Dihedral Groups and Spatio-Chromatic Filter Systems Reiner Lenz, - - PowerPoint PPT Presentation

FWS-2010-Ilmenau Reiner Lenz

Dihedral Groups and Spatio-Chromatic Filter Systems

Reiner Lenz, Martin Solli Linköping University reiner.lenz@liu.se; martin.solli@liu.se

FWS-2010-Ilmenau Reiner Lenz

Take-Home-Message

Can you see the irreducible representations

• f dihedral groups?

Can you hear the shape of a drum? Feynman

FWS-2010-Ilmenau Reiner Lenz

The Algebra of Digital Color Images

• 1. Algebra, Filter design from first principles
• 2. Visual properties of filters
• 3. Discrimination of visual classes, Image retrieval, Emotions

FWS-2010-Ilmenau Reiner Lenz

Digital Color Images

• Consists of pixels
• Pixels are located on a grid
• Pixel value is a 3-D vector

FWS-2010-Ilmenau Reiner Lenz

What can we do with them?

Transform the grid k ● 90 degree rotation reflection on diagonal Form the group D(4)

FWS-2010-Ilmenau Reiner Lenz

What can we do with them?

Transform the channels R G B k ●120 degree rotation reflection on symmetry axis Form the group D(3) = S(3) permutation group

FWS-2010-Ilmenau Reiner Lenz

Dimensionality of Pattern Space

Take N points invariant under D(4) and the 3 color channels invariant under D(3) Coordinates are (x,c) = (position, channel) The possible value distributions form a 3N – dimensional space P P of patterns p(x,c) This space is invariant under D(4) and D(3) Rot, Green = 12 Blue = 24 RGB = 48

FWS-2010-Ilmenau Reiner Lenz

Decomposition of Pattern Space

Representation theory of finite groups: The space can be split into invariant subspaces of minimum dimension P = P P = P0

0 + P

+ P1

1 + … + P

+ … + PK

K

p = (a01 b01 + … + a0k(0) b0k(0)) + … + (aK1 bK1 + … + aKk(K) bKk(K)) akl = p’bkl akl are coefficients, bkl are “basis patterns”

FWS-2010-Ilmenau Reiner Lenz

Some Properties

The spaces P Pk have dimensions 1, 2 or 4 The “basis patterns” or “filter functions” consist of 0,1 and -1 The “feature vectors ” (an1 …ank(n)) have simple transformation properties under D(4)xD(3) transformations of the pattern p (steerable filters) The norm of the “feature vectors ” rn = ||(an1 …ank(n))|| is invariant under D(4)xD(3) transformations

FWS-2010-Ilmenau Reiner Lenz

Simplest example: 2x2 Color Channels

2x2 = 4 pixels 3x4 = 12 dimensionial vectors p Step 1: Combine channels in R+G+B and (2-D vector (R-G,R+G-2B)

FWS-2010-Ilmenau Reiner Lenz

Spatial - Intensity

four pixel vector with intensity values is multiplied by The first is an averaging filter (1D subspace) The second is a line filter (1D subspace) The third+fourth are x- and y-gradients (2D subspace)

FWS-2010-Ilmenau Reiner Lenz

Spatial – Color Channels

4 pixels and 2 channels = 8 – dimensional vector Theory gives the decomposition in invariant subspaces of dimensions up to four

FWS-2010-Ilmenau Reiner Lenz

Implementation

Compute + + + - + + + - This is the “FFT” form of the computation

FWS-2010-Ilmenau Reiner Lenz

4x4 Window

4x4x3 = 48 dimensions

FWS-2010-Ilmenau Reiner Lenz

Signatures

From 4x4x3 = 48D RGB distribution Compute 48 new features (multiplication with a square matrix) Collect the 48 features in 24 sub-vectors For every sub-vector compute the norms r1 … r24 These 24 norms are collected in the signature vector

FWS-2010-Ilmenau Reiner Lenz

Application to Image Retrieval

Databases: Real-world databases from Picsearch search engine a) Objects like beach Monet and Warhol 320 000 images b) Emotions like colorful and elegant 1.2 million images Download at http://diameter.itn.liu.se/emodb

FWS-2010-Ilmenau Reiner Lenz

Descriptors

• 1. An image consists of B blocks of size 4x4
• 2. Every block gives a 24D signature vector
• 3. Describe the image by the histograms over these signature vectors

FWS-2010-Ilmenau Reiner Lenz

Characteristic Images

Intensity line 2x2

FWS-2010-Ilmenau Reiner Lenz

Intensity Edge

FWS-2010-Ilmenau Reiner Lenz

Color Edge

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Homogeneous Color

FWS-2010-Ilmenau Reiner Lenz

Visual properties of Colorful/Elegant

FWS-2010-Ilmenau Reiner Lenz

Training of Support-Vector-Machines for two-class discrimination

Use different combinations of signature vectors train a SVM with half of the data and two classes Use the output of the SVM as indicator of the class membership

FWS-2010-Ilmenau Reiner Lenz

Warhol-Monet

FWS-2010-Ilmenau Reiner Lenz

Intensity-Line-2x2

FWS-2010-Ilmenau Reiner Lenz Edge-Intensity-2x2

FWS-2010-Ilmenau Reiner Lenz Homogeneous color 2x2

FWS-2010-Ilmenau Reiner Lenz Color Edge 2x2

FWS-2010-Ilmenau Reiner Lenz All Color

FWS-2010-Ilmenau Reiner Lenz Colorful-Elegant