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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**

X-Rays Computer Tomography (CT or CAT) MRI (or NMR) PET / SPECT (Positron Emission Tomography, Single Photon Emission Computerized Tomography Ultrasound Computational What about microscopic scanners! What about electron microscopes? What about voxelization / discretization? What about synthetic methods? Provide Links to the numerous online tutorials!

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X-Rays

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**X-Rays - Physics cheap and relatively easy to use**

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**

Add 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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**Non-Intrusive Medical Diagnosis based on Computerized Tomography**

Computer tomography CT (From Jain’s Fig.10.1) An X-ray CT scanning system

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**Non-Intrusive Medical Diagnosis based on Transmission Tomography**

Source and Detector are rotating around human’s body (From Bovik’s Handbook Fig )

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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 transmission (X-ray), emission, magnetic resonance (MRI) Want to recover inner structure from the projections “Computerized Tomography” (CT)

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**Non-Intrusive Medical Diagnosis based on Emission Tomography**

Emission tomography: ET measure emitted gamma rays by the decay of isotopes from radioactive nuclei of certain chemical compounds affixed to body parts. 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) We have now a set of images g(s) which represent g(s,) (From Jain’s Fig.10.2) This is a transform from 2D to 2D spaces

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**Tomography and Reconstruction**

Lecture Overview Applications Background/history of tomography Radon Transform Fourier Slice Theorem Filtered Back Projection Algebraic techniques Measurement of Projection data Example of flame tomography

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**Applications & Types of Tomography**

Medical Applications Type of Tomography Full body scan X-ray Respiratory, digestive systems, brain scanning PET Positron Emission Tomography Respiratory, digestive systems. Radio-isotopes Mammography Ultrasound Whole Body Magnetic Resonance (MRI, NMR) MRI and PET showing lesions in the brain. PET scan on the brain showing Parkinson’s Disease

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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 Analysis Optical Tomography ECT on industrial pipe flows

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**CT or CAT - Principles Computerized (Axial) Tomography Radon again!**

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) Radon again! From 2D to 3D !

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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 (1917) showed how a reconstruction from projections was possible. Cormack (1963,1964) introduced Fourier transforms into the reconstruction algorithms. Hounsfield (1972) invented the X-ray Computer scanner for medical work, (which Cormack and Hounsfield shared a Nobel prize). 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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Backpropagation Principles

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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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**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 Initial Guess Reconstructed model Back- Projection Projection Actual Data Slices 26

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**CT - Reconstruction: FBP Filtered Back Propagation**

METHOD 2 : Filtered Back Projection common method uses Radon transform and Fourier Slice Theorem y F(u,v) f(x,y) Gf(r) x s u gf(s) f Spatial Domain Frequency Domain

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**COMPARISON : CT - FBP vs. ART**

Algebraic Reconstruction Technique Still slow better quality for fewer projections better quality for non-uniform project. “guided” reconstruct. (initial guess!) Filtered Back Projection Computationally cheap Clinically usually 500 projections per slice problematic for noisy projections 28

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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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**attenuation coefficient is used in CT_number of various tissues**

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**

Radon and its inverse easy to use You can do your own projects with CT reconstruction 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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The object An aside The sinogram AN ASIDE

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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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**Matlab example: filtering**

High frequency removed Low frequency removed

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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**

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

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**Matlab Implementation of Radon Transform**

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big noise No noise

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**The Lung Cancer and the reconstruction**

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**The Lung and The CTs 52 [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 [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. [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. 52

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**We want to reconstruct shape of the lungs**

Starting Point Border Detection At the moment two approaches are available. Left the algorithm developed at Pisa Right the algorithm developed at Lecce 53

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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. 54

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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**

We review the principle Discuss various geometries Show the use of filtering

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**Line Integrals and Projections**

The function P = Radon transform object function f(x,y). The function is known as the Radon transform of the function f(x,y).

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**Various types of beams can be used**

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

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**Line Integrals and Projections**

A projection is formed by combining a set of line integrals. Here the simplest projection, a collection of parallel ray integrals i.e constant θ, is shown. 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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Fourier Slice Theorem

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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 For simplicity θ=0 which leads to v=0 Define the projection at angle θ = Pθ(t) Define its transform by 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.

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**The Fourier Slice Theorem**

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

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**The Fourier Slice Theorem**

The Fourier Slice theorem relates the Fourier transform of the object along a radial line. Collection of projections of an object at a number of angles t v u Fourier transform v u θ 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. Space Domain Frequency Domain

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**Backprojection of Radon Transform**

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**Backprojection of Radon Transform**

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**Filtered backpropagation creates crisp edges**

Ideal cylinder Blurred edges Crisp edges Filtered backpropagation creates crisp edges

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**Computerized Tomography Equipment**

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**CT - 2D vs. 3D Linear advancement (slice by slice) helical movement**

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 Positron Emission Tomography Single Photon Emission Computerized Tomography 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 reconstruction 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

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