Quaternions andQuaternion Colour Constancy

Quaternions Quaternions …

Are a member of hypercomplex numbers Are a generalization of complex numbers Has one real part and three imaginary parts

i.e. A RGB colour is represented by a pure

quaternion

kbjgirq

kajaiaaa 3210

1

Quaternions

A picture of quaternions Quaternion axes in 4D space

Pure quaternion for colour

reali

kj

i

kj

Orthogonal in 4D

“pure” = zero real part

Quaternion PCA

QPCA is a generalization of complex PCA

QPCA for dimension reduction Similar to PCA for real numbers Quaternion-valued Texture can be

described in low dim. space

Quaternion PCA

Figure 13: QPCA based image compression. (a) –(d) are the reconstructed images with k(# of basis vectors)=3,16,50,255. Note that (d) is the perfect reconstruction of the original image

(a) (b) (c) (d)

Eg. QPCA For Image Compression Each row of the image is a input variable QPCA on all rows

QPCA for Texture Feature Extraction

Training

QPCA

Image-specific quaternion texture basisSampled sub-windows

Surprisingly, need only the first basis texture element

Feature Extraction Feature Deduction

Single quaternion

A texture patch

1st QPCA Basis texture element

magnitude

real layerred layer

green layerblue layer

T

Classification

Textures By classifying their extracted quaternion

features Images based on content

By recognizing the class of textures they contain

Images based on illumination By identifying the kind of illuminations of

textures they contain

Colour Texture Histogram

An image contains colour textures Colour Texture Histogram

It counts different colour textures Quaternion texture can be used to

build colour histogram An extension of colour histogram

when each pixel is consider as a texture

Quaternion For Colour Constancy Colour Constancy

SVR uses colour histograms Colour Histogram

Contains colour information only Texture Histogram

Contains structural information only Colour Texture Histogram

Integrates both colour and structure info A new representation of images Can SVR do better by Colour Texture Histogram?

K-Medians Clustering for Training Set Reduction

Function Estimation Define a function(curve) that minimizes the energy

function controlled by all training data points

Use this function to estimate new data SVR, TPS

Control Point Reduction

Problem Training set too large to fit into memory Long processing time

Reduce training set using k-medians Partition n control points into k clusters Keep k medians of these clusters Reduce n control points to k

k-Medians k-medians clustering:

Given: N points (x1… xN) in a metric space Find k points C = {c1, c2, …, ck} that minimize

Σ d(xi, C) (the assignment distance)

• In the example above, only 4 control points are needed to define the curve

k-Medians k-medians

Median as the best representative for each cluster

Less sensitive to outliers k can be determined based on memory and

training time requirement