The Mathematics of X-ray Tomography Tatiana A. Bubba Department of Mathematics and Statistics, University of Helsinki tatiana.bubba@helsinki.fi Summer School on Very Finnish Inverse Problems Helsinki, June 3-7, 2019 Finnish Centre of Excellence in Inverse Modelling and Imaging 2018-2025 2018-2025
What is Computed Tomography? CT is a non-invasive device that provides information about the inside of an object by taking measurements from the outside (indirect information). Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
What is Computed Tomography? CT is a non-invasive device that provides information about the inside of an object by taking measurements from the outside (indirect information). Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
What is Computed Tomography? CT is a non-invasive device that provides information about the inside of an object by taking measurements from the outside (indirect information). At the core: measurements are taken exploiting X-ray absorption of tissues; X-ray source and detector are rotated around the object to take measurements at different location. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Toy example: a line inside homogeneous matter I 1 = I 0 e − µs X-ray source I 0 s I 0 : initial intensity of the X-ray s : length of the path of the X-ray inside the body µ > 0 : X-ray attenuation coefficient Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Toy example: two homogeneous blocks I 1 = I 0 e − µ 1 s 1 I 0 X-ray source s 1 I 0 I 1 I 2 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Toy example: two homogeneous blocks I 1 = I 0 e − µ 1 s 1 I 2 = I 1 e − µ 2 s 2 I 0 X-ray source s 1 s 2 I 0 I 1 I 2 I 2 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Beer-Lambert Law Homogeneous material : I 1 I 0 I 1 = I 0 e − µs µ s Non-homogeneous material : I 1 I 0 I 1 = I 0 e − � ℓ µ ( x ) dx X-ray ℓ µ ( x ) Rearranging we get a so-called line integral : � I 1 � � � ℓ µ ( x ) dx I 1 = I 0 e − − log = µ ( x ) dx ⇐ ⇒ I 0 ℓ Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Beer-Lambert Law and Radon Transform The Beer-Lambert law connects the initial and final intensities of an X-ray: � I 1 � � � ℓ f ( x ) dx I 1 = I 0 e − − log = f ( x ) dx ⇐ ⇒ I 0 ℓ and it is connected to the Radon transform � R ( f ) = f ( x ) dx ℓ through the identifications: � I 1 � f ( x ) = µ ( x ) and R ( f ) = − log . I 0 Usually ℓ = ℓ ( θ,τ ) where ( θ, τ ) ∈ [0 , 2 π ) × R . Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Beer-Lambert Law and Radon Transform In general, during a tomographic scan: I 0 is known from calibration and I 1 from measurement. I 1 is measured along many lines ℓ ( θ,τ ) to get many line integral values through the object from which to determine f ( x ) . The intensity I 1 is called the transmission , while the corresponding − log( I 1 /I 0 ) is called absorption or projection . A collection of projections is called sinogram . Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
In Practice: Transmission vs Absorption photon count 1000 • 1000 ✲ X-ray source Detector A digital X-ray detector counts how many photons arrive at each pixel. digital X-ray detector counts how many photons arrive at each pixel. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
In Practice: Transmission vs Absorption photon count 1000 1000 • ✲ 1000 500 • ✲ 1000 250 • ✲ Adding material between the source and detector reveals the exponential X-ray attenuation law. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
In Practice: Transmission vs Absorption photon count log 1000 1000 6.9 • ✲ 1000 500 6.2 • ✲ 1000 250 5.5 • ✲ We take logarithm of the photon counts to compensate for the exponential attenuation law. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
In Practice: Transmission vs Absorption photon line count log integral 1000 1000 6.9 0.0 • ✲ 1000 500 6.2 0.7 • ✲ 1000 250 5.5 1.4 • ✲ The final calibration step is to subtract the logarithms from the empty space value (here 6.9). Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
In Practice: Discretization To represent the scanned object and the projection data in a computer we need to discretize: Sinogram (measured data): these are already discrete (finite set of angles, finite set of detectors) Object : this discretization can be freely chosen, for instance pixels or voxels. Forward model (Radon transform): there are various approaches to discretize (compute or approximate) the line integrals Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Tomography as a Linear System θ Object: target f ∈ R n Data: sinogram y ∈ R m � R 2 δ ( τ − � x, ω θ � ) f ( x ) dx = y ( θ, τ ) = ( R f )( θ, τ ) � �� � K ( x,θ,τ ) = K f y where K ∈ R m × n is the system matrix (discretization of the forward model). Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Briefly About Noise in CT The intensity I 1 in the Beer-Lambert law can be understood as the number of transmitted photons in a interval of time. This can be modelled as a Poisson random variable, namely the probability that k photons are transmitted is: P ( N = k ) = λ k k ! e − λ where λ is the expected value of the number of transmitted photons. In practice, for large number of photons a Poisson distribution can be closely approximated by a Gaussian distribution , with constant mean and varying stan- dard deviation (the standard deviation decays with increasing intensity). This means that in the discrete linear model of CT we have to consider also noise: K f = y + δ = y δ where δ ∈ R m is the noise on the data. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Direct and Inverse Problem Direct problem: K Object f Data y Direct problem: given object f , determine data y Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Direct and Inverse Problem Direct problem: K Object f Data y Inverse problem Direct problem: given object f , determine data y Inverse problem : given noisy data y δ , recover object f Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
How to Solve the Inverse Problem? Solving the inverse problems means reconstructing the object from the measured data: object data K ? Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
How to Solve the Inverse Problem? Solving the inverse problems means reconstructing the object from the measured data: object data K K − 1 ? Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
How to Solve the Inverse Problem? Solving the inverse problems means reconstructing the object from the measured data: object data K K − 1 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Recall the Singular Value Decomposition (SVD) If K ∈ R m × n , there exist unitary matrices U ∈ R m × m and V ∈ R n × n such that σ 1 0 . . . 0 0 σ 2 . . . 0 K = U Σ V T = U V T . . . ... . . . . . . 0 0 . . . σ min( m,n ) where σ 1 ≥ σ 2 ≥ . . . ≥ σ min( m,n ) ≥ 0 are the so-called singular values of K and if we write U = ( u 1 , . . . , u m ) and V = ( v 1 , . . . , v n ) the u i and v i are, respectively, the left and right singular vectors associated with σ i , for i = 1 , . . . , min( m, n ) . The SVD can be written also as a sum of matrices of rank equal to 1: min( m,n ) � K = U Σ V T = σ i u i v T i i =1 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
SVD and Eigenvalue Decomposition The matrices K T K and KK T are symmetric. As a consequence, their eigenval- ues are real and their eigenvectors are real and orthonormal. A simple computa- tion yields: KK T u i = σ 2 K T K v i = σ 2 and i v i i u i or equivalently K T u i = σ i v i and K v i = σ i u i meaning that the square of the singular values of K are the non-zero eigenvalues of K T K and KK T and the left (right, resp.) singular vectors u i ( v i , resp.) are the eigenvectors of KK T ( K T K , resp.). In particular, the singular values of K are unique . The singular vector u i ( v i , i is a simple eigenvalue of KK T ( K T K , resp.). resp.) are unique only when σ 2 For multiple singular values, the corresponding singular vectors can be chosen as any orthonormal basis for the unique subspace that they span. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
SVD and the Fundamental Subspaces of a Matrix The SVD gives complete information about the four fundamental subspaces associated with K . Indeed, if k = rank( K ) we have: ker( K ) = span { v k , . . . , v n } , range( K ) = span { u 1 , . . . , u k } , � K T � � K T � range = span { v 1 , . . . , v k } , ker = span { u k , . . . , u m } , and the following well-known relations hold true: ker( K ) ⊥ = range � K T � , � K T � = range( K ) ⊥ . ker Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
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