Quaternions and Quaternion Colour Constancy. Quaternions Quaternions … Are a member of...

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Quaternions and Quaternion Colour Constancy
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Transcript of Quaternions and Quaternion Colour Constancy. Quaternions Quaternions … Are a member of...

Page 1: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

Quaternions andQuaternion Colour Constancy

Page 2: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 3: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 4: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 5: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 6: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

QPCA for Texture Feature Extraction

Training

QPCA

Image-specific quaternion texture basisSampled sub-windows

Surprisingly, need only the first basis texture element

Page 7: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

Feature Extraction Feature Deduction

Single quaternion

A texture patch

1st QPCA Basis texture element

magnitude

real layerred layer

green layerblue layer

T

Page 8: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 9: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 10: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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?

Page 11: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

K-Medians Clustering for Training Set Reduction

Page 12: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 13: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 14: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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

Page 15: Quaternions and Quaternion Colour Constancy. Quaternions  Quaternions … Are a member of hypercomplex numbers Are a generalization of complex numbers.

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