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

Outline � The problem � The objectives � The solution � Example: circle testing � Experiment: circle testing � Concluding remarks DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 2

Curve Detection Problem • Binary image, • target curve type, • prior information Hypothesis Hypothesis Testing generation • Hypotheses • Detected curves Supported mainly by additional supports number of pixels DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 3

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. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 4

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. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 5

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. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 6

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: s 1 = 0 and s 2 = 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.) DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 7

The transform � From (1), the transform is used to map edge pixels in the binary image into member of the system. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 8

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. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 9

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 . DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 10

An example: circle testing � Let h be a hypothesis with center ( x 0 , y 0 ) and radius r 0 . Choose s 1 =0 and s 2 =0 to be circles with radius sqrt(2) r 0 , with λ (s 1 )=0, λ (s 2 )=1. and λ (h) = ¼ . DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 11

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

The system and the function � The system: � The function: DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 13

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]. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 14

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. DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 15

The input image DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 16

Comparison: definitions (I) DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 17

Comparison: definitions (II) DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 18

Result (I): the proposed DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 19

Result (II): global threshold DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 20

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 0 ) 1 0 . 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 0 . 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 ) 0 0 . 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 ) 0 0 . 5 8 8 9 3 . 5 0 1 C i r c l e P a r a m e t e r ( 2 7 , 2 4 , 1 2 ) 0 0 . 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 ) 0 0 . 5 0 9 8 0 C i r c l e P a r a m e t e r ( 2 5 , 2 5 , 2 0 ) 1 0 . 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 ) 0 0 . 4 9 0 7 0 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 0 . 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 0 ) 1 0 . 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 0 ) 1 0 . 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 ) 0 0 . 5 8 8 9 3 . 5 0 1 C i r c l e P a r a m e t e r ( 3 0 , 2 3 , 1 5 ) 0 0 . 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 ) 0 0 . 4 0 4 8 3 . 3 6 7 C i r c l e P a r a m e t e r ( 2 7 , 2 0 , 1 5 ) 0 0 . 4 3 3 3 3 . 3 0 3 C i r c l e P a r a m e t e r ( 2 7 , 3 0 , 1 5 ) 0 0 . 3 8 8 9 3 . 2 4 9 DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 21

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 0 ) 1 0 . 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 0 . 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 ) 0 0 . 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 ) 0 0 . 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 ) 0 0 . 5 8 3 3 0 C i r c l e P a r a m e t e r ( 2 4 , 2 8 , 1 7 ) 0 0 . 5 5 8 8 0 C i r c l e P a r a m e t e r ( 2 3 , 2 7 , 1 8 ) 0 0 . 5 5 5 6 0 C i r c l e P a r a m e t e r ( 2 3 , 2 8 , 1 8 ) 0 0 . 5 5 5 6 0 C i r c l e P a r a m e t e r ( 2 5 , 2 8 , 1 7 ) 0 0 . 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 ) 0 0 . 5 3 9 2 0 C i r c l e P a r a m e t e r ( 2 4 , 2 4 , 1 5 ) 0 0 . 5 3 3 3 0 C i r c l e P a r a m e t e r ( 2 9 , 2 8 , 1 5 ) 0 0 . 5 3 3 3 0 C i r c l e P a r a m e t e r ( 2 4 , 2 5 , 2 0 ) 0 0 . 5 2 5 0 C i r c l e P a r a m e t e r ( 2 4 , 2 5 , 1 4 ) 0 0 . 5 2 3 8 0 C i r c l e P a r a m e t e r ( 2 8 , 2 9 , 1 5 ) 0 0 . 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 0 ) 1 0 . 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 ) 0 0 . 5 0 9 3 0 DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 22

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 0 . 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 0 ) 1 0 . 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 ) 0 0 . 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 0 ) 1 0 . 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 ) 0 0 . 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 0 , 1 5 ) 0 0 . 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 ) 0 0 . 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 ) 0 0 . 2 4 6 7 3 . 1 5 7 C i r c l e P a r a m e t e r ( 3 0 , 2 3 , 1 5 ) 0 0 . 4 7 7 8 3 . 0 8 8 DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 23

Execution time DICTA, 12 Dec 2003 Y C Cheng, NTUT, Taiwan 24