Lecture 9: Hough Transform and...
Transcript of Lecture 9: Hough Transform and...
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ME5286 – Lecture 9
#2
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
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ME5286 – Lecture 9
Example: Line fitting
• Why fit lines?
Many objects characterized by presence of straight lines
• Can we do it with edge detection? Use edge information
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ME5286 – Lecture 9
• 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
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ME5286 – Lecture 9
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.
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ME5286 – Lecture 9
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.
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ME5286 – Lecture 9
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
x
y
m
b
m0
b0
image space Hough (parameter) space
Slide credit: Steve Seitz
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ME5286 – Lecture 9
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
– What does a point (x0, y0) in the image space map to?
x
y
m
b
image space Hough (parameter) space
– Answer: the solutions of b = -x0m + y0
– this is a line in Hough space
x0
y0
Slide credit: Steve Seitz
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ME5286 – Lecture 9
Finding lines in an image: Hough space
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
x
y
m
b
image space Hough (parameter) space
x0
y0
b = –x1m + y1
(x0, y0)
(x1, y1)
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ME5286 – Lecture 9
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.
x
y
m
b
image space Hough (parameter) space
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ME5286 – Lecture 9
#11
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.
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ME5286 – Lecture 9
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
d
[0,0]
d
x
y
Issues with usual (m,b) parameter space: can take on
infinite values, undefined for vertical lines.
Image columns
Image
row
s
Kristen Grauman
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ME5286 – Lecture 9
#13
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
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ME5286 – Lecture 9
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)
d
Time complexity (in terms of number of votes per pt)?
dyx sincos
Source: Steve Seitz
sincos yxd
sincos yxd
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ME5286 – Lecture 9
#15
Hough Transform for Lines
y
x
0r
1 1( , )x y
0
2 2( , )x y
0 0( , )x y
1 1cos( ) sin( )r x y
-3 -2 -1 1 2 3
-2
-1
1
2
r
1 1( , )x y
0 0( , )x y
2 2( , )x y
0 0( , )r
0 0( , )r
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ME5286 – Lecture 9
#16
Hough Transform for Lines
y
x1
1r
1 1( , )x y
3 3( , )x y4 4( , )x y
-3 -2 -1 1 2 3
-4
-2
2
4
1 1( , )x y
r1 1cos( ) sin( )r x y
3 3( , )x y
4 4( , )x y
1 1( , )r
1 1( , )r
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ME5286 – Lecture 9
#17
Peak in the parametric
space that corresponds
to the line
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ME5286 – Lecture 9
#18
Hough Transform for Lines
• 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
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ME5286 – Lecture 9
Original image Canny edges
Vote space and top peaks
Kristen Grauman
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ME5286 – Lecture 9
Showing longest segments found
Kristen Grauman
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ME5286 – Lecture 9
Impact of noise on Hough
Image space
edge coordinates Votes
x
y d
What difficulty does this present for an implementation?
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ME5286 – Lecture 9
Image space
edge coordinates
Votes
Impact of noise on Hough
In this case, everything appears to be “noise”, or random
edge points, but we still see some peaks in the vote space.
Impact of noise on Hough
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ME5286 – Lecture 9
Extensions Extension 1: Use the image gradient
1. same
2. for each edge point I[x,y] in the image
= 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
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ME5286 – Lecture 9
Extensions Extension 1: Use the image gradient
1. same
2. for each edge point I[x,y] in the image
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
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ME5286 – Lecture 9
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
b
Kristen Grauman
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ME5286 – Lecture 9
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
Intersection:
most votes
for center
occur here.
Kristen Grauman
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ME5286 – Lecture 9
Hough transform for circles
• For an unknown radius r, unknown gradient direction
• Circle: center (a,b) and radius r
222 )()( rbyax ii
Hough space Image space
b
a
r
?
Kristen Grauman
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ME5286 – Lecture 9
Hough transform for circles
• For an unknown radius r, unknown gradient direction
• Circle: center (a,b) and radius r
222 )()( rbyax ii
Hough space Image space
b
a
r
Kristen Grauman
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ME5286 – Lecture 9
HT for Circles: Search with fixed R
Equation of Circle:
222 )()( rbyax ii
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ME5286 – Lecture 9
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.
31
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ME5286 – Lecture 9
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
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ME5286 – Lecture 9
Hough transform for circles
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
end
end
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ME5286 – Lecture 9
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).
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ME5286 – Lecture 9
Original Edges
Example: detecting circles with Hough
Votes: Quarter
Combined detections
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ME5286 – Lecture 9
Example: iris detection
Gradient+threshold Hough space
(fixed
radius)
Max detections
Kristen Grauman
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ME5286 – Lecture 9
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.
Too
coarse
Too fine ?
Kristen Grauman
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ME5286 – Lecture 9
Hough transform: pros and cons
Pros
• 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
Cons
• Complexity of search time increases exponentially with the
number of model parameters
• Non-target shapes can produce spurious peaks in parameter
space
• Quantization: can be tricky to pick a good grid size
Kristen Grauman
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ME5286 – Lecture 9
Generalized Hough Transform
• What if we want to detect arbitrary shapes?
• Detect any arbitrary shape
– Requires specification of the exact shape of the object
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ME5286 – Lecture 9
• Define a model shape by its boundary points and a reference
point.
[Dana H. Ballard, Generalizing the Hough Transform to Detect Arbitrary Shapes, 1980]
x
a
p1
θ
p2
θ
At each boundary point,
compute displacement
vector: r = a – pi.
Store these vectors in a
table indexed by gradient
orientation θ.
Generalized Hough Transform
Offline procedure:
Model shape
θ
θ
…
…
…
Generalized Hough Transform
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ME5286 – Lecture 9
Generalized Hough Transform
R-Table
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ME5286 – Lecture 9
p1
θ θ
For each edge point:
• Use its gradient orientation θ to
index into stored table
• Use retrieved r vectors to vote
for reference point
Generalized Hough Transform
Detection procedure:
Assuming translation is the only transformation here, i.e., orientation and scale are fixed.
x
θ θ
Novel image
θ
θ
…
…
…
θ
x x
x x
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ME5286 – Lecture 9
Generalized Hough Transform
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ME5286 – Lecture 9
Generalized Hough Transform
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ME5286 – Lecture 9
Rotation and Scale Solution
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ME5286 – Lecture 9
Rotation and Scale Solution
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ME5286 – Lecture 9
#47
Segmentation of Objects
Using Thresholding Method
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ME5286 – Lecture 9
#48
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
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ME5286 – Lecture 9
Thresholding Methods #49
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ME5286 – Lecture 9
Thresholding Principles #50
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ME5286 – Lecture 9
Thresholding Example #51
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ME5286 – Lecture 9
Thresholding Examples #52
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ME5286 – Lecture 9
Histogram Calculation #53
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ME5286 – Lecture 9
Histogram Profiles #54
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ME5286 – Lecture 9
Good and Bad Histograms #55
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ME5286 – Lecture 9
Maximum Separation #56
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ME5286 – Lecture 9
Adaptive Thresholding #57
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ME5286 – Lecture 9
Two Types of Variance #58
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ME5286 – Lecture 9
Threshold selection via optimization #59
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ME5286 – Lecture 9
Recursive Procedure #60
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ME5286 – Lecture 9
Properties #61
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ME5286 – Lecture 9
Gaussian Mixture Modeling of Histograms
#62
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ME5286 – Lecture 9
Fitting Model Distribution #63
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ME5286 – Lecture 9
Fitting Model Distribution - 2 #64
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ME5286 – Lecture 9
Derivation of Optimal Threshold #65
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ME5286 – Lecture 9
Derivation of Optimal Threshold - 2 #66
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ME5286 – Lecture 9
Cases for Optimal Threshold #67
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ME5286 – Lecture 9
Algorithm for Gaussian Threshold Detection
#68
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ME5286 – Lecture 9
Properties of Gaussian Mixture Approach
#69
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ME5286 – Lecture 9
Examples #70
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ME5286 – Lecture 9
Otsu vs Gaussian Approach #71
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ME5286 – Lecture 9
Gaussian Gives Poor Results #72
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ME5286 – Lecture 9
Gaussian Mixture – a Fail Case #73
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ME5286 – Lecture 9
#74
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: