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CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

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CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis
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Page 1: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

CS654: Digital Image Analysis

Lecture 36: Feature Extraction and Analysis

Page 2: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Recap of Lecture 35

• JPEG

• DCT

Page 3: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Outline of Lecture 36

• Feature representation

• Shape feature

• Curvature

• Curvature Scale Space

• Other shape features

Page 4: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Introduction

• The goal in digital image analysis is to extract useful information for solving application-based problems.

• The first step to this is to reduce the amount of image data using methods that we have discussed before:

• Image segmentation

• Filtering in frequency domain

• Morphology,

• …

Page 5: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

What next ??

• The next step would be to extract features that are useful in solving computer imaging problems.

• What features to be extracted are application dependent.

• After the features have been extracted, then analysis can be done.

Page 6: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Shape features

Shape

Contour Region

Structural

• Syntactic• Graph• Tree• Model-driven• Data-driven• Shape context

• Perimeter• Compactness• Eccentricity• Fourier Descriptors• Wavelet Descriptors• Curvature Scale Space• Shape Signature• Chain Code• Hausdorff Distance• Elastic Matching

Non-Structural • Area• Euler Number• Eccentricity• Geometric Moments• Zernike Moments• Pseudo-Zernike Mmts• Legendre Moments• Grid Method

Page 7: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Planer curves

Parameterized curve 𝐶 (𝑝 )={𝑥 (𝑝 ) , 𝑦 (𝑝 ) } 𝑝∈ [0,1 ]

y

x

𝐶 (0.1 )

𝐶 (0.5 )𝐶 (0.8 )

Closed curve:

y

x

𝐶 (𝑥 )=¿{𝑥 ,𝐹 (𝑥 ) }

Page 8: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Planer curves: tangent, curvatureParameterized curve

𝐶 (𝑝 )={𝑥 (𝑝 ) , 𝑦 (𝑝 ) } 𝑝∈ [0,1 ]𝐶 (0.1 )

𝐶 (0.5 )𝐶 (0.8 )

Tangent 𝑡=𝐶𝑝

¿𝐶𝑝∨¿¿

𝐶𝑝=𝜕𝐶𝜕𝑝

¿ [𝑥𝑝 , 𝑦 𝑝]

¿𝐶 𝑠

is a parameterization to get tangent of unit length

magnitude of the derivative is unity

⟨𝐶 𝑠 ,𝐶𝑠 ⟩=1 𝜕𝜕𝑠 ⟨𝐶𝑠 ,𝐶 𝑠 ⟩= 𝜕

𝜕 𝑠1 ⇒ 2 ⟨𝐶 𝑠 ,𝐶𝑠𝑠 ⟩=0⇒ ⟨𝑪𝒔 ,𝑪𝒔𝒔 ⟩=𝟎

𝑪𝒔𝒔

Curvature: Magnitude of the second derivative,

Page 9: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Curvature

• Magnitude of the second derivative curvature

• Change in the tangent between two successive point is more

• Curvature is more curve is curving a lot

𝜅=0𝜅=

1𝑟

Page 10: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Linear Transformation

Affine transformation:

𝐴= [𝑢1 ,𝑢2 ] , h𝑤 𝑒𝑟𝑒 ⟨𝑢1 ,𝑢2 ⟩=0 𝑎𝑛𝑑 ⟨𝑢𝑖 ,𝑢𝑖 ⟩=1Euclidean transformation:

Euclidean

Affine Affine

Page 11: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Linear Transformation

Equi- Affine transformation:

Euclidean

Equi-Affine

det ( 𝐴)=1

Equi- Affine

Page 12: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Differential Signature

Euclidean invariant signature {𝑠 ,𝜅 (𝑠 ) }

𝑠

𝜅

Starting point?

Page 13: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Affine transformation

𝐼 2 (𝑥 , 𝑦 )=𝐼 1(𝑇 1 (𝑥 , 𝑦 ) ,𝑇 2 (𝑥 , 𝑦 ))

[𝑇 1 (𝑥 , 𝑦 )𝑇 2 (𝑥 , 𝑦 )]=[𝑎 𝑏

𝑐 𝑑 ][𝑥𝑦 ]+[𝑒𝑓 ]Equi-affine:

What would be the definition of arc length and curvature in case of affine transformation?

Page 14: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Re-parameterization

• The same curve C, can be parameterized with two different parameters

• Magnitude of the derivative changes, not the curve

• Geometric measurement should be invariant to parameterization

• Invariant under group of transformation

Page 15: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Euclidean arc-length

• Only allows for rotation and translation

• Length is preserved

𝑑𝑥

𝑑𝑦𝑑𝑠

𝑑𝑠=√𝑑𝑥2+𝑑𝑦 2

¿𝑑𝑝𝑑𝑝

√𝑑𝑥2+𝑑 𝑦2

¿𝑑𝑝√( 𝑑𝑥𝑑𝑝 )2

+( 𝑑𝑦𝑑𝑝 )2

=¿𝐶𝑝∨𝑑𝑝

𝑠=∫ ¿𝐶𝑝∨𝑑𝑝 𝐿=∫0

𝐿

𝑑𝑠

Page 16: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Equi-affine arclength

• Length is not preserved any more, however area is preserved

𝐶𝑣

𝐶𝑣𝑣

= parameterization for affine transform

|𝐶𝑣 ,𝐶𝑣𝑣|=1

𝑣=∫|𝐶𝑝 ,𝐶𝑝𝑝|13 𝑑𝑝

𝑣=∫|𝑪𝒔 ,𝑪𝒔𝒔|13 𝑑𝑠1

𝜅 𝑣=∫ 𝜅13 𝑑𝑠 𝑑𝑣=𝜅

13 𝑑𝑠

Page 17: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Equi-affine curvature

|𝐶𝑣 ,𝐶𝑣𝑣|=1 ⇒𝜕𝜕𝑣 |𝐶𝑣 ,𝐶𝑣𝑣|=

𝜕𝜕 𝑣1

⇒|𝐶𝑣𝑣 ,𝐶𝑣𝑣|+|𝐶𝑣 ,𝐶𝑣𝑣 𝑣|=0

⇒|𝐶𝑣 ,𝐶𝑣𝑣𝑣|=0

⇒𝐶𝑣∥𝐶𝑣𝑣𝑣

⇒𝐶𝑣𝑣𝑣=𝜇𝐶𝑣

Affine invariant curvature

Page 18: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Differential signature

Affine invariant signature {𝑣 ,𝜇 (𝑣 )}

𝑠

𝜅

Page 19: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Curvature Scale Space

• Defined a unique way of observing and studying 2-D closed shapes

• Trace the outer most closed curve of the  object and thus proceed

• Mapping to a space which represents each point as a curvature w.r.t. the arc length.

• Matching is performed using CSS image

Page 20: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

CSS Image

•  It is a multi-scale organization of the inflection points (zero crossing points) of an evolving contour

• Curvature is a local measure of how fast a planar contour is turning

Page 21: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Process

1. This method convolutes a path (arc length ) based parametric representation of a planar curve with a Gaussian function

2. As the Gaussian width varies from a small to a large value.

3. Plot the curvature vs. the normalized arc length of the planar curve.

4. As  the Gaussian width is increased,  • the scale of the image increases or• the image evolves and thus the amount of noise is reduced and the

curve distortions smoothened.

• The benefits of this representation  are that it is invariant under rotation ,uniform scaling and  translation of the curve.

Page 22: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Illustration

arclength parameter on the original contour

 sta

ndar

d de

viat

ion

of t

he

Gau

ssia

n fil

ter

CSS ImageCurvature zero-crossing segments

Input image

Page 23: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Matching

• Let be the maxima of the query

Where Arc length parameter

= scale parameter, such that

Let be the maxima of the database shape Where Arc length parameter = scale parameter, such that

Page 24: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Illustration

Page 25: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Binary Object Features – Area

• The area of the ith object is defined as follows:

• The area Ai is measured in pixels and indicates the relative size of the object.

1

0

1

0

),(width

ci

height

ri crIA

Page 26: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Binary Object Features – Center of Area

• The center of area is defined as follows:

• These correspond to the row and column coordinate of the center of the i-th object.

1

0

1

0

1

0

1

0

),(1

),(1

width

ci

height

rii

width

ci

height

rii

crcIA

c

crrIA

r

Page 27: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Binary Object Features – Axis of Least Second Moment

• The Axis of Least Second Moment is expressed as - the angle of the axis relatives to the vertical axis.

1

0

21

0

1

0

21

0

1

0

1

01

),()(),()(

),())((2tan2

1width

ci

height

r

width

ci

height

r

width

ci

height

ri

crIcccrIrr

crIccrr

Page 28: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Binary Object Features – Axis of Least Second Moment

• This assumes that the origin is as the center of area.

• This feature provides information about the object’s orientation.

• This axis corresponds to the line about which it takes the least amount of energy to spin an object.

Page 29: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

Binary Object Features – Aspect Ratio

• The equation for aspect ratio is as follows:

• reveals how the object spread in both vertical and horizontal direction.

• High aspect ratio indicates the object spread more towards horizontal direction.

1

1

minmax

minmax

rr

cc

Page 30: CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.

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