曾禎宇

Separated vs. Holistic R G B Color Image

Separated vs. Holistic RGB Color Image

Reference  T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Transactions on Image Processing, vol. 16, pp. 22–35, Jan  T. A. Ell, “Multi-vector color-image filters,” in Proceedings of the IEEE International Conference on Image Processing, vol. V, (San Antonio, Texas),pp. 245– 248, Sept  Geometric Algebra: se99/ se99/

Outline  Geometric Algebra  Multi-Vector Color Images  Applications

Quaternion based Color Image Representation  For 3-D Space: 4 Elements {1, i, j, k}  For Color Image f(n,m) = r(n,m) i + g(n,m) j + b(n,m) k  Hypercomplex Representation i j k

Geometric Algebra for Color Image Representation  Quaternion: f(n,m) = r(n,m) i + g(n,m) j + b(n,m) k  Geometric Algebra f(n,m) = r(n,m) e 1 + g(n,m) e 2 + b(n,m) e 1 ^ e 2  More General  Quaternion is the special case of Geometric Algebra

Geometric Algebra  Inner Product  a · b = |a||b| cos θ  Outer Product  a ^ b : bi-vector  magnitude of a ^ b = |a||b| sin θ  a ^ b = - b ^ a  a^(b+c) = a^b + a^c  Geometric Product  ab = a · b + a^b  ba = a · b - a^b

Geometric Algebra in 2-D  2 orthonormal vectors e 1 and e 2  e 1 2 = e 2 2 = 1  e 1 · e 2 = 0  A full algebra is spanned by  1 scalar1  2 vectors {e 1, e 2 }  1 bi-vector e 1 ^ e 2

Multi-Vector  A = a 0 + a 1 e 1 + a 2 e 2 + a 12 e 1 ^e 2  e 1 ^e 2 = I 2 (pseudo-scalar)  I 2 2 = e 1 e 2 e 1 e 2 = -e 2 e 1 e 1 e 2 = -1  A = a 0 + a 1 e 1 + a 2 e 2 + a 3 I 2  A = (a 0 + a 12 I 2 ) + ( a 1 e 1 + a 2 e 2 )  (a 0 + a 12 I 2 ) : a complex number

Multi-Vector for Color Image Representation  A = (a 0 + a 12 I 2 ) + ( a 1 e 1 + a 2 e 2 )  f = L + v

Color Edge Detection Filters Example: 3 Type Horizontal Edge Detection

Color Edge Detection Filters Example: 3 Type Horizontal Edge Detection

Color Edge Detection Filters H1  detects edge in both luminance and chrominance H2  detects edge in chrominance but smoothes luminance H3  detects edge in luminance but smoothes chrominance