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

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Presentation on theme: "Introduction to Deconvolution Image Processing Introduction to Light and Electron Microscopy Neu259 Spring 2006 Spring 2006 James Bouwer UCSD."— Presentation transcript:

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

2 Outline Convolution:2D: Airy Disk (Light Point Spread Function) Airy Disk (Light Point Spread Function) Convolution with an Image Convolution with an Image3D: 3D Point Spread Function 3D Point Spread Function Convolution with a Volume Convolution with a VolumeDeconvolution: Frequency space decomposition Frequency space decomposition Fourier Transforms Fourier Transforms Application of the Deconvolution Theorem Application of the Deconvolution Theorem Some Examples:

3 The Microscope Optical Train is Complex Every lens element alters Every lens element alters the image in some way the image in some way We call this the image We call this the image transfer function: h(x,y,z) transfer function: h(x,y,z) The image transfer function The image transfer function of the system is a of the system is a convolution of all of the convolution of all of the image transfer functions of image transfer functions of all the lens elements and all the lens elements and apertures in the optical apertures in the optical train 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 Impulse Response Function [ Point Spread Function (psf)] A point in the object space A point convolved with the transfer Microscope function

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 Line Transfer Function

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

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

9 where Theoretical Model of the Impulse Transfer Function, h(x,y) (PSF) J 1 =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 Any image can be decomposed into a series of decomposed into a series of sines and cosines added sines and cosines added together to give the image together to give the image 0.25µm myelin AmplitudesPhase Fourier Transform

12 Low frequency High frequency Fourier Transform of the Myelin Image

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

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

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

16 Deconvolution using the Convolution Theorem The image is a convolution of the impulse function h(x,y) (PSF) and the object (the sample): 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)

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

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

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

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

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22 3D Out-of-focus Point Spread Function 3D Impulse Response x z y 2-D PSFs vs. z-height

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

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


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