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Image

SegmentationBy fitting a model

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What is image segmentation?

Technically speaking, image segmentation refers to the decomposition of a scene into different components (thus to facilitate the task at higher levels such as object detection and recognition)

Scientifically speaking, segmentation is a hypothetical middle-level vision task performed by neurons between low-level and high-level cortical areas

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Fitting or Grouping: (Based on fitting a

geometrical model) here we have a set of distinct data items, and we collect

sets of data items that make sense together according to our model.

a) Collecting together tokens that, taken together, form a line

or other geometry.

b) Collecting together tokens that seem to share a fundamental

matrix.

The key issues here:

To determine what representation is suitable for the problem in

hand. (Supervised Approach)

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Hough transform

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Image spatial space v/s Images Hough Parameter space.

Hough transform means: Transform the image in spatial plane to

hough parameter plane.

i.e. conversion of image from spatial coordinate domain (x, y) to

(m, b) hough parameter plane corresponds to line represented as:

y= mx+b (slope-intercept representation)

(r, ) hough parameter plane corresponds to line represented as:

x cos + y sin =r (polar representation)

m

by

x

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(m, b) hough parameter space:

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Where is the line

that contains both

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

intersection of the

lines b = x0m + y0and b = x1m + y1

-A line in the image

corresponds to a point

in Hough space.

(m, b) hough parameter space.

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

Hough space? There are many lines passing through the point (x0, y0).

Common to them is that they satisfy the equation for some set of

parameters (m, b).

i.e. the solutions of b = x0m + y0 which is a line in hough space.

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(r, ) hough parameter space:

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Problems with the line equation y = mx + b in (m, b) space?

Unbounded parameter domain

Vertical lines require inf inite m so how we can represent the accumulator

array.

-The Alternative: polar representation

The polar (also called normal) representation of straight lines

Each point (x i,yi) in the xy-plane gives a sinusoid in the (, ) parameter

space (or plane).

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-Each curve in the figure represents the family of lines that pass

through a particular point (xi,yi) in the xy-plane.

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(r, ) hough parameter space:

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N nos.of colinear point lying on the line will give N curves that intersect

at (i, j) in the parameter space or plane : i.e.

Sinusoids corresponding to co-linear points intersect at an unique

point.

e.g.

Line: 0.6x + 0.4y = 2.4

Sinusoids intersect at: = 2.4, = 0.9273

Hough Transform Algorithm

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Given the following points and discrete value of and the calculated value of = x.cos +y.sin

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S.No.

(x, y) -450 00 450 900

1 (2, 0) 1.4 2 1.4 0

2 (1,1) 0 1 1.4 1

3 (2, 1) 0.7 2 2.1 1

4 (1, 3) -1.4 1 2.8 3

5 (2, 3) -0.7 2 3.5 3

6 (4, 3) 0.7 4 4.9 3

7 (3, 4) -0.7 3 4.9 4

S.No

.

-450 00 450 900

1 -1.4 12 -0.7 23 0 1 14 0.7 25 1 2 26 1.4 1 27 2 38 2.1 19 2.8 110 3 1 311 3.5 112 4 1 213 4.9 2

Accumulator matrix

The two equal largest values occurs at (, ) = (2, 00) and (3, 900). Then the lines are:

x cos 00 + ysin 00 = 2 i.e. x = 2.x cos 900 + ysin 900 = 3 i.e. y = 3.

Data points

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Data without outliers or noise

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In presence of outliers or noise

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Random data points

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Least Square method to fit a line

Fitting aim: To determine values for the slope "m" and the

intercept "b" in an equation:

y = m x + b

Fitting requires definition of some measure of the error between

the data and the line. The Overall measure of error E(m, b):

Best fit when error belong to Gaussian distribution.

Now find m and b values that could minimize the error for best

fit . So to get minima, find the derivative of E with respect to m

and b.15

Derivative with respect to m:

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Least Square method to fit a line

Eq.1

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Derivative with respect to b:

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Least Square method to fit a line

Eq.2

In standard notation, these two equations can be written as:

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Least Square method to fit a line

Now value for m and b can be given as:

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Least Square method to fit a line

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Least Square method to fit a line

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RANSAC (RANdom SAmpling Consensus)

Ransac is a robust method for fitting a line in the presence of

much more outliers.

View estimation as a two-stage process:

-Classify data points as outliers or inliers

-Fit model to inliers

RANSAC is a re-sampling technique that generates candidate

solutions by using the minimum number observations (data

points) required to estimate the underlying model parameters.

Developed by M. A. Fischler and R. C. Bolles.

Outline of the RANSAC Randomly select a sample of s data points from S and instantiate

the model from this subset.

Determine the set of data points Si which is within a distance threshold t of the model. This set Si, is the consensus set of the sample and defines the inliers of S.

If the size of Si (the number of inliers) is greater than some threshold T, re-estimate the model using all the points in Si and

terminate.

If the size of Si is less than T, select a new subset and repeat the above.

After N trials the largest consensus set Si is selected, and the model is re-estimated using all the points in the subset Si. 22

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Example of RANSAC

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Example of RANSAC

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Example of RANSAC

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Example of RANSAC

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Example of RANSAC

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Example of RANSAC

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Example of RANSAC

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Example of RANSAC

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Best Consensus as per all the sampling in the

complete process.

Example of RANSAC

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Again low consensus due to further random sampling.

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Comparison of Least square & RANSAC

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Least square based fitting RANSAC based fitting

How Many Samples are Necessary (N)? Using all possible samples is often infeasible.

Instead, pick N to assure probability p of at least one sample (containing s points) being all inliers.

Let

a) Probability that any selected data point is an inlier = u

b) Probability of observing an outlier. = v = 1 u.

Then

N iterations of the samples can be given as:

1 p = (1 us)N

Or

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Example: N for the line-fitting problem

n = 12 points. (total nos. of points)

Minimal sample size s = 2.

If 2 outliers then v = 2/12 = 1/6 = 20% (proportion of outliers)

So for probability p = 0.99 (giving us a 99% chance of getting a pure-inlier sample)

The value of N = 5

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Analysis of RANSAC

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