FOURIER TRANSFORM AND IMAGES Nathan Conger Recall Fourier

FOURIER TRANSFORM AND IMAGES Nathan Conger

Recall: Fourier Transform ◦ Transformation from time space to Frequency Space ◦ Decomposed frequencies are spikes in the winding of Transform

Pipeline of Image Processing Fourier Apply Change Inverse Fourier Images from https: //medium. com/@hicraigchen/digital-image-processing-using-fourier-transform-in-python-bcb 49424 fd 82

Image Space Image can be represented as a sum of basis Frequencies along the x and y-directions

Image Processing Greyscale ◦

Centralization ◦ DC Component -M/2 -N/2 0 N/2 N 2 N 0 M/2 M 2 M

Rotation ◦ Images from https: //homepages. inf. ed. ac. uk/rbf/HIPR 2/fourier. htm#: ~: text=The%20 DFT%20 is%20 the%20 sampled, describe%20 the%20 spatial%20 domain%20 image.

Images from https: //medium. com/@hicraigchen/digital-image-processing-using-fourier-transform-in-python-bcb 49424 fd 82 Filtering ◦ Low Pass Filtering: ◦ Filter out higher frequencies ◦ High Pass Filtering: ◦ Filter out lower frequencies Low Pass Filtered Original Image High Pass Filtered

Images from https: //medium. com/@hicraigchen/digital-image-processing-using-fourier-transform-in-python-bcb 49424 fd 82 Ideal Filter ◦ ◦

Ideal Low Pass D=10 D=20 D=40 D=80 Original Ideal High Pass Original image from: https: //pixabay. com/photos/ pin-up-black-and-whiteretro-2188425/

Images from https: //medium. com/@hicraigchen/digital-image-processing-using-fourier-transform-in-python-bcb 49424 fd 82 Butterworth Filter ◦ ◦ Low Pass n=20 Low Pass n = 3 High Pass n = 20 High Pass n = 3

Butterworth Low Pass D=50 n = 5 n=10 n=20 n=40 Original Butterworth High Pass

Images from https: //medium. com/@hicraigchen/digital-image-processing-using-fourier-transform-in-python-bcb 49424 fd 82 Gaussian Filter ◦ ◦

Gaussian Low Pass D=10 D=20 D=40 D=80 Original Gaussian High Pass 10 20 40 80

Low Pass ◦ Up Left: Original ◦ Up Right: Ideal Filter ◦ Bottom Left: Butterworth Filter (n=10) ◦ Bottom Right: Gaussian Filter

High Pass ◦ Up Left: Original ◦ Up Right: Ideal Filter ◦ Bottom Left: Butterworth Filter (n=10) ◦ Bottom Right: Gaussian Filter

Deconvolution ◦

Original image Images from http: //fourier. eng. hmc. edu/e 161/demos/e 161_demo 4. html Convolved image Restored image

Applications: Forensics Images from http: //www. robots. o x. ac. uk/~az/lectures /ia/lect 2. pdf

Medical Imaging Images from https: //biomedpharmajournal. org/vol 7 no 2/image-sharpening-by-gaussian-and-butterworth-high-passfilter/#: ~: text=Fourier%20 transform%20 will%20 reflect%20 the, is%20 called%20 masking%20 or%20 filtering. &text=High%20 pass%20 filters%20 are%20 usually%20 used %20 for%20 sharpening.

Machine Learning

Further Research/Other interesting things ◦ Can we extend this to color images? ◦ Epicycles: Circle drawings with Fourier Series Image from https: //pixabay. com/vectors/intersection-mix-colors-rgb-red-154782/ Animation from https: //www. myfourierepicycles. com/

References ◦ https: //www. di. univr. it/documenti/Occorrenza. Ins/matdid 916567. pdf ◦ https: //homepages. inf. ed. ac. uk/rbf/HIPR 2/fourier. htm#: ~: text=The%20 DFT%20 is%20 the%20 sampled, describe%20 the%20 spatial%20 domain%20 image. ◦ https: //www. cs. auckland. ac. nz/courses/compsci 773 s 1 c/lectures/Image. Processing-html/topic 1. htm ◦ https: //medium. com/@hicraigchen/digital-image-processing-using-fourier-transform-in-python-bcb 49424 fd 82 ◦ http: //www. robots. ox. ac. uk/~az/lectures/ia/lect 2. pdf ◦ https: //biomedpharmajournal. org/vol 7 no 2/image-sharpening-by-gaussian-and-butterworth-high-passfilter/#: ~: text=Fourier%20 transform%20 will%20 reflect%20 the, is%20 called%20 masking%20 or%20 filtering. &text=High%20 p ass%20 filters%20 are%20 usually%20 used%20 for%20 sharpening