signature

November 12, 2013 Computer Vision Lecture 12: Texture

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SignatureAnother popular method of representing shape is called the signature.In order to compute the signature of a contour, we choose an (arbitrary) starting point A on it.Then we measure the shortest distance from A to any other point on the contour, measured in perpendicular direction to the tangent at point A.Now we keep moving along the contour while measuring this distance.The resulting measure of distance as a function of path length is the signature of the contour.


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Signature

Notice that if you normalize the path length, the signature representation is position, rotation, and scale invariant.Smoothing of the contour before computing its signature may be necessary for a useful computation of the tangent.


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Fourier Transform of BoundariesAs we discussed before, a 2D contour can be represented by two 1D functions of a parameter s.Let S be our (arbitrary) starting point on the contour. By definition, we should move in counterclockwise direction (for example, we can adapt the boundary-following algorithm to do that).In our discrete (pixel) space, the points of the contour can then be specified by [v[s], h[s]],where s is the distance from S (measured along the contour)and v and h are the functions describing our vertical and horizontal position on the contour for a given s.


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Fourier Transform of Boundaries


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Fourier Transform of BoundariesObviously, both v[s] and h[s] are periodic functions, as we will pass the same points over and over again in constant time intervals.This gives us a brilliant idea: Why not represent the contour in Fourier space?In fact, this can be done. We simply compute separate 1D Fourier transforms for v[s] and h[s].When we convert the result into the magnitude/phase representation, the transform gives us the functionsMv[l], Mh[l], v[l], and h[l]for frequency l ranging from 0 (offset) to s/2.


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As with images, higher frequencies do not usually contribute much to a function an can be disregarded without any noticeable consequences.Let us see what happens when we delete all functions above frequency lmax and perform the inverse Fourier transform.Will the original contour be conserved even for small values of lmax?


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Fourier Transform of BoundariesThe phase spectrum of the Fourier description usually depends on our chosen starting point.The power (magnitude) spectrum, however, is invariant to this choice if we combine vertical and horizontal magnitude as follows:

We can cut off higher-frequency descriptors and still have a precise and compact contour representation.If we normalize the values of all M[l] so that they add up to 1, we obtain scale-invariant descriptors.

][][][ lMlMlM hv


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Our next topic is…

Texture


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Texture


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Texture

Texture is an important cue for biological vision systems to estimate the boundaries of objects.Also, texture gradient is used to estimate the orientation of surfaces.For example, on a perfect lawn the grass texture is the same everywhere.However, the further away we look, the finer this texture becomes – this change is called texture gradient.For the same reasons, texture is also a useful feature for computer vision systems.


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Texture Gradient


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Texture

The most fundamental question is: How can we “measure” texture, i.e., how can we quantitatively distinguish between different textures?Of course it is not enough to look at the intensity of individual pixels.Since the repetitive local arrangement of intensity determines the texture, we have to analyze neighborhoods of pixels to measure texture properties.


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Frequency Descriptors

One possible approach is to perform local Fourier transforms of the image.Then we can derive information on• the contribution of different spatial frequencies and• the dominant orientation(s) in the local texture.

For both kinds of information, only the power (magnitude) spectrum needs to be analyzed.


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Frequency DescriptorsPrior to the Fourier transform, apply a Gaussian filter to avoid horizontal and vertical “phantom” lines.

In the power spectrum, use ring filters of different radii to extract the frequency band contributions.

Also in the power spectrum, apply wedge filters at different angles to obtain the information on dominant orientation of edges in the texture.


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Frequency Descriptors

The resulting frequency and orientation data can be normalized, for example, so that the sum across frequency or orientation bands is 1.This effectively turns them into histograms that are less affected by monotonic gray-level changes caused by shading etc.However, it is recommended to combine frequency-based approaches with space-based approaches.


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Co-Occurrence Matrices

A simple and popular method for this kind of analysis is the computation of gray-level co-occurrence matrices.To compute such a matrix, we first separate the intensity in the image into a small number of different levels.For example, by dividing the usual brightness values ranging from 0 to 255 by 64, we create the levels 0, 1, 2, and 3.


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Then we choose a displacement vector d = [di, dj].The gray-level co-occurrence matrix P(a, b) is then obtained by counting all pairs of pixels separated by d having gray levels a and b.Afterwards, to normalize the matrix, we determine the sum across all entries and divide each entry by this sum.This co-occurrence matrix contains important information about the texture in the examined area of the image.


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Example (2 gray levels):

011011010010011011010010011011010010

local texture patch

d = (1, 1)

displacement vector

1/25

co-occurrence matrix

0 1

0 1 2 9

10 4


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It is often a good idea to use more than one displacement vector, resulting in multiple co-occurrence matrices.The more similar the matrices of two textures are, the more similar are usually the textures themselves.This means that the difference between corresponding elements of these matrices can be taken as a similarity measure for textures.Based on such measures we can use texture information to enhance the detection of regions and contours in images.


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For a given co-occurrence matrix P(a, b), we can compute the following six important characteristics:

ba

baP,

2 ),(Energy

ba

baPbaP,

2 ),(log),(Entropy

),(maxyprobabilit Maximum,

baPba

ba

baPba,

1 2,usually ,),(||Contrast


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baba ba

baP;, ||

),(moment difference Inverse

yx

bayxbaPab

,

),()(nCorrelatio

ba

x baPa ),(

ab

y baPb ),(

ba

xx baPa ),()( 2

ab

yy baPb ),()( 2


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Co-Occurrence MatricesYou should compute these six characteristics for multiple displacement vectors, including different directions.For example, you could use the following vectors:(-3, 0), (-2, 0), (-1, 0) (1, 0), (2, 0), (3, 0)(0, -3), (0, -2), (0, -1) (0, 1), (0, 2), (0, 3)(-3, -3), (-2, -2), (-1, -1) (1, 1), (2, 2), (3, 3)(-3, 3), (-2, 2), (-1, 1) (1, -1), (2, -2), (3, -3)

The maximum length of your displacement vectors depend on the size of the texture elements.


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Law’s Texture Energy Measures

Law’s measures use a set of convolution filters to assess gray level, edges, spots, ripples, and waves in textures.

This method starts with three basic filters:

• averaging: L3 = (1, 2, 1) • first derivative (edges): E3 = (-1, 0, 1)• second derivative (curvature): S3 = (-1, 2, -1)


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Convolving these filters with themselves and each other results in five new filters:

• L5 = (1, 4, 6, 4, 1) • E5 = (-1, -2, 0, 2, 1)• S5 = (-1, 0, 2, 0, -1)• R5 = (1, -4, 6, -4, 1)• S5 = (-1, 2, 0, -2, 1)


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Now we can multiply any two of these vectors, using the first one as a column vector and the second one as a row vector, resulting in 5 5 Law’s masks.For example:

10201408046012064080410201

55 SLT


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Now you can apply the resulting 25 convolution filters to a given image.The 25 resulting values at each position in the image are useful descriptors of the local texture.Law’s texture energy measures are easy to apply and give good results for most texture types.However, co-occurrence matrices are more flexible; for example, they can be scaled to account for coarse-grained textures.


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Benchmark image for texture segmentation – an ideal segmentation algorithm would divide this image into five segments.For example, a texture-descriptor based variant of split-and-merge may be able to achieve good results.

signature

Documents

computer vision lecture

d contour

original contour

compact contour representation

inverse fourier transform

signature representation

d functions

arbitrary starting point