Apply yourself, Fourier! AP 186 Activity 6

Hello, and welcome to the second edition of the

Deadline rush pt. 2

where I will be discussing the different

Properties and Applications of the 2-D Fourier Transform

In the last activity, I discovered the basics of the 2-D Fourier Transform and how powerful it can really be.

Part 1. Anamorphic property of FT of different 2D patterns

The anamorphic property describes how a wide object in a certain space will be narrow in its inverse space (and vice versa). Here we show the certain examples of the anamorphic property.

Part 1a. Tall rectangle aperture

Part 1b. Wide rectangle aperture

Part 1c. Two dots along x-axis symmetric about center

Part 2. Rotation property of the FT

Part 2a. Basic sinusoid function

Part 2b. Vertical biasing

Notice that the center dot was added

Part 2c. Rotation

Part 2d. Multiplication

Part 2e. Addition

Part 3. Convolution theorem redux

Part 3a. Circle

Part 3b. Square

Part 3c. Gaussian

Part 3d. Random locations

Part 4. Fingerprint

 

Part 5. Lunar image

Original image

FFT. Remove the bias!

Part 6. Painting

References

1. Soriano, M. Properties and Applications of the 2-D Fourier Transform. Applied Physics 186 Activity Manual.
2. Carlo Solibet, Roland Romero, Robbie Esperanza, and Maricor Soriano for their ever-welcome assistance to me.