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Image Reconstruction T-61.182, Biomedical Image Analysis Seminar Presentation 7.4.2005 Seppo Mattila & Mika Pollari

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Overview Reconstruction from projections (general) –projection geometry and radon transform Reconstruction methods –Backprojection, (Fourier slice theorem), Filtered Backprojection, and Algebraic Reconstruction Technique Reconstruction examples MRI reconstruction –examples

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Introduction Only photography (reflection) and planar x-ray (attenuation) measure spatial properties of the imaged object directly. Otherwise, measured parameters are some how related to spatial properties of imaged object. –US (amplitude and time), CT, SPECT and PET (integral projections of parallel rays), MRI (amplitude, frequency and phase of the NMR signal) etc... Construct the object (image) which creates the measured parameters.

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Examples CT Scanner GE ProSpeed Spiral CTCT scanner 1974

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Reconstruction From the Projections Projection is a line integral along the path: CT: measure the projections of passed photons, with different angles. MRI: measure projection of NMR signal with different magnetic gradients (projection based MRI not used anymore). Assumption: no notable diffraction.

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Projection Geometry and Radon Transformation Co-ordinate transformation: Radon transformation

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Radon Space Projections with different angles are stored in sinogram (raw data) Each horizontal line in sinogram is a projection with different angle projection

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Backprojection (BP) Simplest reconstruction method: Integrate all possible rays that pass through the same point. Cause smearing and blurring. Method has nowadays only historical importance.

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Backprojection Graphically

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Point Spread Function (PSF) of BP Reconstructed image is convolution of the real image and the PSF PSF cause blurring of the edges and straight lines - solution filtered back projection (later)

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Fourier Slice Theorem (FST) Fourier slice theorem relates three spaces: image, Fourier, and Radon (projection) together.

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Fourier Slice Theorem 1D FFT INTERPOLATE 2D IFFT

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Filtered Backprojection (FBP) Image reconstruction based on FST is computationally inefficient: computation time and memory consumption FBP utilize FST but is much more efficient Bandlimited ramp filter Weighted with Butterworth lowpass IFT

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FBP ramp vs. smooth filter

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FST and FBP Starting from FST we end up to FBP without any approximations or assumptions

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Remarks of FBP From the previous equations it’s clear that the image is backprojection of filtered signal ( ) and (|w|) is the ramp filter. FBP advantages: –Each projection may be filtered and backprojected while further projections are collected (on-line processing). –No need for 2D inverse Fourier transformation.

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Algebraic Reconstruction Techniques (ART) Each object entity (image pixel/voxel) has physical property (grey-level value) such as attenuation coefficient All pixel in the rays path contribute to sum an amount which equals pixel’s area along the path (weight) times pixels physical property (grey-level value) We end up set of simultaneous equations

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ART Model Each ray sum: Set of simultaneous equations

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Kaczmarz Method – Solution to ART Each equation spans a hyperplane in n- dimensional space. If unique solution exist it is in intersection of the hyperplanes Solution is found iteratively by solving each ray equation at the time

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Geometric Presentation Kaczmarz method ART

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Reconstruction Examples – Number of Projection

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Reconstruction Examples – Projection Angle BPFBP (Butter)ARTPSF

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Reconstruction Examples – BP, FBP, and ART PhantomBPFBP (Butter)ART Reconstruction of phantom with different reconstruction methods using 90 projections from interval 0-180 degree

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Display of CT Images Using Hounsfield Units Attenuation coefficients are normalized with respect of water Now mean and SD of of different tissues are known advanced (measured in HU)

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Estimation of Tissue Composition Apr 2001June 2001 Sept 2001 (A) (B) (C)

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MRI Reconstruction

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MRI Signal

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Transverse magnetisation: Longitudinal magnetisation: Functional Magnetic Resonance Imaging (Huettel, Song, McCarthy) Magnetization MRI signal equation:

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K-space (raw data) (spatial frequency domain) Measured MRI signal (k-space) Magnetization at each voxel (= image): Image space FT

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Reconstruction Examples

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Effects of Sampling the K-space

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Partial k-space coverage (data from AMI GE 3T) IFT

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Multichannel MRI surface coil (data from AMI GE 3T)

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Functional MRI (data from AMI GE 3T) anatomicalfunctional

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Summary I Reconstruct 2D CT image from 1D projections: Backprojection (BP) - Only historical importance Filtered backprojection (FBP) - Most widely used technique - Large number of projections over 0-180 ° required Algebraic Reconstruction Techniques (ART) - Better handling of sparse and non-uniform projections - Slow compared to FBP

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Summary II Reconstruct 2D MR image from measured current (1D) Spatial (x,y) info encoded in frequency and phase Collect data to spatial frequency domain (k-space) Reconstruction by inverse 2D FT + non-invasive + imaging sequences (T1, T2, fMRi etc.) lots of possibilities - image artifacts (distortions, ghosts, etc.) - more expensive

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