Quaternions and Quaternion Colour Constancy

1. Quaternions and Quaternion Colour Constancy

2. 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 Quaternions

3. Quaternions • A picture of quaternions • Quaternion axes in 4D space • Pure quaternion for colour real i Orthogonal in 4D j k i “pure” = zero real part j k

4. 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

5. Quaternion PCA • Eg. QPCA For Image Compression • Each row of the image is a input variable • QPCA on all rows (a) (b) (c) (d) 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

6. Surprisingly, need only the first basis texture element QPCA Image-specific quaternion texture basis QPCA for Texture Feature Extraction • Training Sampled sub-windows

7. Feature Deduction T 1st QPCA Basis texture element Feature Extraction Single quaternion A texture patch

8. 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

9. 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

10. 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?

11. K-Medians Clustering for Training Set Reduction

12. 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

13. 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

14. 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

15. 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