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ME5286 – Lecture 9
#1
Lecture 9: Hough Transform and
Thresholding base SegmentationSaad Bedros
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
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
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
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.
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.
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
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
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) spacex0
y0
b = –x1m + y1
(x0, y0)(x1, y1)
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
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 + bThen plot them on the parameter space (a, b):
b = xi a + yiAll 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.
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
Imag
e ro
ws
Kristen Grauman
ME5286 – Lecture 9
#13
Hough Transform in () plane
To avoid infinity slope, use polar coordinate to represent a line.
Q points on the same straight line gives Q sinusoidal curves in () plane intersecting at the same (ii) cell.
sincos yx
ME5286 – Lecture 9
Hough transform algorithmUsing the polar parameterization:
Basic Hough transform algorithm1. Initialize H[d, ]=02. 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 maximum4. 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
ME5286 – Lecture 9
#15
Hough Transform for Lines
y
x
0r1 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
ME5286 – Lecture 9
#16
Hough Transform for Lines
y
x1
1r1 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
ME5286 – Lecture 9
#17
Peak in the parametric space that corresponds
to the line
ME5286 – Lecture 9
#18
Hough Transform for Lines
• Domain of the parametric space:
M and N image resolution
2 2 2 2, , ,2 2
r M N M N
ME5286 – Lecture 9
Original image Edge Detection
Vote space and top peaks
Kristen Grauman
ME5286 – Lecture 9Showing longest segments found
Kristen Grauman
ME5286 – Lecture 9
Impact of noise on Hough
Image spaceedge coordinates
Votesx
y d
What difficulty does this present for an implementation?
ME5286 – Lecture 9
Image spaceedge 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
ME5286 – Lecture 9
ExtensionsExtension 1: Use the image gradient
1. same2. for each edge point I[x,y] in the image
= gradient at (x,y)
H[d, ] += 13. same4. 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
E t i 4
sincos yxd
ME5286 – Lecture 9
ExtensionsExtension 1: Use the image gradient
1. same2. for each edge point I[x,y] in the image
compute unique (d, ) based on image gradient at (x,y)H[d, ] += 1
3. same4. 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– If you know the range of the angle , look only in that range …
Source: Steve Seitz
ME5286 – Lecture 9
Hough transform for circles
• For a fixed radius r, unknown gradient direction
• Circle: center (a,b) and radius r222 )()( rbyax ii
Image space Hough space a
b
Kristen Grauman
ME5286 – Lecture 9
Hough transform for circles
• For a fixed radius r, unknown gradient direction
• Circle: center (a,b) and radius r222 )()( rbyax ii
Image space Hough space
Intersection: most votes for center occur here.
Kristen Grauman
ME5286 – Lecture 9
Hough transform for circles
• For an unknown radius r, unknown gradient direction
• Circle: center (a,b) and radius r222 )()( rbyax ii
Hough spaceImage space
b
a
r
?
Kristen Grauman
ME5286 – Lecture 9
Hough transform for circles
• For an unknown radius r, unknown gradient direction
• Circle: center (a,b) and radius r222 )()( rbyax ii
Hough spaceImage space
b
a
r
Kristen Grauman
ME5286 – Lecture 9
Hough transform for circles
• For an unknown radius r, known gradient direction
• Circle: center (a,b) and radius r222 )()( rbyax ii
Hough spaceImage space
θ
x
Kristen Grauman
ME5286 – Lecture 9
HT for Circles: Search with fixed R
Equation of Circle: 222 )()( rbyax ii
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
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
ME5286 – Lecture 9
Hough transform for circlesFor 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) // columnb = y + r sin(θ) // rowH[a,b,r] += 1
endend
ME5286 – Lecture 9
Original Edges
Example: detecting circles with HoughVotes: Penny
Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penny).
ME5286 – Lecture 9
Original Edges
Example: detecting circles with HoughVotes: Quarter
Combined detections
ME5286 – Lecture 9
Example: iris detection
Gradient+threshold Hough space (fixed radius)
Max detections
Kristen Grauman
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
ME5286 – Lecture 9
Hough transform: pros and consPros• 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
ME5286 – Lecture 9
#39
Segmentation of Objects Using Thresholding Method
ME5286 – Lecture 9
#40
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
ME5286 – Lecture 9
Thresholding Principles#41
ME5286 – Lecture 9
Thresholding Methods#42
• Principles of gray value thresholding• Histogram base thresholding
– Automatic threshold selection
• Examples
ME5286 – Lecture 9
Thresholding Example#43
ME5286 – Lecture 9
Thresholding Examples#44
ME5286 – Lecture 9
Histogram Calculation#45
ME5286 – Lecture 9
Histogram Profiles#46
ME5286 – Lecture 9
Good and Bad Histograms#47
ME5286 – Lecture 9
Maximum Separation#48
ME5286 – Lecture 9
Otsu’s Method
• Consider an image with L gray levels and its normalized histogram – P(i) is the normalized frequency of i.
• Assuming that we have set the threshold at T, the normalized fraction of pixels that will be classified as background and object will be:
Tbackground object
ME5286 – Lecture 9
Otsu’s Method
• The mean gray-level value of the background and the object pixels will be:
• The mean gray-level value over the whole image is:
ME5286 – Lecture 9
Otsu’s Method
• The variance of the background and the object pixels will be:
• The variance of the whole image is:
ME5286 – Lecture 9
Otsu’s Method#52
ME5286 – Lecture 9
Two Types of Variance#53
ME5286 – Lecture 9
Otsu’s Method:Threshold selection#54
ME5286 – Lecture 9
Otsu’s Method:Threshold selection
Steps to maximize
• rewrite as
• Find the T value that maximizes • Start from the beginning of the histogram and test each gray-
level value for the possibility of being the threshold T that maximizes
ME5286 – Lecture 9
Recursive Procedure#56
ME5286 – Lecture 9
ME5286 – Lecture 9
Using Image Smoothing to improve Global Thresholding
ME5286 – Lecture 9
Otsu’s Method (cont’d)
• Drawbacks of the Otsu’s method– The method assumes that the histogram of the image is bimodal
(i.e., two classes).– The method breaks down when the two classes are very unequal
(i.e., the classes have very different sizes)• In this case, may have two maxima. • The correct maximum is not necessary the global one.
– The method does not work well with variable illumination.
ME5286 – Lecture 9
#60
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:
ME5286 – Lecture 9
Gaussian Mixture Modeling of Histograms#61
ME5286 – Lecture 9
Fitting Model Distribution#62
ME5286 – Lecture 9
Fitting Model Distribution - 2#63
ME5286 – Lecture 9
Derivation of Optimal Threshold#64
Assume model fitting is doneHow do we find the Optimal Threshold ?
ME5286 – Lecture 9
Derivation of Optimal Threshold - 2#65
ME5286 – Lecture 9
Cases for Optimal Threshold#66
ME5286 – Lecture 9
Algorithm for Gaussian Threshold Detection#67
ME5286 – Lecture 9
Properties of Gaussian Mixture Approach#68
ME5286 – Lecture 9
Examples#69
ME5286 – Lecture 9
Otsu vs Gaussian Approach#70
ME5286 – Lecture 9
Gaussian Gives Poor Results#71
ME5286 – Lecture 9
Gaussian Mixture – a Fail Case#72
ME5286 – Lecture 9
Summary#73
• Hough Transform is an efficient method to find lines and other shapes
• Procedure for the hough transform• Thresholding can be used to segment objects from
the scene• Otsu’s method find the optimal threshold to separate
or segment objects• Gaussian Mixture algorithm is another solution to
compute the threshold
Final Exam is due May 10 2017