Fourier Transform

Question 1

Understand waves in 1D, 2D, 3D …

Answer 1

• Wavelength/ frequency is multidimensional
• 3 key parameters of waves:
• Amplitude
• Frequency
• Phase shift
• Frequency is 1/wavelength so if wavelength is 2 units, frequency = ½ units-1
• Waves can be represented in a frequency (reciprocal) space
• 2D waves have amplitude, phase, and a two-dimensional frequency component
• Waves have a certain wavelength along a and b
• Fourier transform will have DC component at 0,0
• Waves have no directionality so if you have a frequency of 0,2 you also have 0,-2
• Brightness of point = amplitude

Question 2

What is a Fourier Transform

Answer 2

• Scattering of waves is a Fourier transform
• Composite waves are made up of waves of many different frequencies
• Fourier transform calculates the component sine waves of a composite wave
• An inverse Fourier transform puts them back together and gets you back to the composite wave
• DC component has infinite wavelength and therefore frequency 0

Question 3

Relate Fourier transform to images and structures

Answer 3

• We use phase vs. Frequency and amplitude vs. Frequency data
• Inverse Fourier transform can be used to get back to the original image
• Fourier transform —> extract individual Fourier components —> inverse Fourier transform —> image
• We can get a feel for the contributions of different spatial frequencies by applying filters in reciprocal space
• Just low spatial frequency components = blurry image
• Dc component increases overall brightness level
• Only high frequency info gives very dark image
• Amplitudes and phases are both important but phases are critical
• Phases are more important to describe the image than amplitude