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Data Acquisition, Representation and Reconstruction of medical images Application of Advanced Spectral Methods

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Acquisition Methods for medical images 1.X-Rays 2.Computer Tomography (CT or CAT) 3.MRI (or NMR) (Positron Emission Tomography, Single Photon Emission Computerized Tomography 4.PET / SPECT (Positron Emission Tomography, Single Photon Emission Computerized Tomography 5.Ultrasound 6.Computational

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X-Rays

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X-Rays - Physics X-Rays are associated with inner shell electrons As the electrons decelerate in the target through interaction, they emit electromagnetic radiation in the form of x-rays. patient is located between an x-ray source and a film -> radiograph cheap and relatively easy to use potentially damaging to biological tissue

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X-Rays X-Rays similar to visible light, but higher energy!

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X-Rays - Visibility bones contain heavy atoms -> with many electrons, which act as an absorber of x-rays commonly used to image gross bone structure and lungs excellent for detecting foreign metal objects main disadvantage -> lack of anatomical structure all other tissue has very similar absorption coefficient for x-rays

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X-Rays - Images X-Rays can be used in computerized tomography

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Computerized (Axial) Tomography

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CT (CAT) scanners and relevant mathematics

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(From Jain’s Fig.10.1) –Computer tomography CT An X-ray CT scanning system Non-Intrusive Medical Diagnosis based on Computerized Tomography

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Non-Intrusive Medical Diagnosis based on Transmission Tomography (From Bovik’s Handbook Fig ) Source and Detector are rotating around human’s body

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Non-Intrusive Medical Diagnosis based on projections Observe a set of projections (integrations) along different angles of a cross-section –Each projection itself loses the resolution of inner structure –Types of measurements transmissiontransmission (X-ray), emissionemission, magnetic resonance (MRI) projectionsWant to recover inner structure from the projections CT –“Computerized Tomography” (CT)

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Non-Intrusive Medical Diagnosis based on Emission Tomography ET –Emission tomography: ET measure emitted gamma rays by the decay of isotopes from radioactive nuclei of certain chemical compounds affixed to body parts. –MRI –MRI : based on that protons possess a magnetic moment and spin. –In magnetic field => align to parallel or antiparallel. –Apply RF => align to antiparallel. Remove RF => absorbed energy is remitted and detected by Rfdetector. f(x,y) is 2D image as before

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Radon Transform Principles A linear transform f(x,y) g(s, ) –Line integral or “ray-sum” –Along a line inclined at angle from y-axis and s away from origin Fix to get a 1-D signal g (s) (From Jain’s Fig.10.2) We have now a set of images g (s) which represent g(s, ) This is a transform from 2D to 2D spaces

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Tomography and Reconstruction Lecture Overview 1.Applications 2.Background/history of tomography 3.Radon Transform 4.Fourier Slice Theorem 5.Filtered Back Projection 6.Algebraic techniques Measurement of Projection data Example of flame tomography

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Applications & Types of Tomography Medical ApplicationsType of Tomography Full body scanX-ray Respiratory, digestive systems, brain scanning PET Positron Emission Tomography Respiratory, digestive systems. Radio-isotopes MammographyUltrasound Whole BodyMagnetic Resonance (MRI, NMR) Parkinson’s Disease PET scan on the brain showing Parkinson’s Disease lesions in the brain MRI and PET showing lesions in the brain.

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Applications & Types of Tomography – non medical Non Medical Applications Type of Tomography Oil Pipe Flow Turbine Plumes Resistive/Capacitance Tomography Flame AnalysisOptical Tomography ECT on industrial pipe flows

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CT or CAT - Principles Computerized (Axial) Tomography introduced in 1972 by Hounsfield and Cormack natural progression from X-rays based on the principle that a three-dimensional object can be reconstructed from its two dimensional projections based on the Radon transform (a map from an n-dimensional space to an (n-1)-dimensional space) From 2D to 3D ! Radon again!

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CT or CAT - Methods measures the attenuation of X-rays from many different angles a computer reconstructs the organ under study in a series of cross sections or planes combine X-ray pictures from various angles to reconstruct 3D structures

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The History of CAT Johan Radon Johan Radon (1917) showed how a reconstruction from projections was possible. Cormack Cormack (1963,1964) introduced Fourier transforms into the reconstruction algorithms. Hounsfield Hounsfield (1972) invented the X-ray Computer scanner for medical work, (which Cormack and Hounsfield shared a Nobel prize). EMI Ltd EMI Ltd (1971) announced development of the EMI scanner which combined X-ray measurements and sophisticated algorithms solved by digital computers.

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BackpropagationPrinciples

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Backpropagation We know that objects are somewhere here in black stripes, but where?

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Example of Simple Backprojection Reconstruction Given are sums, we have to reconstruct values of pixels A, B, C and D

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ART Image Reconstruction: ART or Algebraic Reconstruction Technique

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CT - Reconstruction: ART or Algebraic Reconstruction Technique METHOD 1: Algebraic Reconstruction Technique –iterative technique –attributed to Gordon Reconstructed model Actual Data Slices Projection Back- Projection Initial Guess

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CT - Reconstruction: FBP Filtered Back Propagation METHOD 2 : Filtered Back ProjectionMETHOD 2 : Filtered Back Projection –common method and –uses Radon transform and Fourier Slice Theorem f(x,y) y x s g (s) G()G() u F(u,v) Spatial DomainFrequency Domain

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COMPARISON : CT - FBP vs. ART Computationally cheap Clinically usually 500 projections per slice problematic for noisy projections Still slow better quality for fewer projections better quality for non- uniform project. “guided” reconstruct. (initial guess!) FBP ART Algebraic Reconstruction Technique Filtered Back Projection

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Fourier Slice Theorem and FFT review Patient’s body is described by spatial distribution of attenuation coefficient

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Properties of attenuation coefficient Our transform: f(x,y) p(r, )

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1.attenuation coefficient is used in CT_number of various tissues 2.These numbers are represented in HU = Hounsfield Units CT_number uses attenuation coefficients

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RADON TRANSFORM Properties REMEMBER: f(x,y) p(r, )

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Radon Transform is available in Matlab 1.Radon and its inverse easy to use 2.You can do your own projects with CT reconstruction 3.Data are available on internet sinogram

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Inverse Radon Transform

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Matlab examples

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Sinogram versus Hough

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An aside AN ASIDE The sinogram The object

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Review and notation – Fourier Transform of Image f(x,y)

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Matlab example In Matlab there are implemented functions that use Fourier Slice Theorem

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Low frequency removed High frequency removed Matlab example: filtering

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Matlab example - convolution

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Remainder of main theorem of spectral imaging Matlab example – filtering by convolution in spectral domain

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Radon Transform and a Head Phantom

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Reconstructing with more and more rays

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Example of Image Radon Transform (From Matlab Image Processing Toolbox Documentation) [Y-axis] distance, [X-axis] angle

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Matlab Implementation of Radon Transform

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No noise big noise

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The Lung Cancer and the reconstruction

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The Lung and The CTs [LUNG] 1.Either of the pair of organs occupying the cavity of the thorax that effect the aeration of the blood. 2.Balloon-like structures in the chest that bring oxygen into the body and expel carbon dioxide from the body [LUNG CANCER] Lung Cancer happens when cells in the lung begin to grow out of control and can than invade nearby tissues or spread throughout the body; Large collections of this out of control tissues are called tumors. [TYPES] 1.Small Cell Lung Cancer (SCLC) - 20% of all lung cancers 2.Non Small Cell Lung Cancer (NSCLC) - 80% of all lung cancer [Risks] In the United States alone, it is estimated that 154,900 died from lung cancer in In comparison,is estimated that 126,800 people died from colon, breast and prostate cancer combined, in 2002.

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Starting Point -At the moment two approaches are available. -Left the algorithm developed at Pisa -Right the algorithm developed at Lecce Border Detection We want to reconstruct shape of the lungs

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Image Interpolation - Theory [IDEA] In order to provide a richer environment we are thinking of using interpolation methods that will generate “artificial images” thus revealing hidden information. [RADON RECONSTRUCTION] Radon reconstruction is the technique in which the object is reconstructed from its projections. This reconstruction method is based on approximating the inverse Radon Transform. [RADON Transform] The 2-D Radon transform is the mathematical relationship which maps the spatial domain (x,y) to the Radon domain (p,phi). The Radon transform consists of taking a line integral along a line (ray) which passes through the object space. The radon transform is expressed mathematically as: [FILTERED BACK PROJECTION - INVERSE R.T.] It is an approximation of the Inverse Radon Transform. [The principle] Several x-ray images of a real-world volume are acquired [The Data] X-ray images (projections) of known orientation, given by data samples. [The Goal] Reconstruct a numeric representation of the volume from these samples. [The Mean] Obtain each voxel value from its pooled trace on the several projections. [Resampling] At this point one can obtain the “artificial slices” [Reslicing] An advantage of the volume reconstruction is the capability of obtaining new perpendicular slices on the original ones.

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Image Interpolation - Graphical Representation (I)

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Image Interpolation - Graphical Representation (II)

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Line Integrals and Projections 1.We review the principle 2.Discuss various geometries 3.Show the use of filtering

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Line Integrals and Projections The functionis known as the Radon transform of the function f(x,y). The function P = Radon transform object function f(x,y).

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Parallel Beams Fan Beams Parallel beams projections are taken by measuring a set of parallel rays for a number of different angles A fan beam projection is taken if the rays meet in one location Various types of beams can be used

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A projection is formed by combining a set of line integrals. parallel ray integrals Here the simplest projection, a collection of parallel ray integrals i.e constant θ, is shown. Line Integrals and Projections Notation for calculations in these projections

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Line Integrals and Projections A simple diagram showing the fan beam projection

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FourierSliceTheorem

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Fourier Slice Theorem The Fourier slice theorem is derived by taking the one-dimensional Fourier transform of a parallel projection and noting that it is equal to a slice of the two-dimensional Fourier transform of the original object. It follows that given the projection data, it should then be possible to estimate the object by simply performing the 2D inverse Fourier transform. Start by defining the 2D Fourier transform of the object function as Define the projection at angle θ = P θ (t) Define its transform by For simplicity θ=0 which leads to v=0 As the phase factor is no-longer dependent on y, the integral can be split.

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Fourier Slice Theorem As the phase factor is no-longer dependent on y, the integral can be split. The part in brackets is the equation for a projection along lines of constant x Substituting in Thus the following relationship between the vertical projection and the 2D transform of the object function:

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Fourier Slice Theorem Stanley and Kak Full details of derivation, not for now.Full details of derivation, not for now.

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t θ v u Space Domain Frequency Domain Fourier transform The Fourier Slice theorem relates the Fourier transform of the object along a radial line. The Fourier Slice Theorem

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t θ v u Space DomainFrequency Domain Fourier transform The Fourier Slice theorem relates the Fourier transform of the object along a radial line. The Fourier Slice Theorem v u Collection of projections of an object at a number of angles For the reconstruction to be made it is common to determine the values onto a square grid by linear interpolation from the radial points. But for high frequencies the points are further apart resulting in image degradation.

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Backprojection of Radon Transform

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Crisp edges Blurred edges Filtered backpropagation creates crisp edges Ideal cylinder

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Computerized Tomography Equipment

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CT - 2D vs. 3D Linear advancement (slice by slice) –typical method –tumor might fall between ‘cracks’ –takes long time helical movement –5-8 times faster –A whole set of trade-offs

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Evolution of CT technology

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CT or CAT - Advantages significantly more data is collected superior to single X-ray scans far easier to separate soft tissues other than bone from one another (e.g. liver, kidney) data exist in digital form -> can be analyzed quantitatively adds enormously to the diagnostic information used in many large hospitals and medical centers throughout the world

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CT or CAT - Disadvantages significantly more data is collected soft tissue X-ray absorption still relatively similar still a health risk *MRI is used for a detailed imaging of anatomy – no Xrays involved.

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Nuclear Magnetic Resonance (NMR) Magnetic Resonance Imaging (MRI)

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MRI Nuclear Magnetic Resonance (NMR) (or Magnetic Resonance Imaging - MRI) most detailed anatomical information high-energy radiation is not used, i.e. this is “safe method” based on the principle of nuclear resonance (medicine) uses resonance properties of protons

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Magnetic Resonance Imaging MRI - polarized all atoms (core) with an odd number of protons have a ‘spin’, which leads to a magnetic behavior Hydrogen (H) - very common in human body + very well magnetizing Stimulate to form a macroscopically measurable magnetic field

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MRI - Signal to Noise Ratio proton density pictures - measures H MRI is good for tissues, but not for bone signal recorded in Frequency domain!! Noise - the more protons per volume unit, the more accurate the measurements - better signal to noise ratio (SNR) through decreased resolution

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PET/SPECT Positron Emission Tomography Single Photon Emission Computerized Tomography

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PET/SPECT –recent technique involves the emission of particles of antimatter by compounds injected into the body being scanned follow the movements of the injected compound and its metabolism Filter Back Projectionreconstruction techniques similar to CT - Filter Back Projection & iterative schemes

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Ultrasound

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Ultrasound the use of high-frequency sound (ultrasonic) waves to produce images of structures within the human body above the range of sound audible to humans (typically above 1MHz) piezoelectric crystal creates sound waves aimed at a specific area of the body change in tissue density reflects waves echoes are recorded

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Ultrasound (2) Delay of reflected signal and amplitude determines the position of the tissue still images or a moving picture of the inside of the body there are no known examples of tissue damage from conventional ultrasound imaging commonly used to examine fetuses in utero in order to ascertain size, position, or abnormalities also for heart, liver, kidneys, gallbladder, breast, eye, and major blood vessels

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Ultrasound (3) by far least expensive very safe very noisy 1D, 2D, 3D scanners irregular sampling - reconstruction problems

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Typical Homework

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Sources of slides and information Badri Roysam Jian Huang, Machiraju, Torsten Moeller, Han-Wei Shen Kai Thomenius Badri Roysam

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