The Mathematics of X-ray Tomography
Tatiana A. BubbaDepartment of Mathematics and Statistics, University of Helsinki
Summer School on Very Finnish Inverse ProblemsHelsinki, June 3-7, 2019
Finnish Centre of Excellence in
Inverse Modelling and Imaging 2018-20252018-2025
TSVD as Spectral Filtering
We can regard the TSVD also as the result of a filtering operation, namely:
fTSVD =
r∑i=1
uTi yδ
σivi =
min(m,n)∑i=1
φTSVDi
uTi yδ
σivi
where r is the truncation parameter and
φTSVDi =
{1 i = 1, . . . , r
0 elsewhere
are the filter factors associated with the method.
These are called spectral filtering methods because the SVD basis can be re-garded as a spectral basis, since the vectors ui and vi are the eigenvectors ofKTK and KKT .
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Tikhonov Method
Let’s now consider the following filter factors:
φTIKHi =
σ2i
σ2i +α
2 i = 1, . . . ,min(m,n)
0 elsewhere
which yield the reconstruction method:
fTIKH =
min(m,n)∑i=1
φTIKHi
uTi yδ
σivi =
min(m,n)∑i=1
σi (uTi yδ)
σ2i + α2
vi.
This choice of the filters result in a regularization technique called Tikhonovmethod and α > 0 is the so-called regularization parameter.
The parameter α acts in the same way as the parameter r in the TSVDmethod: it controls which SVD components we want to damp or filter.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Tikhonov Regularization
Similarly to SVD being the solution of the least squares problem, also Tikhonovregularization can be understood as the solution of a minimization problem:
fTIKH = argminf
{∥∥Kf − yδ∥∥22
+ α∥∥f∥∥2
2
}.
This problem is motivated by the fact that we clearly want∥∥Kf − yδ
∥∥22
to besmall, but we also wish to avoid that it becomes zero. Indeed, by taking theMoore-Pensore solution f† we would have
‖f†‖22 =
k∑i=1
(uTi yδ)2
σ2i
which could become unrealistically large when the magnitude of the noise insome direction ui greatly exceeds the magnitude of the singular value σi.
The above minimization problem ensures that both the norm of the residualKfTIKH − yδ and the norm of the solution fTIKH are somewhat small and αbalances the trade-off between the two terms.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Normal Equation and Stacked Form for Tikhonov Regularization
The Tikhonov solution can be also formulated as a linear least squares problem:
fTIKH = argminf
∥∥∥∥∥[
K√α1
]−[yδ
0
] ∥∥∥∥∥2
2
.
This is called stacked form. If we denote by K =
[K√α1
]and yδ =
[yδ
0
]then
the least square solution of the stacked form satisfies the normal equations:
KT Kf = KT yδ.
It is easy to check that
KT K = KTK + α1 and KT yδ = KTyδ.
Hence we also have
fTIKH = (KTK + α1)−1KTyδ.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)
Truncated SVD RegularizationTikhonov Regularization
Original phantom f†: RE = 100%
fTSVD: RE = 35%fTIKH: RE = 32%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)
Truncated SVD Regularization
Tikhonov Regularization
Original phantom
f†: RE = 100%
fTSVD: RE = 35%
fTIKH: RE = 32%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)Truncated SVD Regularization
Tikhonov Regularization
Original phantom
f†: RE = 100%fTSVD: RE = 35%
fTIKH: RE = 32%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Where Tikhonov Solution Stands in The Geometry of Ill-ConditionedProblems
Object Space
Rn = span{v1, . . . ,vn}Data Space
Rm = span{u1, . . . ,um}
= f true y = Kf true =
yδ = y + δ
f† = K†yδ
fTSVD
fTIKH
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
About the Regularization Parameter
By looking at the minimization problem formulation of the Tikhonov solution
fTIKH = argminf
{∥∥Kf − yδ∥∥22
+ α∥∥f∥∥2
2
}it is clear that:
a large α results in strong regularity and possible over smoothing
a small α small yields a good fitting, with the risk of over fitting.
In general, choosing the regularization parameter for an ill-posed problem is nota trivial task and there are no rule of thumbs. Usually, it is a combination ofgood heuristics and prior knowledge of the noise in the observations.
Delving into this is out of the scope, but there are methods that can be foundin the literature (Morozov’s discrepancy principle, generalized cross validation,L-curve criterion), and more recent approaches tailored to specific problems.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom fTIKH: α = 103
fTIKH: α = 102fTIKH: α = 10fTIKH: α = 1fTIKH: α = 10−1fTIKH: α = 10−2fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom
fTIKH: α = 103
fTIKH: α = 102
fTIKH: α = 10fTIKH: α = 1fTIKH: α = 10−1fTIKH: α = 10−2fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom
fTIKH: α = 103fTIKH: α = 102
fTIKH: α = 10
fTIKH: α = 1fTIKH: α = 10−1fTIKH: α = 10−2fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom
fTIKH: α = 103fTIKH: α = 102fTIKH: α = 10
fTIKH: α = 1
fTIKH: α = 10−1fTIKH: α = 10−2fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom
fTIKH: α = 103fTIKH: α = 102fTIKH: α = 10fTIKH: α = 1
fTIKH: α = 10−1
fTIKH: α = 10−2fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom
fTIKH: α = 103fTIKH: α = 102fTIKH: α = 10fTIKH: α = 1fTIKH: α = 10−1
fTIKH: α = 10−2
fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Influence of the Choice of α in Tikhonov Regularization
Original phantom
fTIKH: α = 103fTIKH: α = 102fTIKH: α = 10fTIKH: α = 1fTIKH: α = 10−1fTIKH: α = 10−2
fTIKH: α = 10−3
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Generalized Tikhonov Regularization
Sometimes we have a priori information about the solution of the inverse prob-lem. This can be incorporated in the minimization formulation of the Tikhonovmethod. For instance:
f is close to a know f∗
fGTIKH = argminf
{∥∥Kf − yδ∥∥22
+ α∥∥f − f∗
∥∥22
}f is known to be smooth
fGTIKH = argminf
{∥∥Kf − yδ∥∥22
+ α∥∥Lf∥∥2
2
}f has similar smoothing properties as f∗
fGTIKH = argminf
{∥∥Kf − yδ∥∥22
+ α∥∥L(f − f∗)
∥∥22
}where L is a suitable operator.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Generalized Tikhonov Regularization
A common choice for generalized Tikhonov regularization is to take L as adiscretized differential operator. For example, using forward differences:
L =1
∆s
−1 1 0 0 0 . . . 00 −1 1 0 0 . . . 00 0 −1 1 0 . . . 0...
. . ....
.... . .
...0 . . . 0 −1 1 00 . . . 0 0 −1 11 . . . 0 0 0 −1
where ∆s is the length of the discretization interval.
This choice promotes smoothness in the reconstruction.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Variational Regularization
In general, a minimization problem of the form:
Γα(yδ) = argminf
{1
2
∥∥Kf − yδ∥∥22
+ α R(f)
}is called variational formulation:
The data fidelity (or data fitting) term∥∥Kf − yδ
∥∥22
keeps the estimationof the solution close to the data under the forward physical system.
The regularization parameter α > 0 controls the trade-off between a goodfit and the requirements from the regularization.
R(f) incorporates a priori information or assumptions on the unknown f .A non exhaustive list:
Tikhonov regularization: ‖f‖22Generalized Tikhonov regularization: ‖Lf‖22Compress sensing or sparse regularization: ‖f‖0 or ‖f‖1 or ‖Lf‖1Indicator functions of constraints sets: ιR+
(f)
A combination of the above
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
`p Norms for Rn
Let f ∈ Rn. The `p norms for 1 ≤ p <∞ are defined by
‖f‖p =
( n∑j=1
|f j |p
)1/p
.
Also important, but not a norm:
‖f‖0 = limp→0‖f‖pp =
∣∣{j : fj 6= 0}∣∣.
The `0 “norm” counts the number of non-zeros components in f : this is usedto measure sparsity.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Sparse Regularization
Finding the sparsest solution:
argminf
{1
2
∥∥Kf − yδ∥∥22
+ α‖Lf‖0}
is known as Compressed Sensing (CS). However, the problem above is NP-hard,since it requires a combinatorial search of exponential size for considering allpossible supports.
Under certain conditions on Lf and K, replacing `0 with `1 yields “similar”results. This relaxation leads to a convex problem:
argminf
{1
2‖Kf − yδ‖22 + α‖Lf‖1
}.
which is at the basis of optimization-based methods for CS.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
About the Convex Relaxation
The formulation
argminf
{1
2
∥∥Kf − yδ∥∥22
+ α‖Lf‖1}.
it is more easily solvable, but still nonsmooth. Also, it is convex, but not strictlyconvex. So why not using Tikhonov regularization?
x1
x2
x1
x2
|x1|2 + |x2|2 = const |x1|+ |x2| = const
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
About the Convex Relaxation
The formulation
argminf
{1
2
∥∥Kf − yδ∥∥22
+ α‖Lf‖1}.
it is more easily solvable, but still nonsmooth. Also, it is convex, but not strictlyconvex. So why not using Tikhonov regularization?
x1
x2
x1
x2
|x1|2 + |x2|2 = const |x1|+ |x2| = const
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Total Variation Regularization
If we take L = ∇ as the discrete differentiation matrix, the variational formula-tion
fTV = argminf
{1
2
∥∥Kf − yδ∥∥22
+ α ‖∇f‖1}
is called Total Variation.
Total Variation (TV) regularization promotes sparsity in the derivative, in otherwords favouring piece-wise constantness.
TV, first introduced to face denoising problems (in 1992, the so-called ROFmodel) became a popular approach in many imaging processing tasks (includingCT) due to its ability to preserve or even favour reconstructions with sharpedges.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Beyond Classical TV
Total Generalized Variation (TGVk)defines a whole family of priors depending on the order of the derivative kTGV2 is suitable for piecewise smooth targets
Total p-Variation (TpV)0 < p < 1 refers to the normdesigns a nonsmooth and nonconvex problem
Many many more: Higher Order TV, Directional TV, Anisotropic TV, . . .
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Wavelet-based Regularization
If we take L = W as the matrix associated to a certain wavelet transform, thevariational formulation
fWLET = argminf
{1
2
∥∥Kf − yδ∥∥22
+ α ‖Wf‖1}
promotes sparsity on the wavelet coefficients.
The idea behind wavelet-based regularization is that wavelet coefficients comewith different magnitudes and the smallest ones are associated with noise. The`1-norm suppresses the small coefficients in favor of the largest ones, which areassociated with edges and images dominant features.
Wavelets (widely used in image processing since 1990s) are a very common choicein CS approaches since they model images quite adequately.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
A Bit About Wavelets
Wavelets arose in 1980s to overcome some of the limitations of Fourier analysis.
Similarly to Fourier series, the idea is to “break” a signal into building blocks,but unlike Fourier series the building blocks are localized not only in the frequencydomain but also in the space domain.
Time-frequency plane for the Time-frequency plane for the
Fourier Transform. wavelet transform.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
A Bit About Wavelets
Different families of wavelets can be generated by considering different “parents”:
the scaling function φ ∈ L2(R2), a low-pass filter, provides a rougherversion of the signal itself;
the (mother) wavelet ψ ∈ L2(R2), a high-pass filter, describes the detailsin the signal.
A wavelet system is generated by applying to both parents two operators:
Isotropic dilation:
DMψ(x) = 2−j2ψ(2jx),
where j ∈ Z is the scaling parameter.
Translation:Tkψ(x) = ψ(x− k)
where k ∈ Z is the location parameter.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
A Bit About Wavelets
The elements of a wavelet system are given by:
ψjk(x) ={TkDjψ(x) = 2−
j2ψ(2jx− k) : (j, k) ∈ Z× Z
}and similarly for the scaling function.
The wavelets coefficients are the result of the wavelet transform:
W : f −→ Wf(j, k) = 〈f, ψj,k〉.
In practical applications these are computed using the language of filters, withconvolutions and downsampling and upsampling operations.
In particular in 2D, by considering tensor products, from the scaling andwavelet functions we get one scaling function but three wavelet functions(horizontal, vertical and diagonal):
Φ(x) = φ(x1)φ(x2),
and
Ψ1(x) = φ(x1)ψ(x2), Ψ2(x) = ψ(x1)φ(x2), Ψ3(x) = ψ(x1)ψ(x2).
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
An Example: Haar Wavelets
0 1
(x)
1
φ(x) =
{1 0 < x < 1
0 elsewhere
0 1/2 1
1
-1
(x)
ψ(x) =
{1 0 ≤ x < 1
2
0 12≤ x < 1
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
An Example: Haar Wavelet Transform of the Square Phantom
1-level Haar wavelet transform
2-level Haar wavelet transform3-level Haar wavelet transform
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
An Example: Haar Wavelet Transform of the Square Phantom
1-level Haar wavelet transform
2-level Haar wavelet transform
3-level Haar wavelet transform
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
An Example: Haar Wavelet Transform of the Square Phantom
1-level Haar wavelet transform2-level Haar wavelet transform
3-level Haar wavelet transform
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
An Example: Haar Wavelet Transform of a Walnut
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Constrained Regularization
In many cases, and CT is one of them, it is beneficial to include in the model anonnegativity constraint:
argminf
{1
2
∥∥Kf − yδ∥∥22
+ ιR+(f)
}or argmin
f>0
{1
2
∥∥Kf − yδ∥∥22
},
where the inequality is meant component-wise.
The nonnegative constraint can also be coupled with other regularizers:
Nonnegativity constrained Tikhonov regularization:
fTIKH+ = argmin
f>0
{1
2
∥∥Kf − yδ∥∥22
+ α ‖f‖22}
Nonnegativity constrained sparse regularization:
argminf>0
{1
2
∥∥Kf − yδ∥∥22
+ α ‖Lf‖1}
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
How to Solve `1-type Problems?
Approximating the absolute value function by
|t|β =√t2 + β.
Then the problem becomes smooth and we can use gradient-basedminimization algorithms. This is often done for TV regularization(smoothed TV ).
Using algorithms for nonsmooth objective functions (primal-dual,forward-backward, Bregman iteration, . . . ). In general, these requires thecomputation of the proximal operator and depending wether there is ornot an analytical closed form for it, the minimization problem can berather challenging.
fTV and fWLET are special cases for which the proximal operator is easy andfast to compute, because is given by the soft-thresholding operator.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
How to Solve `1-type Problems?
Approximating the absolute value function by
|t|β =√t2 + β.
Then the problem becomes smooth and we can use gradient-basedminimization algorithms. This is often done for TV regularization(smoothed TV ).
Using algorithms for nonsmooth objective functions (primal-dual,forward-backward, Bregman iteration, . . . ). In general, these requires thecomputation of the proximal operator and depending wether there is ornot an analytical closed form for it, the minimization problem can berather challenging.
fTV and fWLET are special cases for which the proximal operator is easy andfast to compute, because is given by the soft-thresholding operator.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
How to Solve `1-type Problems?
Approximating the absolute value function by
|t|β =√t2 + β.
Then the problem becomes smooth and we can use gradient-basedminimization algorithms. This is often done for TV regularization(smoothed TV ).
Using algorithms for nonsmooth objective functions (primal-dual,forward-backward, Bregman iteration, . . . ). In general, these requires thecomputation of the proximal operator and depending wether there is ornot an analytical closed form for it, the minimization problem can berather challenging.
fTV and fWLET are special cases for which the proximal operator is easy andfast to compute, because is given by the soft-thresholding operator.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Hard- and Soft-Thresholding
Sα(x) =
x+ α
2if x ≤ −α
2
0 if |x| < α2
x− α2
if x ≥ α2
Hα(x) =
x if x ≤ −α
2
0 if |x| < α2
x if x ≥ α2
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1Orig. Signal
Hard Thr. H
Soft Thr. S
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Iterative Soft-Thresholding Algorithm (ISTA)
For instance, when L = W is the matrix associated with an orthogonal wavelettransform (e.g., Haar wavelets), problems of the form:
argminf∈Rn
{1
2
∥∥Kf − yδ∥∥22
+ α ‖Wf‖1}, (1)
can be solved using an algorithm called Iterative Soft-Thresholding (ISTA) andthe approximate solution is given by:
f (i+1) = WTSαW(f (i) + KT (yδ −Kf (i)))
where Sα is the soft-thresholding operation.
There are many variants of ISTA to gain faster convergence (FISTA) or to extendit to non-orthogonal bases (or frames), or to include the non-negativity constraint(primal-dual fixed point, PDFP).
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Iterative Soft-Thresholding Algorithm (ISTA)
For instance, when L = W is the matrix associated with an orthogonal wavelettransform (e.g., Haar wavelets), problems of the form:
argminf∈Rn
{1
2
∥∥Kf − yδ∥∥22
+ α ‖Wf‖1}, (1)
can be solved using an algorithm called Iterative Soft-Thresholding (ISTA) andthe approximate solution is given by:
f (i+1) = WTSαW(f (i) + KT (yδ −Kf (i)))
where Sα is the soft-thresholding operation.
There are many variants of ISTA to gain faster convergence (FISTA) or to extendit to non-orthogonal bases (or frames), or to include the non-negativity constraint(primal-dual fixed point, PDFP).
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)
Truncated SVD RegularizationTikhonov RegularizationNonnegativity Constrained Tikhonov RegularizationNonnegativity Constrained Total Variation RegularizationNonnegativity Constrained Wavelet-based Regularization
Original phantom f†: RE = 100%
fTSVD: RE = 35%fTIKH: RE = 32%fTIKH+ : RE = 13%fWLET+ : RE = 26%fTV
+ : RE = 3%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)
Truncated SVD Regularization
Tikhonov RegularizationNonnegativity Constrained Tikhonov RegularizationNonnegativity Constrained Total Variation RegularizationNonnegativity Constrained Wavelet-based Regularization
Original phantom
f†: RE = 100%
fTSVD: RE = 35%
fTIKH: RE = 32%fTIKH+ : RE = 13%fWLET+ : RE = 26%fTV
+ : RE = 3%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)Truncated SVD Regularization
Tikhonov Regularization
Nonnegativity Constrained Tikhonov RegularizationNonnegativity Constrained Total Variation RegularizationNonnegativity Constrained Wavelet-based Regularization
Original phantom
f†: RE = 100%fTSVD: RE = 35%
fTIKH: RE = 32%
fTIKH+ : RE = 13%fWLET+ : RE = 26%fTV
+ : RE = 3%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)Truncated SVD RegularizationTikhonov Regularization
Nonnegativity Constrained Tikhonov Regularization
Nonnegativity Constrained Total Variation RegularizationNonnegativity Constrained Wavelet-based Regularization
Original phantom
f†: RE = 100%fTSVD: RE = 35%fTIKH: RE = 32%
fTIKH+ : RE = 13%
fWLET+ : RE = 26%fTV
+ : RE = 3%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)Truncated SVD RegularizationTikhonov RegularizationNonnegativity Constrained Tikhonov RegularizationNonnegativity Constrained Total Variation Regularization
Nonnegativity Constrained Wavelet-based Regularization
Original phantom
f†: RE = 100%fTSVD: RE = 35%fTIKH: RE = 32%fTIKH+ : RE = 13%
fWLET+ : RE = 26%
fTV+ : RE = 3%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Naive Reconstruction (Moore-Penrose Pseudoinverse)Truncated SVD RegularizationTikhonov RegularizationNonnegativity Constrained Tikhonov Regularization
Nonnegativity Constrained Total Variation Regularization
Nonnegativity Constrained Wavelet-based Regularization
Original phantom
f†: RE = 100%fTSVD: RE = 35%fTIKH: RE = 32%fTIKH+ : RE = 13%fWLET+ : RE = 26%
fTV+ : RE = 3%
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Take-home message
Uniqueness does not save us.Even with an injective forward map, failure of Hadamard’s condition 3means that we need regularization for solving the inverse problem.
Non-uniqueness can be handled.Stable regularization strategy just needs enough a priori information forpicking out a unique object among those with same data.
Caveat. Regularization is not the only “cure” to ill-posedness: bayesian in-version and analytical strategies (designed ad hoc on the problem) are possibleapproaches as well.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Take-home message
Uniqueness does not save us.Even with an injective forward map, failure of Hadamard’s condition 3means that we need regularization for solving the inverse problem.
Non-uniqueness can be handled.Stable regularization strategy just needs enough a priori information forpicking out a unique object among those with same data.
Caveat. Regularization is not the only “cure” to ill-posedness: bayesian in-version and analytical strategies (designed ad hoc on the problem) are possibleapproaches as well.
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Some References
Inverse Problems:Engl, Hanke & Neubauer, Regularization of inverse problems, 1996Hansen, Rank-deficient and discrete ill-posed problems, 1998Bertero & Boccacci, Introduction to inverse problems in imaging, 1998Vogel, Computational methods for inverse problems, 2002Hansen, Discrete inverse problems, 2010Mueller & Siltanen, Linear and Nonlinear Inverse Problems with PracticalApplications, 2012
X-ray Tomography:Deans, The Radon Transform and Some of Its Applications, 1983Natterer, The mathematics of computerized tomography, 1986Kak & Slaney, Principles of computerized tomographic imaging, 1988Buzug: Computed Tomography: From Photon Statistics to ModernCone-Beam CT, 2008Natterer & Wubbeling, Mathematical Methods in Image Reconstruction,2001Epstein, Introduction to the mathematics of medical imaging, 2008
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Some References
Total variation and waveletsBurger & Osher, A guide to the TV zoo (chapter 1 in Burger & Osher,Level-Set and PDE-based Reconstruction Methods, 2013)Rudin, Osher & Fatemi, Nonlinear total variation based noise removalalgorithms, 1992Boggess & Narcowich, A first course in wavelets with Fourier analysis, 2009Mallat, A wavelet tour of signal processing, 1999
OptimizationBoyd & Vandenberghe, Convex optimization, 2004Nocedal & Wright, Numerical optimization, 2006Rockafellar, Convex optimization, 1996Daubechies, Defrise & De Mol, An iterative thresholding algorithm for linearinverse problems with a sparsity constraint, 2004
Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019