Divergence of Jensen Shanon for texture classification

1. Membrane Detector by Texture Analisys
An Analysis of Edge Detection by Using the Jensen-Shannon Divergence,
G´omez-Lopera, Juan Francisco and Mart´ınez-Aroza, Jos´e and
Robles-P´erez, Aureliano M. and Rom´an-Rold´an, Ram´on
Rodrigo Rojas Moraleda
July 4, 2012
Rodrigo Rojas Moraleda — Membrane Detector by Texture Analisys 1/24

2. Outline
1 Introduction
2 The system
3 Conclusions
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3. Outline
1 Introduction
2 The system
3 Conclusions
Rodrigo Rojas Moraleda — Membrane Detector by Texture Analisys 3/24

4. Introduction
Texture analisys
Definition
Texture and texture analisys is the most important visual clue in identifying types of
homogeneous regions. This is called texture classification. The goal of texture
classification then is to produce a classification map of the input image where each
uniform textured region is identified with the texture class it belongs to.
Problem features
In many machine vision and image processing algorithms, simplifying assumptions
are made about the uniformity of intensities in local image regions. However,
images of real objects often do not exhibit regions of uniform intensities.
The patterns in a image can be the result of physical surface properties such as
roughness or oriented strands which often have a tactile quality, or they could be
the result of reflectance differences such as the color on a surface.
One immediate application of image texture is the recognition of image regions
using texture properties.
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5. Introduction
Jensen–Shannon divergence
Jensen–Shannon divergence
Jensen–Shannon divergence is a popular method of measuring the similarity
between two probability distributions. It is also known as information radius
(IRad) or total divergence to the average.
JSD(P Q) =
1
2
D(P M) +
1
2
D(Q M)
M = 1/2(P + Q)
D(P Q) = DKL(P Q) =
i
P(i)log
P(i)
Q(i)
The average of the logarithmic difference between the probabilities P and Q,
where the average is taken using the probabilities P.
Divergence grows as the differences between its arguments (the probability
distributions involved) increase, and vanishes when all the probability
distributions are identical.
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6. Introduction
Jensen–Shannon divergence
Texture and texture analisys is the most important visual clue in identifying
types of homogeneous regions. Texture analisys aim to produce a classification
map of the input image where each uniform textured region is identified.
Considerations about texture analisys and the real world
In image processing is possible made assumptions about the uniformity of
intensities in local regions. Despite of in real objects often do not exhibit
regions of uniform intensities.
The patterns in a image can be the result of physical surface properties
such as roughness, oriented strands or reflectance differences such as the
color on a surface.
Image Intensities and probabilities
Image histograms represents how frequent brightness levels from 0 to 255
appear in the image, showing a visual impression of the distribution of data. It
is an estimate of the probability distribution of a continuous variable. The total
area of a histogram used for probability density is always normalized to 1.
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7. Introduction
Jensen–Shannon divergence
Jensen–Shannon divergence is a popular method of measuring the cohesion of
a finite set of probability distributions having the same number of possible
events.Its value grows as the differences between its arguments (the probability
distributions involved) increase, and vanishes when all the probability
distributions are identical.
If we consider a window W made up of two identical subwindows W1 and W2,
sliding over a straight horizontal edge between two different homogeneous
regions a and b, Jensen-Shannon divergence between the normalised
histograms of the subwindows reaches maximum value just when each
subwindow lies completely within one region.
W1
W2
W1
W2
W1
W2
W1
W2
W1
W2
W1
W2
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8. Introduction
Jensen–Shannon divergence
Trying several window orientations for each pixel is possible to obtain an
estimate for the edge orientation which maximize the divergence value.
W1
W2
W
1
W
2
W1
W2
W
1
W
2
JS1 JS2 JS3 JS4
Figure: The values JS1,JS2,JS3 and JS4 are calculated for the fixed window
orientations 0, π/4, π/2and3π/4
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9. Outline
1 Introduction
2 The system
3 Conclusions
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10. The system
Texture analisys
Steps
Step 1. Calculation of divergence and direction matrices.
Step 2. Edge-pixel selection.
Step 3. Edge linking.
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11. Step 1
Calculation of divergence and direction matrices
Window sliding
W1
W2
W1
W2
W1
W2
W1
W2
W1
W2
W1
W2
Figure: Behavior of Jensen Shanon divergence versus sliding window over an perfect
edge
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12. Step 1
Calculation of divergence and direction matrices
Window sliding
W1
W2
W
1
W
2
W1
W2
W
1
W
2
JS1 JS2 JS3 JS4
Figure: The values JS1,JS2,JS3 and JS4 are calculated for the fixed window
orientations 0, π/4, π/2and3π/4
Problem
How to obtain an estimate of the direction fromthese four values that maximizes the
JS and then the value of this maximum, JSmax . For a given pixel, the JS value is a
π − periodic function of window orientation over the image. It reaches its maximum
value for a given orientation, β, and a minimum in β + π. A periodic function can be
expressed as:
JS(x) = a + bcos(β + 2πx), x ∈ [0, 1]
Here β ∈ [0, π) is the edge direction in the pixel, a,b are constants used to specify the
amplitude.
JS(x) = c + msen(2πx) + ncos(2πx), x ∈ [0, 1]
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13. Step 1
Calculation of divergence and direction matrices
Maximum JS
f (x) ≈ sen(2πx), g(x) ≈ cos(2πx)
With a least-squares fir over the points JS1 + JS2 + JS3 + JS4
JS(x) =
JS1 + JS2 + JS3 + JS4
4
+
JS2 + JS4
2
f (x) +
JS1 + JS3
2
g(x)
Maximum JS
The direction, x, having the maximun JS can be obtained by:
if JS1 − JS3 ≥ 0, JS2 − JS4 ≥ 0 ⇒
x =
JS2 − JS4
4[(JS1 − JS3) − (JS2 − JS4)]
∈ [0, 1/4]
if JS1 − JS3 ≥ 0, JS2 − JS4 ≤ 0 ⇒
x =
4(JS1 − JS3) − 3(JS2 − JS4)
4[(JS1 − JS3) − (JS2 − JS4)]
∈ [3/4, 1]
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14. Step 1
Calculation of divergence and direction matrices
Maximum JS
if JS1 − JS3 ≤ 0, JS2 − JS4 ≥ 0 ⇒
x =
2(JS1 − JS3) − (JS2 − JS4)
4[(JS1 − JS3) − (JS2 − JS4)]
∈ [1/4, 1/2]
if JS1 − JS3 ≤ 0, JS2 − JS4 ≤ 0 ⇒
x =
2(JS1 − JS3) + 3(JS2 − JS4)
4[(JS1 − JS3) − (JS2 − JS4)]
∈ [1/2, 3/4]
Finally δ = πx ∈ [0, π) as the estimated edge direction. The x direction
maximizes the JS.
Now each pixel is labelled with a pair of values, (estimated edge direction, and
the estimated JSmax)
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15. Step 1
Calculation of divergence and direction matrices
Attenuation Factor
Due the JS is too sensitive to any change in grey levels between regions is necesary
include extra information, as an attenuation factor.
JS∗
i,j = JSi,j (1 − α + αWi,j )
Where
Wi,j =|Nw1 − Nw2|/Nw
Nw1, Nw2 are the average grey level of subwindows W1 and W2
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16. Step 2
Edge-pixel selection
Edge-pixel selection
In this step the procedure selects which pixels from the divergence matrix are edge
pixels.
Thresholding the divergence matrix is not always useful, since maximum JS values
depend on the composition of adjacent textures, and will thus vary according to
texture. Consequently, it would seem more appropriate to use a local criterion.
Accordingly, each edge-pixel candidate is the centre of an odd-length monodimensional
window, placed perpendicular to the estimated edge direction in that pixel
Estimated edge
direction
Pixel under
study
Figure: Monidimensional Window
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17. Step 2
Edge-pixel selection
Edge-pixel selection
JScentre − JSj Td
Any other pixel j in that particular monodimensional window, where Td is a threshold.
Pixels marked as edge pixels are then outstanding local maxima of the divergence
matrix. Obviously, detection results depend directly on the parameter Td, which can
be modified by the user if necessary.
This local edge-pixel detection method requires simple divergence matrix
pre-processing. The divergence matrix is therefore smoothed out by repeatedly
applying a 3 × 3 mean filter.
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18. Step 3
Edge-linking
Edge-linking
This step attempts to join the various sets of edge pixels using information from the
divergence matrix associated with the image, together with knowledge of the direction
in which maximum JS is produced. In broad terms, the linking procedure consists in
extracting edge pixels unmarked since they did not satisfy the condition, but nearly
did. Not all the pixels in the image are candidates for filling the gaps, only those
classified as neighbour candidates of end pixels.
Figure: End points and neighbour candidates for edge prolongation. E, end point; C,
neighbour candidates. The remaining grey pixels are edge pixels.
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19. Step 3
Edge-linking
Join End-points
End pixel criteria, is a pixel having one or two marked pixels joined together.
Neighbour candidate must have a JS reasonably high.
The estimated edge direction of the end pixel Dirend , the edge-direction
neighbour candidate and the edge-direction of the physical line joining them
must not differ more than a specified amount.
Join End-points
JSend − JSneighbourcandidate τd
(Dir(end, neighbourcandidate)) − Dirend )2
+(Dir(end, neighbourcandidate)
−Dirneighbourcandidate )2
τθ
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20. Results Theoretical
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21. Results Theoretical
Figure:
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22. Outline
1 Introduction
2 The system
3 Conclusions
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23. Discussion
Discussion
Although this work is still in preliminary stages, we have seen the Monfroy framework
is suitable for use in the modeling and prototype a dynamic composition of Web
Services in the import of goods constrained problem.
Solve the backtracking problem in an totally distributed environment is still a problem,
and must be resolved for use in a real environment
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24. Questions ?
Rodrigo Rojas Moraleda
rodrigo.rojas@postgrado.usm.cl
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