Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial and Workshop March 14, 2007 Fourier Transforms, Filtering and Convolution

Fourier Transform is an invertible operator ImageFourier Transform v2 will display image or its transform FT

Fourier Transform is an invertible operator Image f(x,y) F(k x,k y ) x y 0NxNx NyNy {F(k x,k y )} = f(x,y)} {f(x,y)} = F(k x,k y ) N x ⁄ 2 N y ⁄ 2 Fourier Transform

Continuous Fourier Transform f(x) = F(s) Euler’s Formula

Some Conventions Image Domain Forward Transform f(x,y,z) g(x) F Fourier Domain –Reciprocal space –Fourier Space –K-space –Frequency Space Reverse Transform, Inverse Transform F(k x,k y,k z ) G(s) F

Math Review - Periodic Functions If there is some a, for a function f(x), such that f(x) = f(x + na) then function is periodic with the period a 0 a2a3a

Math Review - Attributes of cosine wave Amplitude Phase f(x) = cos (x) f(x) = 5 cos (x) f(x) = 5 cos (x )

Math Review - Attributes of cosine wave Amplitude Phase Frequency f(x) = 5 cos (x) f(x) = 5 cos (x ) f(x) = 5 cos (3 x )

Math Review - Attributes of cosine wave f(x) = cos (x) Amplitude, Frequency, Phase f(x) = A cos (kx +  )

Math Review - Complex numbers Real numbers:  Complex numbers i – 6.7 i -5.2 ( i )

Math Review - Complex numbers Complex numbers i – 6.7 i -5.2 ( i ) General Form Z = a + b i Re(Z) = a Im(Z) = b Amplitude A = | Z | = √ (a 2 + b 2 ) Phase  =  Z = tan -1 (b/a)

Math Review – Complex Numbers Polar Coordinate Z = a + bi Amplitude A = √ (a 2 + b 2 ) Phase  = tan -1 (b/a) a b   A 

Math Review – Complex Numbers and Cosine Waves Cosine wave has three properties –Frequency –Amplitude –Phase Complex number has two properties –Amplitude –Wave Complex numbers to represent cosine waves at varying frequency –Frequency 1: Z 1 = 5 +2i –Frequency 2: Z 2 = i –Frequency 3: Z 3 = 1.3 – 1.6i

Fourier Analysis Decompose f(x) into a series of cosine waves that when summed reconstruct f(x)

Fourier Analysis in 1D. Audio signals (Hz) (Hz) Amplitude Only

Fourier Analysis in 1D. Audio signals (Hz) Your ear performs fourier analysis.

Fourier Analysis in 1D. Spectrum Analyzer. iTunes performs fourier analysis.

Fourier Synthesis Summing cosine waves reconstructs the original function

Fourier Synthesis of Boxcar Function Boxcar function Periodic Boxcar Can this function be reproduced with cosine waves?

k=1. One cycle per period A 1 ·cos(2  kx +  1 ) k=1 A k ·cos(2  kx +  k )  k=1 1

k=2. Two cycles per period A 2 ·cos(2  kx +  2 ) k=2 A k ·cos(2  kx +  k )  k=1 2

k=3. Three cycles per period A 3 ·cos(2  kx +  3 ) k=3 A k ·cos(2  kx +  k )  k=1 3

A k ·cos(2  kx +  k ) N Fourier Synthesis. N Cycles A 3 ·cos(2  kx +  3 ) k=3  k=1

Fourier Synthesis of a 2D Function An image is two dimensional data. Intensities as a function of x,y White pixels represent the highest intensities. Greyscale image of iris 128x128 pixels

Fourier Synthesis of a 2D Function F(2,3)

Fourier Filters Change the image by changing which frequencies of cosine waves go into the image Represented by 1D spectral profile 2D Profile is rotationally symmetrized 1D profile

Low frequency terms –Close to origin in Fourier Space –Changes with great spatial extent (like ice gradient), or particle size High frequency terms –Closer to edge in Fourier Space –Necessary to represent edges or high- resolution features

Frequency-based Filters Low-pass Filter (blurs) –Restricts data to low-frequency components High-pass Filter (sharpens) –Restricts data to high-frequency-componenets Band-pass Filter –Restrict data to a band of frequencies Band-stop Filter –Suppress a certain band of frequencies

Cutoff Low-pass Filter Image is blurred Sharp features are lost Ringing artifacts

Butterworth Low-pass Filter Flat in the pass-band Zero in the stop-band No ringing

Gaussian Low-pass Filter

Butterworth High-pass Filter Note the loss of solid densities

How the filter looks in 2D unprocessed lowpass highpass bandpass

Filtering with EMAN2 LowPass Filters filtered=image.process(‘filter.lowpass.guass’, {‘sigma’:0.10}) filtered=image.process(‘filter.lowpass.butterworth’, {‘low_cutoff_frequency’:0.10, ‘high_cutoff_frequency’:0.35}) filtered=image.process(‘filter.lowpass.tanh’, {‘cutoff_frequency’:0.10, ‘falloff’:0.2}) HighPass Filters filtered=image.process(‘filter.highpass.guass’, {‘sigma’:0.10}) filtered=image.process(‘filter.highpass.butterworth’, {‘low_cutoff_frequency’:0.10, ‘high_cutoff_frequency’:0.35}) filtered=image.process(‘filter.highpass.tanh’, {‘cutoff_frequency’:0.10, ‘falloff’:0.2}) BandPass Filters filtered=image.process(‘filter.bandpass.guass’, {‘center’:0.2,‘sigma’:0.10}) filtered=image.process(‘filter.bandpass.butterworth’, {‘low_cutoff_frequency’:0.10, ‘high_cutoff_frequency’:0.35}) filtered=image.process(‘filter.bandpass.tanh’, {‘cutoff_frequency’:0.10, ‘falloff’:0.2})

Convolution Convolution of some function f(x) with some kernel g(x) *= Continuous Discrete

xx Convolution in 2D x  xx = x  xx = xx xx xx xx

Microscope Point-Spread-Function is Convolution

Convolution Theorem f  g = {FG} f = FG G Convolution in image domain Is equivalent to multiplication in fourier domain

Contrast Theory observed image f(x) for true particle point-spread function envelope function noise obs(x) = f(x)  psf(x)  env(x) + n(x) Incoherant average of transform F 2 (s) CTF 2 (s) Env 2 (s) + N 2 (s) Power spectrum PS =

Lowpass Filtering by Convolution f  g = {FG} Camera shake Crystallographic B-factor

Review Fourier Transform is invertible operator Math Review Periodic functions Amplitude, Phase and Frequency Complex number Amplitude and Phase Fourier Analysis (Forward Transform) Decomposition of periodic signal into cosine waves Fourier Synthesis (Inverse Transform) Summation of cosine waves into multi-frequency waveform Fourier Transforms in 1D, 2D, 3D, ND Image Analysis Image (real-valued) Transform (complex-valued, amplitude plot) Fourier Filters Low-pass High-pass Band-pass Band-stop Convolution Theorem Deconvolute by Division in Fourier Space All Fourier Filters can be expressed as real-space Convolution Kernels Lens does Foureir transforms Diffraction Microscopy

Further Reading Wikipedia Mathworld The Fourier Transform and its Applications. Ronald Bracewell

Lens Performs Fourier Transform

Gibbs Ringing 5 waves 25 waves 125 waves