CAP 5415 Computer Vision Fall 2013 - crcv.ucf.edu · the max cell goes up, too Alper Yilmaz, Mubarak Shah, Fall 2011 UCF. Difficulties What is the increments for and p. – too large?

Alper Yilmaz, Mubarak Shah, Fall 2011 UCF

CAP 5415 Computer VisionFall 2013

Hough TransformLecture-17Sections 4.2, 4.3 Fundamentals of Computer Vision

Image Feature Extraction

Edges (edge pixels)– Sobel, Roberts, Prewit– Laplacian of Gaussian– Canny

Interest Points– Harris– SIFT

Descriptors– SIFT– HOG

Alper Yilmaz, Fall 2004 UCF

Shape Features

Straight Lines Circles and Ellipses Arbitrary Shapes


How to Fit A Line?


cmxy

How to Fit A Line?

Least squares Fit (over constraint) RANSAC (constraint) Hough Transform (under constraint)


Least Squares Fit

Standard linear solution to estimating unknowns.– If we know which points belong to which line– Or if there is only one line

cmxfcmxy ,,

i

ii cmxfyE 2,, Minimize

Take derivatives wrt m and c set them to 0Alper Yilmaz, Mubarak Shah, Fall 2011

UCF

Line Fitting

cmxy

cmxy

cmxycmxy

nn

22

11

D

A

n

B

n

cm

x

xx

y

yy

1111

2

1

2

1

ADB

BAAAD

DAAAABAAA

ADABA

TT

TTTT

TT

1

11


RANSAC: Random Sampling and Consensus


1. Randomly select two points to fit a line2. Find the error between the estimated solution and all

other points. • If the error is less than tolerance, then quit, else go

to step (1).

Line Fitting: Segmentation


• Several Lines• How do we Know which points belong to

which lines?

Hough Transform

METHOD AND MEANS FOR RECOGNIZING COMPLEX PATTERNS, Paul V. C. Hough et al– Inventors: Paul V. C. Hough, Paul V. C. Hough

Current U.S. Classification: 382/281; 342/176; 342/190; 382/202

– http://www.google.com/patents?vid=3069654


Line Fitting: Hough Transform


Line equation

Rewrite this equation

For particular edge point i this becomes

This is an equation of a line in (c,m) space.

intercept- is slope, is ycmcmxy

ymxc )(

ii ymxc )(

Line Fitting: Hough Transform


ii ymxc )(

Hough Transform Algorithm for Fitting Straight Lines


Polar Form of Equation of Line

sincos yxp


Problematic for vertical linesm and c grow to infinity

ymxc ji )(

Use from gradient

Image Gradient


x

y

yx

yx

SS

SS

SS

1

22

tan

)(

),(

direction

magnitude

VectorGradient yx ,

yx ,

yx ,

Hough Transform for Polar Form of Equation of Line


Line Fitting

p


Line Fitting


Line Fitting Examples

noisy

ideal

very noisy


Noise Factor

This is the number of votes that the real lineof 20 points gets with increasing noise

Noise Factor

as the noise increases in a picture without a line, the number of points in the max cell goes up, too


Difficulties

What is the increments for and p.– too large? We cannot distinguish between

different lines– too small? noise causes lines to be missed


Circle Fitting

Similar to line fitting– Three unknowns

Construct a 3D accumulator array A– Dimensions: x0, y0, r

Fix one of the parameters and loop for the others Increment corresponding entry in A. Find the local maxima in A

0)()( 222 ryyxx oo


More Practical Circle Fitting

Use the tangent direction at the edge point

Compute x0, y0 given x, y, r

sincos

0

0

ryyrxx




Examples

Generalized Hough Transform

Used for shapes with no analytical expression Requires training

– Object of known shape– Generate model

R-table

Similar approach to line and circle fitting during detection


Generating R-table

Compute centroid For each edge compute its distance

to centroid

Find edge orientation (gradient angle) Construct a table of angles and r

values

(xo,yo)

(x,y)

0

0

yyxx

yx

r


Generating R-table

(xo,yo)

(x,y)

(x,y)

Ф1

Ф2Ф1Ф2

Ф3

Ф4

r1, r2, r3 …r14, r21, r23 …r41, r42, r33 …r10, r12, r13 …


r1

Detecting shape

known – Edge points (x,y)– Gradient angle at every edge point – R-table of the shape needs to be determined

For each edge point find store it in corresponding row of R-table

Create an accumulator array of 2D (x,y)



Rotation and Scale Invariance

Rotation around Z-axis

Scaling

Rotation+scaling

cossinsincos

yxyyxx

syysxx

cossinsincos

yxsyyxsx


Rotation and Scale Invariance

Replace equations 4.13 and 4.14 in Algorithm 4.8 by following and loop for scale and rotation angles.

Alpr Yilmaz, Mubarak Shah, Fall 2011 UCF


Documents

CAP 5415 Computer Vision Fall 2013 - crcv.ucf.edu · the max cell goes up, too Alper Yilmaz, Mubarak Shah, Fall 2011 UCF. Difficulties What is the increments for and p. – too large?