CS654: Digital Image Analysis Lecture 15: Image Transforms with Real Basis Functions.

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Presentation transcript:

CS654: Digital Image Analysis Lecture 15: Image Transforms with Real Basis Functions

Recap of Lecture 14 Discrete Fourier Transform Orthogonal sinusoidal waveform Computational complexity is high Involves complex multiplication

Outline of Lecture 15 Basis function with real (Integer) values Hadamard Transform Haar Transform KL Transform

Hadamard Transform Core matrix

Generation of transformation matrix Using Kronecker product recursion Example

Unitary Hadamard Transform General unitary transformation equation Using Hadamard transform Forward transformation Inverse transformation Forward transformation Inverse transformation What happens in case of images?

Summation expression Forward transformation Inverse transformation LSB, MSB ?

Properties of Hadamard Transformation Sequency

Natural Ordering vs. Sequency Ordering Natural Order (h) Sequency (s) Natural order of the Hadamard transform coefficients = bit reversed gray code representation of its sequency

Haar Transform

Haar Function

Haar Basis Function Computation Determine the order of N Calculate the Haar function

Haar Basis Function Computation

Haar basis for N=2

KL Transform Exploits the statistical properties of an image Basis functions are orthogonal Eigen vectors of the covariance matrix Optimally de-correlates the input data Energy compaction Input dependent, and high computational complexity

Eigen analysis Inverse Transform is defined as:

Thank you Next Lecture: Convolution and Correlation