10/19/200611ENGG 167 MEDICAL IMAGINGLecture 10: Oct. 13Image Processing II Frequency Domain & Transform ProcessingReferences: Chapter 10, The Essential Physics of Medical Imaging, BushbergRadiation Detection and Measurement, Knoll, 2nded.Intermediate Physics for Medicine and Biology, Hobbie, 3rded.Principles of Computerized Tomographic Imaging, Kak and Slaney.(http://rvl4.ecn.purdue.edu/~malcolm/pct/pct-toc.html)2Preparation -Review Imaging Processing Toolbox Help Manual(on your computer)Download NIHImage (Mac) / ImageJ (Windows)Create a Macro which will analyze images and save a processed version of the images(http://rsbweb.nih.gov/ij/developer/macro/macros.html)Complete Image Processing Assignment10/19/200623The Fourier TheorumRef: Gonzalez et al, TextAll signals can be decomposed into pure sinusoidal signals+++=4The Fourier transformRef: Gonzalez et al, TextSpatial signalCorrespondingFrequency-domain signalAll signals can be decomposed into pure sinusoidal signalsThis theorum is especially appropriate for periodic signals, but can be used for discrete signals if enough frequencies are used to capture the relevant information.10/19/2006351) Basics – 1-dimensional sinusoidal representation of signalsRef: RizzoniWhere magnitude and phase of the coefficients are given by:or:61) Basics – complex numbersSinusoids are related to complex exponential expressions.A complex number is one which is of the form:z = a + ib, where I is the square root of -1, an imaginary numberRecall that the magnitude and phase of z can be calculated by:magnitude => I = [a2+ b2] ½phase => θ = tan-1(b/a)So that another way to express z is :z = I eiθNow, we can make use of a definition called Euler’s formula:eiθ= cos(θ) + i sin(θ) And e-iθ= cos(θ) - i sin(θ) Or written for the sinusoid signals: cos(θ) = [eiθ+ e-iθ]/2 and sin(θ) = [eiθ-e-iθ]/(2i)10/19/2006471) Basics – complex numbersSo that equivalent expressions for a time domain signal are :x(t) = ∑ ancos(ωnt) + bnsin (ωnt)where anand bnare the magnitudes of the signal at each frequency ωn, and the summation is carried out over all values of n.x(t) = ∑ Inexp(iωnt+φn)where Inand φnare the amplitude and phase at each frequency ωn.81) Basics – fourier transform of a signalX(ω)= ∑ x(tn) exp(-iωtn)where the signal X(ω) is now in the frequency domain (recall ω=2πf = 2π/T), whereas the original signal x(t) was a time resolved signal with N total data points. Summation is from n=1 to n=N.Alternatively a spatial data set can be transformed to spatial frequency data set by the same approach:F(k) = ∑ f(xn) exp(-i 2πkxn)where the signal F(k) is now in spatial frequency, k, and describes the exact discritized shape of the original signal f(xn).1N1N10/19/2006591) Basics – 1-D Fourier Transforms f(x) Æ F(kx)Ref: RizzoniInput analytic function Fourier Transformed function10Fourier Transform SummaryRef: Gonzalez et al, Text10/19/2006611Fourier Transform SummaryRef: Gonzalez et al, Text12Fourier Transform SummaryRef: Gonzalez et al, Text10/19/2006713Fourier Transform SummaryRef: Gonzalez et al, Text142D Fourier TransformsWhere is the information in (u,v) space?Where are the low frequencies?Where are the high frequencies?Ref: Gonzalez et al, Text10/19/2006815The Fourier space image (k-space)Lowest frequencyRef: Gonzalez et al, TextHighest positivekxfrequencyHighest negativekxfrequencyHighest positivekyfrequencyHighest negativekyfrequency16Fourier space filteringSpatial frequency changes in the Fourier domain are simply done with linear mathematics!Ref: Gonzalez et al, Text10/19/2006917Fourier space filteringSpatial frequency changes in the Fourier domain are simply done with linear mathematics!Edge enhancement example (what is the shape of this filter)Ref: Gonzalez et al, Text18Fourier space filteringRef: Gonzalez et al, Text10/19/20061019Try the online DEMOhttp://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertransform/20Fourier space filteringRef: Gonzalez et al, Text10/19/20061121Special filters – Butterworth & GaussianRef: Gonzalez et al, Text22Special filters – Butterworth & GaussianRef: Gonzalez et al, Text10/19/20061223Special filters – Butterworth & GaussianRef: Gonzalez et al, Text24NoiseRef: Gonzalez et al, Text10/19/20061325NoiseRef: Gonzalez et al, Text26Noise: simple linear filtering can helpRef: Gonzalez et al, Text10/19/20061427Noise: frequency domain filteringRef: Gonzalez et al, Text28Noise: frequency domain filteringRef: Gonzalez et al, Text10/19/20061529Online resources for more info about image processinghttp://micro.magnet.fsu.edu/primer/digitalimaging/imageprocessingintro.html30Geometric Linear TransformationsRef: Gonzalez et al, TextAffine transform10/19/20061631Geometric Linear TransformationsRef: Gonzalez et al, TextAffine transform32Image compression : changing the binary code used to reduce # of bits requiredRef: Gonzalez et al, TextUse fewer bits to encodeInformation that occurs a lot and then use more bits to encode information that occurs little in the imagep(r) is the probability of occurrenceI2is a more efficient coding of the bits than I110/19/20061733Lossy Image compression : transforms to fewer bits across the entire imageRef: Gonzalez et al, Text8 bits4 bits2 bits34Minimum Lossy Image compression : bit reduction at the local level, not globally!Ref: Gonzalez et al, Text10/19/20061835Minimum Lossy Image compression : bit reduction at the local level, not globally!Ref: Gonzalez et al, Text36Minimum Lossy Image compression : bit reduction at the local level, 8x8 blocks -JPEGRef: Gonzalez et al, Textpredictederrorcompressedimageclose