 5 5 5 6 C i r c l e P a r a m e t e r ( 2 3 , 2 8 , 1 8 ) . 5 5 5 6 C i r c l e P a r a m e t e r ( 2 5 , 2 8 , 1 7 ) . 5 3 9 2 2 . 4 9 3 C i r c l e P a r a m e t e r ( 2 6 , 2 8 , 1 7 ) . 5 3 9 2 C i r c l e P a r a m e t e r ( 2 4 , 2 4 , 1 5 ) . 5 3 3 3 C i r c l e P a r a m e t e r ( 2 9 , 2 8 , 1 5 ) . 5 3 3 3 C i r c l e P a r a m e t e r ( 2 4 , 2 5 , 2 ) . 5 2 5 C i r c l e P a r a m e t e r ( 2 4 , 2 5 , 1 4 ) . 5 2 3 8 C i r c l e P a r a m e t e r ( 2 8 , 2 9 , 1 5 ) . 5 2 2 2 1 . 8 6 9 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 2 ) 1 . 5 1 6 7 4 . 1 1 9 C i r c l e P a r a m e t e r ( 2 5 , 2 8 , 1 8 ) . 5 9 3

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At 7% noise: proposed

true(1)/false(0) GT proposed C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 5 ) 1 . 7 2 2 2 7 . 3 5 5 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 1 ) 1 . 8 6 . 5 4 7 C i r c l e P a r a m e t e r ( 2 9 , 3 1 , 1 7 ) . 4 1 1 8 4 . 2 3 3 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 2 ) 1 . 5 1 6 7 4 . 1 1 9 C i r c l e P a r a m e t e r ( 2 9 , 2 4 , 1 9 ) . 3 5 9 6 3 . 6 7 C i r c l e P a r a m e t e r ( 2 7 , 2 , 1 5 ) . 4 8 8 9 3 . 4 3 5 C i r c l e P a r a m e t e r ( 2 5 , 2 6 , 1 5 ) . 6 2 2 2 3 . 3 6 8 C i r c l e P a r a m e t e r ( 3 5 , 2 6 , 2 5 ) . 2 4 6 7 3 . 1 5 7 C i r c l e P a r a m e t e r ( 3 , 2 3 , 1 5 ) . 4 7 7 8 3 . 8 8

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Execution time

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Concluding remarks

A hypothesis testing strategy for curve testing

is proposed.

It is independent of the curve detection process. It adds an image space statistic support to a

hypothesis.

It is as efficient as the global threshold.

Preliminary results on circle testing are

• btained. Results in testing real images are

being obtained.

Implementation of ellipse testing and line

testing are underway.

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Thank you!