Automation

Mike MarshNational Center for Macromolecular Imaging

Baylor College of Medicine

Single-Particle Reconstructions and VisualizationEMAN Tutorial and Workshop

March 14, 2007

Fourier Transforms, Filtering and Convolution

Fourier Transform is an invertible operator

Image Fourier Transform

v2 will display image or its transform

FT

Fourier Transform is an invertible operator

Image

f(x,y)F(kx,ky)

x

y

0 Nx

Ny

{F(kx,ky)} = f(x,y)} {f(x,y)} = F(kx,ky)

Nx ⁄ 2

Ny ⁄ 2Fourier Transform

Continuous Fourier Transform

dsesFxf

dxexfsF

xsi

xsi

2

2

)()(

)()(

dsesFxf

dxexfsF

ixs

ixs

)(2

1)(

)()(

dsesFxf

dxexfsF

ixs

ixs

)(2

1)(

)(2

1)(

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(kx,ky,kz)• 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

a 2a 3a

Math Review - Attributes of cosine wave

Amplitude

Phase

f(x) = cos (x)

f(x) = 5 cos (x)

f(x) = 5 cos (x + 3.14)

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Math Review - Attributes of cosine wave

Amplitude

Phase

Frequency

f(x) = 5 cos (x)

f(x) = 5 cos (x + 3.14)

f(x) = 5 cos (3 x + 3.14)

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Math Review - Attributes of cosine wave

f(x) = cos (x)

Amplitude, Frequency, Phase

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f(x) = A cos (kx + )

Math Review - Complex numbers

• Real numbers:1-5.2

• Complex numbers4.2 + 3.7i9.4447 – 6.7i-5.2 (-5.2 + 0i)

1i

Math Review - Complex numbers

• Complex numbers4.2 + 3.7i9.4447 – 6.7i-5.2 (-5.2 + 0i)

• General FormZ = a + biRe(Z) = a

Im(Z) = b

• AmplitudeA = | Z | = √(a2 + b2)

• Phase = Z = tan-1(b/a)

Math Review – Complex Numbers

• Polar CoordinateZ = a + bi

• AmplitudeA = √(a2 + b2)

• 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: Z1 = 5 +2i– Frequency 2: Z2 = -3 + 4i– Frequency 3: Z3 = 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

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5 10 15(Hz)

5 10 15(Hz)

Amplitude OnlyAmplitude Only

Fourier Analysis in 1D. Audio signals

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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

A1·cos(2kx + 1)k=1

Ak·cos(2kx + k)k=1

1

k=2. Two cycles per period

A2·cos(2kx + 2)k=2

Ak·cos(2kx + k)k=1

2

k=3. Three cycles per period

A3·cos(2kx + 3)k=3

Ak·cos(2kx + k)k=1

3

Ak·cos(2kx + k)N

Fourier Synthesis. N Cycles

A3·cos(2kx + 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 iris128x128 pixels

Fourier Synthesis of a 2D Function

F(2,3)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

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Butterworth Low-pass Filter

Flat in the pass-band

Zero in the stop-band

No ringing

Gaussian Low-pass Filter

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Butterworth High-pass Filter

• Note the loss of solid densities

How the filter looks in 2D

unprocessed

lowpass

highpass

bandpass

Filtering with EMAN2

LowPass Filtersfiltered=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 Filtersfiltered=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 Filtersfiltered=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

x x

Convolution in 2D

x

x x

=

x

x x

=x x

x xx xx x

Microscope Point-Spread-Function is Convolution

Convolution Theorem

f g = {FG}

f = FG

G

Convolution in image domainIs equivalent to multiplication in fourier domain

Contrast Theory

observed imagef(x) for true particle

point-spread function

envelope functionnoise

obs(x) = f(x) psf(x) env(x) + n(x)

Incoherant average of transform

F2(s) CTF2(s) Env2(s) + N2(s)

Power spectrum

PS =

Lowpass Filtering by Convolution

f g = {FG}

• Camera shake• Crystallographic B-factor

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Review

Fourier Transform is invertible operator

Math ReviewPeriodic functions

Amplitude, Phase and Frequency

Complex numberAmplitude 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 AnalysisImage (real-valued)Transform (complex-valued,

amplitude plot)

Fourier FiltersLow-pass High-pass Band-passBand-stop

Convolution TheoremDeconvolute 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