Digital Image Processing Lecture 8: Image Enhancement in Frequency Domain II Naveed Ejaz

6/25/20162 Fourier Transform Review Notice how we get closer and closer to the original function as we add more and more frequencies

6/25/20163 Fourier Transform Review Frequency domain signal processing example in Excel

6/25/20164 Properties of Fourier Transform

6/25/ D DFT

6/25/20166 Translation Property of 2-D DFT

6/25/20167 Shifting the Origin to the Center

6/25/20168 Shifting the Origin to the Center

6/25/20169 Properties of Fourier Transform

6/25/ Properties of Fourier Transform

6/25/ Reciprocality of Lengths due to Scaling Property

6/25/ Properties of Fourier Transform

6/25/ Rotation Examples

6/25/ Properties of Fourier Transform

6/25/ Properties of Fourier Transform

6/25/ Properties of Fourier Transform  As we move away from the origin in F(u,v) the lower frequencies corresponding to slow gray level changes  Higher frequencies correspond to the fast changes in gray levels (smaller details such edges of objects and noise)  The direction of amplitude change in spatial domain and the amplitude change in the frequency domain are orthogonal (see the examples)

6/25/ DFT Examples

6/25/ DFT Examples

6/25/ DFT Examples

6/25/ Masking, Correlation and Convolution

6/25/ Properties of Fourier Transform

6/25/ Filtering using Fourier Transforms

6/25/ Example of Gaussian LPF and HPF

6/25/ Example of Modified HPF

6/25/ Relationship b/w Spatial domain filters and Fourier domain filters

6/25/ Relationship b/w Spatial domain filters and Fourier domain filters

6/25/ Filters to be Discussed

6/25/ Ideal Low Pass Filter

6/25/ Ideal Low Pass Filter

6/25/ Ideal Low Pass Filter (example)

6/25/ Why Ringing Effect

6/25/ Ringing Effect (example)

6/25/ Butterworth Low Pass Filter

6/25/ Butterworth Low Pass Filter

6/25/ Butterworth Low Pass Filter (example)

6/25/ Butterworth Low Pass Filter

6/25/ Gaussian Low Pass Filters

6/25/ Gaussian Low Pass and High Pass Filters

6/25/ Gaussian Low Pass Filters

6/25/ Gaussian Low Pass Filters (example)

6/25/ Gaussian Low Pass Filters (example)

6/25/ Sharpening Fourier Domain Filters

6/25/ Sharpening Spatial Domain Representations

6/25/ Sharpening Fourier Domain Filters (Examples)

6/25/ Sharpening Fourier Domain Filters (Examples)

6/25/ Sharpening Fourier Domain Filters (Examples)

6/25/ Laplacian in Frequency Domain

6/25/ Laplacian in Frequency Domain

6/25/ Unsharp Masking, High Boost Filtering

6/25/ High Frequency Emphasis

6/25/ Example of Modified High Pass Filtering

6/25/ Homomorphic Filtering

6/25/ Homomorphic Filtering

6/25/ Homomorphic Filtering

6/25/ Homomorphic Filtering

6/25/ Homomorphic Filtering (Example)

6/25/ Basic Filters And scaling rest of values.

6/25/ Example (Notch Function)

6/25/ Properties of Fourier Transform

6/25/ Implementation/Optimization of Fourier Transform  First multiply the factor (-1) (x+y) to the original function f(x,y) to shift the origin in DFT to center of the image.  Find Fourier transform by decomposing into 2-D function into 1-D transformations.  Inverse DFT can be computed using the forward DFT algorithm with slight modification.  Optimization of Fourier transform – Brute Force method (decomposition into 1-D Fourier transform) – Fast Fourier Transform (FFT)

6/25/ Implementation/Optimization of Fourier Transform

6/25/ Comparison of Number of Operations for DFT Techniques

6/25/ Need of Padding Due to Symmetrical Properties of DFT

6/25/ Need of Padding Due to Symmetrical Properties of DFT  To overcome this problem sue periodicity of DFT Extended/padded functions are used, given by (in 1-D)

6/25/ Need of Padding Due to Symmetrical Properties of DFT

6/25/ Need of Padding Due to Symmetrical Properties of DFT

6/25/ Need of Padding Due to Symmetrical Properties of DFT

6/25/ Padding During Filtering in Frequency Domain