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Introduction to Deconvolution Image Processing

Introduction to Light and Electron Microscopy. Neu259 Spring 2006. Introduction to Deconvolution Image Processing. James Bouwer UCSD. Outline. Convolution: 2D: Airy Disk (Light Point Spread Function) Convolution with an Image 3D: 3D Point Spread Function

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Introduction to Deconvolution Image Processing

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  1. Introduction to Light and Electron Microscopy Neu259 Spring 2006 Introduction to Deconvolution Image Processing James Bouwer UCSD

  2. Outline • Convolution: • 2D: • Airy Disk (Light Point Spread Function) • Convolution with an Image 3D: • 3D Point Spread Function • Convolution with a Volume Deconvolution: • Frequency space decomposition • Fourier Transforms • Application of the Deconvolution Theorem Some Examples:

  3. The Microscope Optical Train is Complex • Every lens element alters • the image in some way • We call this the image • transfer function: h(x,y,z) • The image transfer function • of the system is a • convolution of all of the • image transfer functions of • all the lens elements and • apertures in the optical • train

  4. The Definition of Convolution: Con-vo-lut-ed: adj.1.Having numerous overlapping coils or folds: a convoluted seashell.2.Intricate; complicated: convoluted legal language; convoluted reasoning. The American Heritage® Dictionary of the English Language, Fourth Edition

  5. Formation of an Airy Disk Pattern A point convolved with the transfer Microscope function A point in the object space Impulse Response Function [ Point Spread Function (psf)]

  6. Impulse Response Function Applied to a line A great advantage is afforded by the ability to express the response of the optical system to an arbitrary input in terms of the response to certain “elementary” functions into which the input has been decomposed Line Transfer Function

  7. How the psf (Impulse Response) Effects Resolution slide courtesy of Edgar Garduno

  8. Image Object Image Transfer Function h(x,y)

  9. Theoretical Model of the Impulse Transfer Function, h(x,y) (PSF) where J1=Bessel function of the first kind

  10. The Definition of Convolution: Con-vo-lut-ed: adj.1.Having numerous overlapping coils or folds: a convoluted seashell.2.Intricate; complicated: convoluted legal language; convoluted reasoning. The American Heritage® Dictionary of the English Language, Fourth Edition Convoluted Mess!

  11. Spatial Frequency Decomposition Fourier Transform • Any image can be • decomposed into a series of • sines and cosines added • together to give the image 0.25µm myelin Amplitudes Phase Fourier Transform

  12. Low frequency High frequency Fourier Transform of the Myelin Image

  13. Inverse 2-D Fourier Transform: Frequency Space to.. Real Space Mathematical Formulation of the Fourier Transform 2-D Fourier Transform: Real Space to..Frequency Space

  14. Fourier transform of myelin F -1 = Inverse Fourier Transform of the Fourier Transform Returns the original Image Very Powerful Tool !

  15. and Then: Remembering that the image intensity is a convolution of the impulse function h(x,y) and the object Iobject(x,y) The object intensity can be easily de-convolved from the Smear of the impulse function (PSF) The Wonderful, Great and Amazing Convolution Theorem If

  16. Therefore, the Fourier transform of the image is just the Fourier transform of object times the Fourier transform of the impulse function (PSF) in frequency space So, to obtain the object, we simply divide by the Fourier transform of the impulse function (PSF) Deconvolution using the Convolution Theorem The image is a convolution of the impulse function h(x,y) (PSF) and the object (the sample):

  17. A simple division in frequency space yields the Object intensity ! Transform back to real space Finally, we obtain the deconvolved object (sample) by applying an inverse Fourier transform

  18. Let’s Run though it… Step 1: Acquire the image and it’s Fourier transform Convoluted Image Fourier Transform

  19. Steps to Deconvolution… Step 2: Obtain the impulse function (PSF) and the Fourier transform of the impulse function (PSF) Point spread function Fourier transform

  20. Steps in the Deconvolution… Step 3: Divide and inverse Fourier transform F -1

  21. z x y 3D Out-of-focus Point Spread Function 3D Impulse Response 2-D PSFs vs. z-height

  22. A Real 3-D PSF From the RTS-2000 100nm Diameter Fluorescent Latex Bead Imaged with 50nm Steps in z

  23. … Plenty more to this story … Many commercial software packages are available in at the NCMIR

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