23 Convolution theorem Periodic sine-like functions are the Eigenfunctions of the convolution operation. This means, convolution changes the amplitude of a sine wave but nothing else. Hence, convolution is completely described by the transfer function or frequency response (Fourier transform of the PSF), which determines how much amplitude is transmitted for different frequencies
25 -filtered layergram reconstruction 1.Backproject measured projections, and integrate over 2.Fourier transform in 2D 3.Multiply with distance from the origin, | |, in the frequency space 4.Inverse Fourier transform
35 Reconstruction from fan projections k: counter of source positions l: counter of projection lines k+l = 7: parallel
36 Summary CT image reconstruction 1.In CT we measure projections, i.e., line integrals 2.The set of projection lines from all directions is called Radon transform or sinogram 3.Backprojection leads back into image space but introduces severe 1/r blurring
37 Summary CT image reconstruction, cont’d 4.Image can be de-blurred with deconvolution techniques 5.Deconvolution can be done in projection space using the central slice theorem 6.A common filter function is the Ram-Lak filter 7.Filters have negative components (“eraser”) to remove blurring
Image Reconstruction and Inverse Treatment Planning – Sharpening the Edge – Thomas Bortfeld I.CT Image Reconstruction II.Inverse Treatment Planning
44 Image Reconstruction (Filtered Backprojection) Conformal Radiotherapy (Filtered Projection) x-ray Projection (CT-Scanner) x-ray Projection (CT-Scanner) IMRT with Filtered Projections IMRT with Filtered Projections Projection (Computer) Projection (Computer) 1D Filtering of the Projections 1D Filtering of the Projections Backprojection 1D Filtering of the Projections 1D Filtering of the Projections Density Distribution of the Tissue Set of Prescribed 2D Dose Distributions Dose DistributionSet of 2D Slice Images
G. Birkhoff: On drawings composed of uniform straight lines Journ. de Math., tome XIX, - Fasc. 3, 1940.
46 After filtering: x Intensity Negative Intensities -
Consider the inverse problem as an optimization problem. Define the objectives of the treatment and let the computer determine the parameters giving optimal results. The inverse problem has no solution!
Objective Function F(x) = i (d i - p i ) 2, d i = f(x 1,.., x n ) F(x) = NTCP (1-TCP) Constraints d i 0 DVH constraints NTCP < 5% Parameters x = (x 1,..., x n ) (e.g., intensity values) Mathematical optimization: Minimization of objective functions actual doseprescribed dose Optimization basics
“Decision variables” Intensity profiles Beam weights, segment weights Beam angles (gantry angle, table angle) Number of beams Energy (especially in charged particle therapy) Type of radiation (photons, electrons,...)
50 The “standard model” of inverse planning : dose contribution of pencil beam j to voxel i all the physics is here beam intensities dose values minimize
small weight (w) Volume DoseD max large weight (w) DVH Volume D max Dose DVH Dose-volume histogram for OAR
52 Critical structure (organ at risk) costlet weight importance “penalty” dose at voxel i in OAR k tolerance dose
Volume Dose DVH D max D min Dose-volume histogram for the target
54 The “standard model” of inverse planning Minimize weights, penalties, importance factors objectives, costlets, indicators
55 The “standard model” of inverse planning High-dimensional problem: Ray intensities b j : 10,000 Dose voxels d i : 500,000 D ij matrix: 10,000 x 500,000 entries 20 GByte
56 x1x1 x0x0 x3x3 x F(x) local min. global min. 1D: Optimization with gradient descent x2x2
57 Optimization algorithms Projecting back and forth between dose distribution and intensity maps
84 Summary inverse treatment planning 1.Intensity-modulated radiation therapy (IMRT) uses non-uniform beam intensities from various (5-9) beam directions 2.“Inverse planning” is the calculation of intensities that will give the desired spatial dose distribution 3.CT reconstruction techniques cannot be (directly) applied here because we cannot deliver negative intensities 4.Today “inverse planning” is usually defined as an optimization problem, which is “solved” with gradient techniques
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