Introduction

3D Computer Visionand Video Computing Introduction

Part IFeature Extraction (2)

Edge Detection

CSc I6716Spring 2013

Zhigang Zhu, City College of New York [email protected]

3D Computer Visionand Video Computing Edge Detection

n What’s an edge?l “He was sitting on the Edge of his seat.”l “She paints with a hard Edge.”l “I almost ran off the Edge of the road.”l “She was standing by the Edge of the woods.”l “Film negatives should only be handled by their Edges.”l “We are on the Edge of tomorrow.”l “He likes to live life on the Edge.”l “She is feeling rather Edgy.”

n The definition of Edge is not always clear.n In Computer Vision, Edge is usually related to a

discontinuity within a local set of pixels.

3D Computer Visionand Video Computing Discontinuities

n A: Depth discontinuity: abrupt depth change in the worldn B: Surface normal discontinuity: change in surface orientationn C: Illumination discontinuity: shadows, lighting changesn D: Reflectance discontinuity: surface properties, markings

A

C

B

D

3D Computer Visionand Video Computing Illusory Edges

n Illusory edges will not be detectable by the algorithms that we will discuss

n No change in image irradiance - no image processing algorithm can directly address these situations

n Computer vision can deal with these sorts of things by drawing on information external to the image (perceptual grouping techniques)

Kanizsa Triangles

3D Computer Visionand Video Computing Another One

3D Computer Visionand Video Computing Goal

n Devise computational algorithms for the extraction of significant edges from the image.

n What is meant by significant is unclear.l Partly defined by the context in which the edge detector

is being applied

3D Computer Visionand Video Computing Edgels

n Define a local edge or edgel to be a rapid change in the image function over a small areal implies that edgels should be detectable over a local

neighborhoodn Edgels are NOT contours, boundaries, or lines

l edgels may lend support to the existence of those structuresl these structures are typically constructed from edgels

n Edgels have propertiesl Orientationl Magnitudel Position

3D Computer Visionand Video Computing Outline

n First order edge detectors (lecture - required)l Mathematicsl 1x2, Roberts, Sobel, Prewitt

n Canny edge detector (after-class reading)n Second order edge detector (after-class reading)

l Laplacian, LOG / DOGn Hough Transform – detect by voting

l Linesl Circlesl Other shapes

3D Computer Visionand Video Computing Locating Edgels

Rapid change in image => high local gradient => differentiation

f(x) = step edge

1st Derivative f ’(x)

2nd Derivative -f ’’(x)

maximum

zero crossing

3D Computer Visionand Video Computing Reality

3D Computer Visionand Video Computing Properties of an Edge

OrientationOriginal

Orientation

Magnitude

Position

3D Computer Visionand Video Computing Quantitative Edge Descriptors

n Edge Orientationl Edge Normal - unit vector in the direction of

maximum intensity change (maximum intensity gradient)

l Edge Direction - unit vector perpendicular to the edge normal

n Edge Position or Centerl image position at which edge is located

(usually saved as binary image)n Edge Strength / Magnitude

l related to local contrast or gradient - how rapid is the intensity variation across the edge along the edge normal.

3D Computer Visionand Video Computing Edge Degradation in Noise

Ideal step edge Step edge + noise

Increasing noise

3D Computer Visionand Video Computing Real Image

3D Computer Visionand Video Computing Edge Detection: Typical

n Noise Smoothingl Suppress as much noise as possible while retaining

‘true’ edgesl In the absence of other information, assume ‘white’

noise with a Gaussian distributionn Edge Enhancement

l Design a filter that responds to edges; filter output high are edge pixels and low elsewhere

n Edge Localizationl Determine which edge pixels should be discarded as

noise and which should be retainedu thin wide edges to 1-pixel width (nonmaximum

suppression)u establish minimum value to declare a local maximum from

edge filter to be an edge (thresholding)

3D Computer Visionand Video Computing Edge Detection Methods

n 1st Derivative Estimatel Gradient edge detectionl Compass edge detectionl Canny edge detector (*)

n 2nd Derivative Estimatel Laplacianl Difference of Gaussians

n Parametric Edge Models (*)

3D Computer Visionand Video Computing Gradient Methods

F(x)

xF’(x)

x

Edge= sharp variation

Large first derivative

3D Computer Visionand Video Computing Gradient of a Function

n Assume f is a continuous function in (x,y). Then

n are the rates of change of the function f in the x and y directions, respectively.

n The vector (Dx, Dy) is called the gradient of f.n This vector has a magnitude: and an orientation:

n q is the direction of the maximum change in f.n S is the size of that change.

s = Dx2+Dy

2

Dx

Dyq = tan-1 ( )

yf

xf

yx

D

D ,

3D Computer Visionand Video Computing Geometric Interpretation

n Butl I(i,j) is not a continuous function.

n Thereforel look for discrete approximations to the gradient.

y

x

f(x,y) S

f

Dy

qDx

3D Computer Visionand Video Computing Discrete Approximations

df(x)dx = lim

Dx 0

f(x + Dx) - f(x)Dx

df(x)dx

f(x) - f(x-1)1@ xx-1

f(x)

Convolve with -1 1

3D Computer Visionand Video Computing In Two Dimensions

n Discrete image function I

n Derivatives Differences

i

j

Image

row i-1

row i

row i+1

col j-1 col j col j+1

I(i-1,j-1)

I(i,j-1)

I(i+1,j-1)

I(i-1,j)

I(i,j)

I(i+1,j)

I(i-1,j+1)

I(i,j+1)

I(i+1,j+1)

-1 11

-1DjI = DiI =

3D Computer Visionand Video Computing 1x2 Example

1x2 Vertical

1x2 HorizontalCombined

3D Computer Visionand Video Computing Smoothing and Edge Detection

n Derivatives are 'noisy' operationsl edges are a high spatial frequency phenomenonl edge detectors are sensitive to and accent noise

n Averaging reduces noisel spatial averages can be computed using masks

n Combine smoothing with edge detection.

1 1 1

1 1 1

1 1 1

1 / 9 x

1 1 1

1 1 1

1 1 1

1 1 1

1 0 1

1 1 1

1 / 8 x

3D Computer Visionand Video Computing Effect of Blurring

Original Orig+1 Iter Orig+2 Iter

Image

Edges

ThresholdedEdges

3D Computer Visionand Video Computing Combining the Two

n Applying this mask is equivalent to taking the difference of averages on either side of the central pixel.

-1 -1 -1

11 1

0 0

Average

Average

3D Computer Visionand Video Computing Many Different Kernels

n Variablesl Size of kernell Pattern of weights

n 1x2 Operator (we’ve already seen this one

-1 11

-1DjI = DiI =

3D Computer Visionand Video Computing Roberts Cross Operator

n Does not return any information about the orientation of the edge 22 ),1()1,()1,1(),( yxIyxIyxIyxI

or

1 00 -1

0 1-1 0+

| I(x, y) - I(x+1, y+1) | + | I(x, y+1) - I(x+1, y) |

[ I(x, y) - I(x+1, y+1) ]2 + [ I(x, y+1) - I(x+1, y) ]2S =

S =

3D Computer Visionand Video Computing Sobel Operator

-1 -2 -1 0 0 0 1 2 1

-1 0 1-2 0 2 -1 0 1

S1= S2 =

Edge Magnitude =

Edge Direction =

S1 + S22 2

tan-1S1

S2

3D Computer Visionand Video Computing Anatomy of the Sobel

1 2 1 0 0 0

-1 -2 -1

-1 0 1 -2 0 2-1 0 1

= 1/4 * [ 1 2 1] 1 0 -1

Sobel kernel is separable!

1

-1

-2 1

1

2 Averaging done parallel to edge

1/4

1/4

= 1/4 * [-1 0 -1] 1 2 1

+

3D Computer Visionand Video Computing Prewitt Operator

-1 -1 -1 0 0 0 1 1 1

-1 0 1-1 0 1 -1 0 1

P1= P2 =

Edge Magnitude =

Edge Direction =

P1 + P22 2

tan-1P1

P2

3D Computer Visionand Video Computing Large Masks

1 x 2

1 x 5

1 x 9

1 x 9 uniform weights

-1 0 0 0 0 0 0 0 1

-1 -1 -1 -1 0 1 1 1 1

-1 0 0 0 1

-1 1

What happens as the mask size increases?

3D Computer Visionand Video Computing Large Kernels

7x7 Horizontal Edges only

13x13 Horizontal Edges only

3D Computer Visionand Video Computing Compass Masks

n Use eight masks aligned with the usual compass directions

n Select largest response (magnitude)n Orientation is the direction associated with the largest

response

SE

E

NENNW

W

SE S(-)

(+)

3D Computer Visionand Video Computing Many Different Kernels

1

111

1

-1 -1

Prewitt 1

-2

-1

-3

555

-3

-3 -3 -3

Kirsch

0

0

111

0

-1 -1 -1

Prewitt 2

0 0

121

0

-1 -2 -1

Sobel

0

-1 -1

11

0 0 0

- 2

2

Frei & Chen

3D Computer Visionand Video Computing Robinson Compass Masks

-1 0 1-2 0 2 -1 0 1

0 1 2-1 0 1 -2 -1 0

1 2 1 0 0 0 -1 -2 -1

2 1 0 1 0 -1 0 -1 -2

1 0 -1 2 0 -2 1 0 -1

0 -1 -2 1 0 -1 2 1 0

-1 -2 -1 0 0 0 1 2 1

-2 -1 0-1 0 1 0 1 2

3D Computer Visionand Video Computing Analysis of Edge Kernels

n Analysis based on a step edge inclined at an angle q (relative to y-axis) through center of window.

n Robinson/Sobel: true edge contrast less than 1.6% different from that computed by the operator.

n Error in edge directionl Robinson/Sobel: less than 1.5 degrees errorl Prewitt: less than 7.5 degrees error

n Summaryl Typically, 3 x 3 gradient operators perform better than 2 x 2.l Prewitt2 and Sobel perform better than any of the other 3x3 gradient

estimation operators.l In low signal to noise ratio situations, gradient estimation operators of

size larger than 3 x 3 have improved performance.l In large masks, weighting by distance from the central pixel is

beneficial.

3D Computer Visionand Video Computing Prewitt Example

Santa Fe Mission Prewitt Horizontal and Vertical Edges Combined

3D Computer Visionand Video Computing Edge Thresholding

n Global approach

See Haralick paper for thresholding based on statistical significance tests.

0

1000

2000

3000

4000

5000

Num

ber o

f Pix

els

Edge Gradient Magnitude

64 128

T=64T=128

Edge Histogram

3D Computer Visionand Video Computing Reading and Homework

- Go through slides 40-71 after class

- Reading: Lecture notes on feature extraction

- Homework 2: Due after two weeks

You may try different filters in Matlab, but do your homework by your programming … …

3D Computer Visionand Video Computing Canny Edge Detector

n Probably most widely usedn LF. Canny, "A computational approach to edge detection",

IEEE Trans. Pattern Anal. Machine Intelligence (PAMI), vol. PAMI vii-g, pp. 679-697, 1986.

n Based on a set of criteria that should be satisfied by an edge detector:l Good detection. There should be a minimum number of false

negatives and false positives. l Good localization. The edge location must be reported as

close as possible to the correct position. l Only one response to a single edge.

Cost function which could be optimized using variational methods

3D Computer Visionand Video Computing Canny Results

I = imread(‘image file name’);BW1 = edge(I,'sobel');BW2 = edge(I,'canny');imshow(BW1)figure, imshow(BW2)

s=1, T2=255, T1=1

‘Y’ or ‘T’ junction problem with Canny operator

3D Computer Visionand Video Computing Canny Results

s=1, T2=255, T1=220 s=1, T2=128, T1=1 s=2, T2=128, T1=1

M. Heath, S. Sarkar, T. Sanocki, and K.W. Bowyer, "A Robust Visual Method for Assessing the Relative Performance of Edge-Detection Algorithms" IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 19, No. 12, December 1997, pp. 1338-1359.

http://marathon.csee.usf.edu/edge/edge_detection.html

3D Computer Visionand Video Computing

n Second derivatives…

3D Computer Visionand Video Computing Edges from Second Derivatives

n Digital gradient operators estimate the first derivative of the image function in two or more directions.

f(x) = step edge

1st Derivative f’(x)

2nd Derivative f’’(x)

maximum

zero crossing

GR

AD

IEN

TM

ETH

OD

S

3D Computer Visionand Video Computing Second Derivatives

n Second derivative = rate of change of first derivative.n Maxima of first derivative = zero crossings of second

derivative.n For a discrete function, derivatives can be approximated by

differencing.n Consider the one dimensional case:

..... f(i-2) f(i-1) f(i) f(i+1) f(i+2) .....

Df(i-1) Df(i) Df(i+1) Df(i+2)

D f(i-1)2 2D f(i+1)D f(i)2

D f(i) = D f(i+1) - D f(i)2

= f(i+1) - 2 f(i) + f(i-1)

-2 11Mask:

3D Computer Visionand Video Computing Laplacian Operator

n Now consider a two-dimensional function f(x,y).n The second partials of f(x,y) are not isotropic.n Can be shown that the smallest possible isotropic

second derivative operator is the Laplacian:

n Two-dimensional discrete approximation is:

1

1 1

1

-4

2

2

2

22

yf

xff

3D Computer Visionand Video Computing

-1 -1 24 -1 -1-1 -1 -1 -1 -1-1 -1 -1 -1 -1

-1 -1 -1 -1 -1-1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 +8 +8 +8 -1 -1 -1 -1 -1 -1 +8 +8 +8 -1 -1 -1 -1 -1 -1 +8 +8 +8 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Example Laplacian Kernels

n Note that these are not the optimal approximations to the Laplacian of the sizes shown.

5X5

9X9

3D Computer Visionand Video Computing Example Application

5x5 Laplacian Filter 9x9 Laplacian Filter

3D Computer Visionand Video Computing Detailed View of Results

3D Computer Visionand Video Computing Interpretation of the Laplacian

n Consider the definition of the discrete Laplacian:

n Rewrite as:

n Factor out -5 to get:

n Laplacian can be obtained, up to the constant -5, by subtracting the average value around a point (i,j) from the image value at the point (i,j)!l What window and what averaging function?

I = I(i+1,j)+I(i-1,j)+I(i,j+1)+I(i,j-1) - 4I(i,j)

looks like a window sum

I = I(i+1,j)+I(i-1,j)+I(i,j+1)+I(i,j-1)+I(i,j) - 5I(i,j)

I = -5 {I(i,j) - window average}

3D Computer Visionand Video ComputingEnhancement using the Laplacian

n The Laplacian can be used to enhance images:

n If (i,j) is in the middle of a flat region or long ramp: I-2I = In If (i,j) is at low end of ramp or edge: I-2I < In If (i,j) is at high end of ramp or edge: I-2I > I

n Effect is one of deblurring the image

I(i,j) - I(i,j) =5 I(i,j) -[I(i+1,j) + I(i-1,j) + I(i,j+1) + I(i,j-1)]

3D Computer Visionand Video Computing Laplacian Enhancement

Blurred Original 3x3 Laplacian Enhanced

3D Computer Visionand Video Computing Noise

n Second derivative, like first derivative, enhances noise

n Combine second derivative operator with a smoothing operator.

n Questions:l Nature of optimal smoothing filter.l How to detect intensity changes at a given

scale.l How to combine information across multiple

scales.n Smoothing operator should be

l 'tunable' in what it leaves behindl smooth and localized in image space.

n One operator which satisfies these two constraints is the Gaussian:

3D Computer Visionand Video Computing 2D Gaussian Distribution

n The two-dimensional Gaussian distribution is defined by:

n From this distribution, can generate smoothing masks whose width depends upon s:

e(x + y )2 2

2 s 2s 2

1G(x,y) =

x

y

3D Computer Visionand Video Computing s Defines Kernel ‘Width’

s2 = .25 s2 = 1.0 s2 = 4.0

3D Computer Visionand Video Computing Creating Gaussian Kernels

n The mask weights are evaluated from the Gaussian distribution:

n This can be rewritten as:

n This can now be evaluated over a window of size nxn to obtain a kernel in which the (0,0) value is 1.

n k is a scaling constant

W(i,j) = k * exp (- )i2 + j22 s2

= exp (- )i2 + j22 s2

W(i,j)k

3D Computer Visionand Video Computing Example

n Choose s = 2. and n = 7, then:2

.011 .039 .082 .105 .082 .039 .011

.011 .039 .082 .105 .082 .039 .011

.039 .135 .287 .368 .287 .135 .039

.105 .039 .779 1.000 .779 .368 .105.082 .287 .606 .779 .606 .287 .082

.082 .287 .606 .779 .606 .287 .082

.039 .135 .287 .368 .287 .135 .039

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

i

j

W(1,2)k = exp(-12+ 22

2*2 ) To make this value 1, choose k = 91.


1 4 7 10 7 4 1

7 26 55 71 55 26 7

4 12 26 33 26 12 4

1 4 7 10 7 4 1

7 26 55 71 55 26 7

4 12 26 33 26 12 4

10 33 71 91 71 33 10

W(i,j) = 1,115j = -3

3

i = -3

3

7x7 Gaussian Filter

Plot of Weight Values

3D Computer Visionand Video Computing Kernel Application

7x7 Gaussian Kernel 15x15 Gaussian Kernel

3D Computer Visionand Video Computing Why Gaussian for Smoothing

n Gaussian is not the only choice, but it has a number of important propertiesl If we convolve a Gaussian with another Gaussian, the

result is a Gaussianu This is called linear scale space

l Efficiency: separablel Central limit theorem

3D Computer Visionand Video Computing Why Gaussian for Smoothing

n Gaussian is separable

3D Computer Visionand Video Computing Why Gaussian for Smoothing – cont.

n Gaussian is the solution to the diffusion equation

n We can extend it to non-linear smoothing

3D Computer Visionand Video Computing 2G Filter

n Marr and Hildreth approach:1. Apply Gaussian smoothing using s's of increasing size:

2. Take the Laplacian of the resulting images:

3. Look for zero crossings.

n Second expression can be written as:

n Thus, can take Laplacian of the Gaussian and use that as the operator.

G I*

(G I)*

(2G ) * I

3D Computer Visionand Video Computing Mexican Hat Filter

n Laplacian of the Gaussian

n 2G is a circularly symmetric operator.n Also called the hat or Mexican-hat operator.

-1s 4

G (x,y) =2s2

(x + y )2 21 - e 2s2

(x + y )2 22

3D Computer Visionand Video Computing s2 Controls Size

s2 = 0.5 s2 = 1.0 s2 = 2.0

3D Computer Visionand Video Computing Kernels

n Remember the center surround cells in the human system?

0 0 -1 0 0

-1 -2 16 -2 -1 0 -1 -2 -1 0

0 -1 -2 -1 0 0 0 -1 0 0

0 0 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0

0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0

0 0 -1 -1 -1 -2 -3 -3 -3 -3 -3 -2 -1 -1 -1 0 0

0 0 -1 -1 -1 -2 -3 -3 -3 -3 -3 -2 -1 -1 -1 0 0

0 0 -1 -1 -2 -3 -3 -3 -3 -3 -3 -3 -2 -1 -1 0 0

0 0 -1 -1 -2 -3 -3 -3 -3 -3 -3 -3 -2 -1 -1 0 0

0 -1 -1 -2 -3 -3 -3 -2 -3 -2 -3 -3 -3 -2 -1 -1 0 0 -1 -2 -3 -3 -3 0 2 4 2 0 -3 -3 -3 -2 -1 0

0 -1 -1 -2 -3 -3 -3 -3 -3 -3 -3 -3 -3 -2 -1 -1 0 0 -1 -2 -3 -3 -3 0 2 4 2 0 -3 -3 -3 -2 -1 0

-1 -1 -3 -3 -3 0 4 10 12 10 4 0 -3 -3 -3 -1 -1

-1 -1 -3 -3 -3 0 4 10 12 10 4 0 -3 -3 -3 -1 -1

-1 -1 -3 -3 -2 2 10 18 21 18 10 2 -2 -3 -3 -1 -1

-1 -1 -3 -3 -2 2 10 18 21 18 10 2 -2 -3 -3 -1 -1 -1 -1 -3 -3 -3 4 12 21 24 21 112 4 -3 -3 -3 -1 -1

5x5

17 x 17


13x13 Kernel


13 x 13 Hat Filter Thesholded Positive

Thesholded Negative Zero Crossings

3D Computer Visionand Video Computing Scale Space

Thresholded Negative

17x17 LoG Filter Thresholded Positive

Zero Crossings

3D Computer Visionand Video Computing Scale Space

s2 = 2 s2 = 2

s2 =2 2 s2 = 4

3D Computer Visionand Video Computing Multi-Resolution Scale Space

n Observations:l For sufficiently different s 's, the zero crossings will be

unrelated unless there is 'something going on' in the image.l If there are coincident zero crossings in two or more

successive zero crossing images, then there is sufficient evidence for an edge in the image.

l If the coincident zero crossings disappear as s becomes larger, then either:

u two or more local intensity changes are being averaged together, oru two independent phenomena are operating to produce intensity changes

in the same region of the image but at different scales.

n Use these ideas to produce a 'first-pass' approach to edge detection using multi-resolution zero crossing data.

n Never completely worked outn See Tony Lindbergh’s thesis and papers

3D Computer Visionand Video Computing Color Edge Detection

n Typical Approachesl Fusion of results on R, G, B separately

l Multi-dimensional gradient methods

l Vector methodsl Color signatures: Stanford (Rubner and Thomasi)

3D Computer Visionand Video Computing Hierarchical Feature Extraction

n Most features are extracted by combining a small set of primitive features (edges, corners, regions)

l Grouping: which edges/corners/curves form a group?u perceptual organization at the intermediate-level of vision

l Model Fitting: what structure best describes the group?

n Consider a slightly simpler problem…..

3D Computer Visionand Video Computing From Edgels to Lines

n Given local edge elements:

n Can we organize these into more 'complete' structures, such as straight lines?

n Group edge points into lines?n Consider a fairly simple technique...

3D Computer Visionand Video Computing Edgels to Lines

n Given a set of local edge elementsl With or without orientation information

n How can we extract longer straight lines?n General idea:

l Find an alternative space in which lines map to pointsl Each edge element 'votes' for the straight line which it

may be a part of.l Points receiving a high number of votes might

correspond to actual straight lines in the image.n The idea behind the Hough transform is that a change

in representation converts a point grouping problem into a peak detection problem

3D Computer Visionand Video Computing Edgels to Lines

n Consider two (edge) points, P(x,y) and P’(x’,y’) in image space:

n The set of all lines through P=(x,y) is y=mx + b, for appropriate choices of m and b.l Similarly for P’

n But this is also the equation of a line in (m,b) space, or parameter space.

P

P 'L

x

y

3D Computer Visionand Video Computing Parameter Space

b

m

b = -mx+y

b’ = -m’x'+y'

x,y; x',y' are fixed

(m,b)

L 1

L2

n The intersection represents the parameters of the equation of a line y=mx+b going through both (x,y) and (x',y').

n The more colinear edgels there are in the image, the more lines will intersect in parameter space

n Leads directly to an algorithm

3D Computer Visionand Video Computing General Idea

n General Idea: l The Hough space (m,b) is a representation of every

possible line segment in the planel Make the Hough space (m and b) discretel Let every edge point in the image plane ‘vote for’ any

line it might belong to.

3D Computer Visionand Video Computing Hough Transform

n Line Detection Algorithm: Hough Transforml Quantize b and m into appropriate 'buckets'.

u Need to decide what’s ‘appropriate’l Create accumulator array H(m,b), all of whose

elements are initially zero.l For each point (i,j) in the edge image for which the edge

magnitude is above a specific threshold, increment all points in H(m,b) for all discrete values of m and b satisfying b = -mj+i.

u Note that H is a two dimensional histograml Local maxima in H corresponds to colinear edge points

in the edge image.

3D Computer Visionand Video Computing Quantized Parameter Space

n Quantization

b single votes two votes

m

The problem of line detection in image space has been transformed into the problem of clusterdetection in parameter space


n The problem of line detection in image space has been transformed into the problem of cluster detection in parameter space

Image Edges

AccumulatorArray

Result

3D Computer Visionand Video Computing Problems

n Vertical lines have infinite slopesl difficult to quantize m to take this into account.

n Use alternative parameterization of a linel polar coordinate representation

r1

q1

q2

r2

y

x

r 1 x 1 cos q y 1 sin q+=

r = x cos q + y sin q

3D Computer Visionand Video Computing Why?

n (r,q) is an efficient representation:l Small: only two parameters (like y=mx+b)l Finite: 0 £ r £ Ö(row2+col2), 0 £ q £ 2l Unique: only one representation per line

3D Computer Visionand Video Computing Alternate Representation

n Curve in (r,q) space is now a sinusoid l but the algorithm remains valid.

r

2 q

r1 x1 cos q y 1 sin q+=

r 2 x 2 cos q y2 sin q+=


x

y

P = (4, 4)

P = (-3, 5)

1

2

P2 P1

r

q

r 4 c 4 s+=r 3 c 5 s+=

s 2 c 2+ 1=

s 750

50=

c 150

50=

q 1.4289=

r 4.5255=

Solve for r and q

r 3 cos q 5 sin q+=

r 4 cos q 4 sin q+=

Two Constraints

(r, q )(r, q ) Space

3D Computer Visionand Video Computing Real Example

Image Edges

AccumulatorArray

Result

3D Computer Visionand Video Computing Modifications

n Note that this technique only uses the fact that an edge exists at point (i,j).

n What about the orientation of the edge?l More constraints!

n Use estimate of edge orientation as q.n Each edgel now maps to a point in Hough space.

Image

Origin is arbitrary

The three edgels have same (r, q)

3D Computer Visionand Video Computing Gradient Data

n Colinear edges in Cartesian coordinate space now form point clusters in (m,b) parameter space.

E1

E2

E3

L1L2

L3

b

L1

L2 L3

mm

3D Computer Visionand Video Computing Gradient Data

n ‘Average’ point in Hough Space:

n Leads to an ‘average’ line in image space:

b

L1

L2 L3

m m

ba = -max + y

Average line in coordinate space

3D Computer Visionand Video Computing Post Hough

n Image space localization is lost:

n Consequently, we still need to do some image space manipulations, e.g., something like an edge 'connected components' algorithm.

n Heikki Kälviäinen, Petri Hirvonen, L. Xu and Erkki Oja, “Probabilistic and nonprobabilistic Hough Transforms: Overview and comparisons”, Image and vision computing, Volume 13, Number 4, pp. 239-252, May 1995.

both sets contribute to the same Hough maxima.

3D Computer Visionand Video Computing Hough Fitting

n Sort the edges in one Hough clusterl rotate edge points according to ql sort them by (rotated) x coordinate

n Look for Gapsl have the user provide a “max gap” thresholdl if two edges (in the sorted list) are more than max gap

apart, break the line into segmentsl if there are enough edges in a given segment, fit a

straight line to the points

3D Computer Visionand Video Computing Generalizations

n Hough technique generalizes to any parameterized curve:

n Success of technique depends upon the quantization of the parameters:l too coarse: maxima 'pushed' togetherl too fine: peaks less defined

n Note that exponential growth in the dimensions of the accumulator array with the the number of curve parameters restricts its practical application to curves with few parameters

parameter vector (axes in Hough space)

f(x,a) = 0

3D Computer Visionand Video Computing Example: Finding a Circle

n Circles have three parametersl Center (a,b)l Radius r

n Circle f(x,y,r) = (x-a)2+(y-b)2-r2 = 0n Task:

n Given an edge point at (x,y) in the image, where could the center of the circle be?

Find the center of a circle with known radius r given an edge image with no gradient direction information (edge location only)

3D Computer Visionand Video Computing Finding a Circle

(i-a)2+(j-b)2-r2 = 0

Image

fixed (i,j)

Parameter space (a,b)



Circle Center(lots of votes!)

3D Computer Visionand Video Computing Finding Circles

n If we don’t know r, accumulator array is 3-dimensionaln If edge directions are known, computational

complexity if reducedl Suppose there is a known error limit on the edge

direction (say +/- 10o) - how does this affect the search?n Hough can be extended in many ways….see, for

example:l Ballard, D. H. Generalizing the Hough Transform to

Detect Arbitrary Shapes, Pattern Recognition 13:111-122, 1981.

l Illingworth, J. and J. Kittler, Survey of the Hough Transform, Computer Vision, Graphics, and Image Processing, 44(1):87-116, 1988

Introduction

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Transcript of Introduction