Lecture 9: Hough Transform and...

transcript

ME5286 – Lecture 9

Lecture 9:

Hough Transform and Thresholding

Saad Bedros

sbedros@umn.edu

Hough Transform

• Robust method to find a shape in an image

• Shape can be described in parametric form

• A voting scheme is used to determine the correct

parameters

Example: Line fitting

• Why fit lines?

Many objects characterized by presence of straight lines

• Can we do it with edge detection? Use edge information

• Extra edge points (clutter),

multiple models:

– which points go with which

line, if any?

• Only some parts of each line

detected, and some parts

are missing:

– how to find a line that bridges

missing evidence?

• Noise in measured edge

points, orientations:

– how to detect true underlying

parameters?

Difficulty of line fitting

Voting

• It’s not feasible to check all combinations of features by fitting

a model to each possible subset.

• Voting is a general technique where we let the features vote

for all models that are compatible with it.

– Cycle through features, cast votes for model parameters.

– Look for model parameters that receive a lot of votes.

• Noise & clutter features will cast votes too, but typically their

votes should be inconsistent with the majority of “good”

features.

Fitting lines: Hough transform

• Given points that belong to a line, what is

the line?

• How many lines are there?

• Which points belong to which lines?

• Hough Transform is a voting technique

that can be used to answer all of these

questions.

Main idea:

1. Record vote for each possible line on

which each edge point lies.

2. Look for lines that get many votes.

Finding lines in an image: Hough space

Connection between image (x,y) and Hough (m,b)

spaces

– A line in the image corresponds to a point in Hough space

– To go from image space to Hough space: • given a set of points (x,y), find all (m,b) such that y = mx + b

image space Hough (parameter) space

Slide credit: Steve Seitz

Connection between image (x,y) and Hough (m,b) spaces

– A line in the image corresponds to a point in Hough space

– To go from image space to Hough space: • given a set of points (x,y), find all (m,b) such that y = mx + b

– What does a point (x0, y0) in the image space map to?

– Answer: the solutions of b = -x0m + y0

– this is a line in Hough space

Slide credit: Steve Seitz

What are the line parameters for the line that

contains both (x0, y0) and (x1, y1)?

– It is the intersection of the lines b = –x0m + y0 and

b = –x1m + y1

(x0, y0)

(x1, y1)

Finding lines in an image: Hough algorithm

How can we use this to find the most likely parameters (m,b) for

the most prominent line in the image space?

• Let each edge point in image space vote for a set of possible

parameters in Hough space

• Accumulate votes in discrete set of bins; parameters with the

most votes indicate line in image space.

Hough Transform for Line Detection

Find a subset of n points on an image that lie on the same straight line.

Write each line formed by a pair of these points as

yi = axi + b

Then plot them on the parameter space (a, b):

b = xi a + yi

All points (xi, yi) on the same line will pass the same parameter space point (a, b).

Quantize the parameter space and tally # of times each points fall into the same accumulator cell. The cell count = # of points in the same line.

Polar representation for lines

: perpendicular distance

from line to origin

: angle the perpendicular

makes with the x-axis

Point in image space sinusoid segment in Hough space

dyx sincos

Issues with usual (m,b) parameter space: can take on

infinite values, undefined for vertical lines.

Image columns

Kristen Grauman

Hough Transform in (r, ) plane

To avoid infinity slope,

use polar coordinate to

represent a line.

Q points on the same

straight line gives Q

sinusoidal curves in (r,

) plane intersecting at

the same (ri, i) cell.

r sincos yx

Hough transform algorithm Using the polar parameterization:

Basic Hough transform algorithm

1. Initialize H[d, ]=0

2. for each edge point I[x,y] in the image

for = [min to max ] // some quantization

H[d, ] += 1

3. Find the value(s) of (d, ) where H[d, ] is maximum

4. The detected line in the image is given by

H: accumulator array (votes)

Time complexity (in terms of number of votes per pt)?

dyx sincos

Source: Steve Seitz

sincos yxd

Hough Transform for Lines

1 1( , )x y

2 2( , )x y

0 0( , )x y

1 1cos( ) sin( )r x y

-3 -2 -1 1 2 3

1 1( , )x y

0 0( , )x y

2 2( , )x y

0 0( , )r

1 1( , )x y

3 3( , )x y4 4( , )x y

-3 -2 -1 1 2 3

1 1( , )x y

r1 1cos( ) sin( )r x y

3 3( , )x y

4 4( , )x y

1 1( , )r

Peak in the parametric

space that corresponds

to the line

• Domain of the parametric space:

M and N image resolution

Not just lines, any parametric curve!

However increase of dimensions of the

parametric space

2 2 2 2, , ,2 2

r M N M N

Original image Canny edges

Vote space and top peaks

Kristen Grauman

Showing longest segments found

Kristen Grauman

Impact of noise on Hough

Image space

edge coordinates Votes

What difficulty does this present for an implementation?

Image space

edge coordinates

In this case, everything appears to be “noise”, or random

edge points, but we still see some peaks in the vote space.

Extensions Extension 1: Use the image gradient

1. same

= gradient at (x,y)

H[d, ] += 1

3. same

4. same

(Reduces degrees of freedom)

Extension 2

– give more votes for stronger edges

Extension 3

– change the sampling of (d, ) to give more/less resolution

Extension 4

– The same procedure can be used with circles, squares, or any

other shape

sincos yxd

Extensions Extension 1: Use the image gradient

1. same

compute unique (d, ) based on image gradient at (x,y)

H[d, ] += 1

3. same

4. same

(Reduces degrees of freedom)

Extension 2

– give more votes for stronger edges (use magnitude of gradient)

Extension 3

– change the sampling of (d, ) to give more/less resolution

Extension 4

– The same procedure can be used with circles, squares, or any other

shape…

Source: Steve Seitz

Hough transform for circles

• For a fixed radius r, unknown gradient direction

• Circle: center (a,b) and radius r

222 )()( rbyax ii

Image space Hough space a

Kristen Grauman

• For a fixed radius r, unknown gradient direction

222 )()( rbyax ii

Image space Hough space

Intersection:

most votes

for center

occur here.

Kristen Grauman

• For an unknown radius r, unknown gradient direction

222 )()( rbyax ii

Hough space Image space

Kristen Grauman

• For an unknown radius r, unknown gradient direction

222 )()( rbyax ii

Hough space Image space

Kristen Grauman

HT for Circles: Search with fixed R

Equation of Circle:

222 )()( rbyax ii

Multiple Circles with known R

• Multiple circles with the same radius can be found with the same

technique. The centerpoints are represented as red cells in the

parameter space drawing.

• Overlap of circles can cause spurious centers to also be found, such as

at the blue cell. Spurious circles can be removed by matching to circles

in the original image.

HT for Circles: Search with unknown R

Equation of Circle:

222 )()( rbyax ii

If radius is not known: 3D Hough Space!

Use Accumulator array

),,( rbaA

For every edge pixel (x,y) :

For each possible radius value r:

For each possible gradient direction θ:

// or use estimated gradient at (x,y)

a = x – r cos(θ) // column

b = y + r sin(θ) // row

H[a,b,r] += 1

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).

Original Edges

Example: detecting circles with Hough

Votes: Quarter

Combined detections

Example: iris detection

Gradient+threshold Hough space

(fixed

radius)

Max detections

Kristen Grauman

Voting: practical tips

• Minimize irrelevant tokens first

• Choose a good grid / discretization

• Vote for neighbors, also (smoothing in accumulator

array)

• Use direction of edge to reduce parameters by 1

• To read back which points voted for “winning” peaks,

keep tags on the votes.

coarse

Too fine ?

Kristen Grauman

Hough transform: pros and cons

• All points are processed independently, so can cope with

occlusion, gaps

• Some robustness to noise: noise points unlikely to contribute

consistently to any single bin

• Can detect multiple instances of a model in a single pass

• Complexity of search time increases exponentially with the

number of model parameters

• Non-target shapes can produce spurious peaks in parameter

• Quantization: can be tricky to pick a good grid size

Kristen Grauman

Generalized Hough Transform

• What if we want to detect arbitrary shapes?

• Detect any arbitrary shape

– Requires specification of the exact shape of the object

• Define a model shape by its boundary points and a reference

point.

[Dana H. Ballard, Generalizing the Hough Transform to Detect Arbitrary Shapes, 1980]

At each boundary point,

compute displacement

vector: r = a – pi.

Store these vectors in a

table indexed by gradient

orientation θ.

Offline procedure:

Model shape

R-Table

For each edge point:

• Use its gradient orientation θ to

index into stored table

• Use retrieved r vectors to vote

for reference point

Detection procedure:

Assuming translation is the only transformation here, i.e., orientation and scale are fixed.

Novel image

Rotation and Scale Solution

Segmentation of Objects

Using Thresholding Method

Thresholding based Segmentation

• Goal is to identify an object based on uniform

intensity

• Use the Histogram to compute the best threshold that

can separate the object intensity

Thresholding Methods #49

Thresholding Principles #50

Thresholding Example #51

Thresholding Examples #52

Histogram Calculation #53

Histogram Profiles #54

Good and Bad Histograms #55

Maximum Separation #56

Adaptive Thresholding #57

Two Types of Variance #58

Threshold selection via optimization #59

Recursive Procedure #60

Properties #61

Gaussian Mixture Modeling of Histograms

Fitting Model Distribution #63

Fitting Model Distribution - 2 #64

Derivation of Optimal Threshold #65

Derivation of Optimal Threshold - 2 #66

Cases for Optimal Threshold #67

Algorithm for Gaussian Threshold Detection

Properties of Gaussian Mixture Approach

Examples #70

Otsu vs Gaussian Approach #71

Gaussian Gives Poor Results #72

Gaussian Mixture – a Fail Case #73

Issues with Thresholding

• Histogram based thresholding is very effective

• Even with low noise, if one class is much smaller

than the other we might still be in trouble.

• Remember also that both these images have the same

histogram:

Lecture 9: Hough Transform and...

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