CS654: Digital Image Analysis Lecture 13: Discrete Fourier Transformation

Recap of Lecture 12 Unitary transform Separable transform Kronecker Product Improvement of computational complexity

Outline of lecture 13 Discrete Fourier transformation 1D and 2D Separable DFT Fast Fourier Transform

Kronecker Products Computational complexity??Fast image transforms

Validation using Basis images Verification using:

Basis images Real part of the Fourier transform basis images.

Properties of Unitary transform Energy Conservation Energy compaction Decorrelation

Introduction 1-D Unitary transform Forward transformation Reverse transformation Transformation matrix to be chosen appropriately

Discrete Fourier Transformation (DFT) Let the transformation matrix be defined as For ease of notation

Inverse DFT Then the inverse DFT will be defined as: Is the transformation unitary?

Unitary DFT Unitary forward and reverse DFT equations are defined as Using matrix notation where,

Is matrix used for DFT Unitary? Magnitude of each row is equal to 1 Rows are orthogonal to each other

2-D DFT Forward transformation Reverse transformation

Unitary 2-D DFT Forward transformation Reverse transformation

Separable 2-D DFT

Significance of Separability 1-D case: Using the 1D analogy

Visualization of separability (0,0) Transform over column for each row (0,0) Transform over rows for each columns Input image DFT image

Magnitude and Phase of DFT Magnitude: Phase: Input image MagnitudePhase angle

Illustration of reconstruction Input Image 1 (Woman) Phase angle of Input (IPA1) Reconstructed only using IPA1 Reconstructed only using the magnitude

Thank you Next Lecture: Properties of DFT