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Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Chapter 1Examples of Ill-Posed ProblemsIll-Posed Problems in Image and Signal ProcessingWS 2014/2015
Michael MoellerOptimization and Data Analysis
Department of MathematicsTU Munchen
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What are ill-posed problems?
Definition (Well-posed problems (Hadamard))
A problem is well-posed if the following three properties hold.1 Existence: For all suitable data, a solution exists.2 Uniqueness: For all suitable data, the solution is unique.3 Stability: The solution depends continuously on the data.
Definition (Ill-posed problems)
A problem that violates any of the three properties ofwell-posedness is called an ill-posed problem.
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Differentiation
Data from: Microsoft Research GeoLife GPS Trajectories
Time ’12:44:12’ ’12:44:13’ ’12:44:15’Latitude 39.974408918 39.974397078 39.973982524Longitude 116.30352210 116.30352693 116.30362184
How fast did this person go?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Differentiation
0 100 200 300 400 500 600 700 800 900 10000
5
10
15
Measurements
met
ers
per
seco
nd
Usain Bolt World Record
New world record? Top speed of 161.78 km/h?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Differentiation is ill-posed
Computation:
The solution does not depend continuously on the data.
Ill-posedness of differentiation
For f , f δ ∈ C1([0,1]), although the error in the data
‖f − f δ‖ ≤ δ
is arbitrary small, the error between the derivatives
‖∂x f − ∂x f δ‖
can be arbitrary large!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Why is that interesting in practice?
In practice, measurements are NEVER exact!
0 50 100 150 200 2500
0.5
1
1.5
2Measurements when dropping a ball
Hei
ght z
(t)
Measurements50 100 150
−2
−1.5
−1
−0.5
0
0.5
1x 10
−3 Acceleration during free fall
Acc
eler
atio
n ∂ ttz(
t)
Measurements
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What can we do?
We need more information or additional assumptions!
Goal: Continuous dependence of ‖∂x f δ − ∂x f‖X on ‖f δ − f‖Y .
Option 1: Bound the noise in a stronger norm, e.g.
‖nδ‖2Y =
∫ 1
0|nδ(x)|2 dx +
∫ 1
0|∂xnδ(x)|2 dx ,
(a norm in the Sobolev space H1([0,1])).
→ Unrealistic in practice!→ In our example this would assume the prior knowledge thatthe frequency k of the noise is bounded!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What can we do?
We need more information or additional assumptions!
Goal: Continuous dependence of ‖∂x f δ − ∂x f‖X on ‖f δ − f‖Y .
Option 1: Bound the noise in a stronger norm, e.g.
‖nδ‖2Y =
∫ 1
0|nδ(x)|2 dx +
∫ 1
0|∂xnδ(x)|2 dx ,
(a norm in the Sobolev space H1([0,1])).
→ Unrealistic in practice!→ In our example this would assume the prior knowledge thatthe frequency k of the noise is bounded!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What can we do?
We need more information or additional assumptions!
Goal: Continuous dependence of ‖∂x f δ − ∂x f‖X on ‖f δ − f‖Y .
Option 1: Bound the noise in a stronger norm, e.g.
‖nδ‖2Y =
∫ 1
0|nδ(x)|2 dx +
∫ 1
0|∂xnδ(x)|2 dx ,
(a norm in the Sobolev space H1([0,1])).
→ Unrealistic in practice!→ In our example this would assume the prior knowledge thatthe frequency k of the noise is bounded!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What can we do?
We need more information or additional assumptions!
Goal: Continuous dependence of ‖∂x f δ − ∂x f‖X on ‖f δ − f‖Y .
Option 2: Assume additional regularity of the estimatedsolution fα and use regularization. Solve
−α∂xx fα(x) + fα(x) = f δ(x) (RD)
for fα.
→ Allows us to bound the error in the derivatives.
→ Computation on the board.
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What can we do?
We need more information or additional assumptions!
Goal: Continuous dependence of ‖∂x f δ − ∂x f‖X on ‖f δ − f‖Y .
Option 2: Assume additional regularity of the estimatedsolution fα and use regularization. Solve
−α∂xx fα(x) + fα(x) = f δ(x) (RD)
for fα.
→ Allows us to bound the error in the derivatives.
→ Computation on the board.
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
What can we do?
We need more information or additional assumptions!
Goal: Continuous dependence of ‖∂x f δ − ∂x f‖X on ‖f δ − f‖Y .
Option 2: Assume additional regularity of the estimatedsolution fα and use regularization. Solve
−α∂xx fα(x) + fα(x) = f δ(x) (RD)
for fα.
→ Allows us to bound the error in the derivatives.
→ Computation on the board.
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Error estimation for regularized derivatives
For fα determined by
−α∂xx fα(x) + fα(x) = f δ(x) (RD)
and twice continuously differentiable f , we can choose α suchthat
‖∂x fα − ∂x f‖2 ≤√
C√δ,
although‖f δ − f‖2 ≤ δ.
Observation
Even when regularization is used, the order of thereconstruction error is worse than the order of the error in thedata.
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Error estimation for regularized derivatives
For fα determined by
−α∂xx fα(x) + fα(x) = f δ(x), (RD)
let us make a (seemingly small) change and only assume thatf is one time continuously differentiable.
Computation on the board shows:
Observation
Without additional smoothness assumptions on the exactsolution, the convergence of the regularized solutions isarbitrarily slow!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Error estimation for regularized derivatives
For fα determined by
−α∂xx fα(x) + fα(x) = f δ(x), (RD)
let us make a (seemingly small) change and only assume thatf is one time continuously differentiable.
Computation on the board shows:
Observation
Without additional smoothness assumptions on the exactsolution, the convergence of the regularized solutions isarbitrarily slow!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Looking ahead...
What did we do by computing −α∂xx fα(x) + fα(x) = f δ(x)?
In this lecture we will learn that fα as computed above, solves
fα = arg minu‖u − f δ‖2
2 + α‖∂xu‖22.
→ Tikhonov regularization!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Looking ahead...
What did we do by computing −α∂xx fα(x) + fα(x) = f δ(x)?
In this lecture we will learn that fα as computed above, solves
fα = arg minu‖u − f δ‖2
2 + α‖∂xu‖22.
→ Tikhonov regularization!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Looking ahead...
What did we do by computing −α∂xx fα(x) + fα(x) = f δ(x)?
In this lecture we will learn that fα as computed above, solves
fα = arg minu‖u − f δ‖2
2 + α‖∂xu‖22.
→ Tikhonov regularization!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Discrete differentiation by finite differences
In practice: Evaluations fi of the function at grid points xi withxi+1 − xi = h. Typically, one computes
∂x f (xi ) ≈fi − fi−1
hleft sided differences
∂x f (xi ) ≈fi+1 − fi
hright sided differences
∂x f (xi ) ≈fi+1 − fi−1
2hcentral differences
We will see in the exercises:• The finite difference approximation of the derivative gets
better as h decreases.• The error due to taking finite differences of noisy data
increases as h decreases.→ The step size has to be chosen carefully!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Discrete differentiation by finite differences
In practice: Evaluations fi of the function at grid points xi withxi+1 − xi = h. Typically, one computes
∂x f (xi ) ≈fi − fi−1
hleft sided differences
∂x f (xi ) ≈fi+1 − fi
hright sided differences
∂x f (xi ) ≈fi+1 − fi−1
2hcentral differences
We will see in the exercises:• The finite difference approximation of the derivative gets
better as h decreases.• The error due to taking finite differences of noisy data
increases as h decreases.→ The step size has to be chosen carefully!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Discrete differentiation by finite differences
In practice: Evaluations fi of the function at grid points xi withxi+1 − xi = h. Typically, one computes
∂x f (xi ) ≈fi − fi−1
hleft sided differences
∂x f (xi ) ≈fi+1 − fi
hright sided differences
∂x f (xi ) ≈fi+1 − fi−1
2hcentral differences
We will see in the exercises:• The finite difference approximation of the derivative gets
better as h decreases.• The error due to taking finite differences of noisy data
increases as h decreases.→ The step size has to be chosen carefully!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Summary: Differentiation
What we have learned about the differentiation of a function:
1 Without regularization an arbitrarily small error in the datacan lead to an arbitrarily large error in the derivative!
2 Even with regularization the order with which the error inthe derivative decays is worse than the order of the error inthe data!
3 We need additional smoothness assumption on the truefunction f to even derive an estimate on the error!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Forward diffusion
Heat Equation
The heat equation with zero Dirichlet boundary conditions isgiven by the following partial differential equation (PDE):
∂tu(x , t) = ∂xxu(x , t) for x ∈]0, π[, t ∈ R+
u(0, t) = 0 ∀t ∈ R+
u(π, t) = 0 ∀t ∈ R+
u(x ,0) = f (x) for x ∈]0, π[
Naive idea:
ui,j+1 − ui,j
∆t≈
ui+1,j − 2ui,j + ui−1,j
∆x
Does it work both ways - forward and backward?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
The 1d heat equation
First homework:f (y) = sin(m y), ∀δ,T , ∃m : ‖u(·,T )− 0‖2 ≤ δ, but ‖f‖∞ = 1.
Conclusion from homework
The backward heat equation is ill-posed!
A computation on the board shows:
Solution of the heat equation
The solution of the 1d heat equation with zero Dirichletboundary conditions is given by
u(x , t) =
∫ π
0k(x , y , t)f (y) dy ,
k(x , y , t) =2π
∞∑n=1
e−n2t sin(nx) sin(ny).
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
The 1d heat equation
First homework:f (y) = sin(m y), ∀δ,T , ∃m : ‖u(·,T )− 0‖2 ≤ δ, but ‖f‖∞ = 1.
Conclusion from homework
The backward heat equation is ill-posed!
A computation on the board shows:
Solution of the heat equation
The solution of the 1d heat equation with zero Dirichletboundary conditions is given by
u(x , t) =
∫ π
0k(x , y , t)f (y) dy ,
k(x , y , t) =2π
∞∑n=1
e−n2t sin(nx) sin(ny).
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
Original image
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
Blurry image f = k ∗ u
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
Reconstructed image u = F−1(F(f )/F(k))
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
Blurry noisy image f = k ∗ u + n,⇒ F(f ) ≈ F(k) · F(u)
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
Reconstruction by F−1(F(f )/F(k))
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Some things about image processing
What is an image?
Continuous representation: Function u : Ω→ R (grayscale)or u : Ω→ R3 (color), where Ω ⊂ R2 (typically open andbounded).
Discrete representation: Matrix u ∈ Rn×m (grayscale) orthree matrices u1, u2, u3 ∈ Rn×m (color). The discretepoints/entries of the matrix are called pixels.
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Blurring
Continuous model for a blurred image:
f (x , y) =
∫R2
k(s − x , t − y)u(s, t) ds dt
with a convolution kernel k , e.g.
k(s, t) =1
2πσ2 exp(−s2 + t2
2σ2
)
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
The Fourier Theorem states that
f = k ∗ u ⇒ F(f ) = F(k)F(u).
Riemann-Lebesgue Lemma
Let k : R2 → R be absolutely integrable, i.e.∫ ∞−∞
∫ ∞−∞|k(x , y)| dx dy <∞.
Then |F(k)(µ, ν)| → 0 for ‖(µ, ν)‖ → ∞.
The reconstruction of
u = F−1(F(f )
F(k)
)becomes unstable!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring
The Fourier Theorem states that
f = k ∗ u ⇒ F(f ) = F(k)F(u).
Riemann-Lebesgue Lemma
Let k : R2 → R be absolutely integrable, i.e.∫ ∞−∞
∫ ∞−∞|k(x , y)| dx dy <∞.
Then |F(k)(µ, ν)| → 0 for ‖(µ, ν)‖ → ∞.
The reconstruction of
u = F−1(F(f )
F(k)
)becomes unstable!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
How can we discretize a blur?
Image: u ∈ Rn×m.
Blur kernel: k ∈ Rr×r , typically r << minn,m. Assume zerovalues outside.
For example
k =
0.0030 0.0133 0.0219 0.0133 0.00300.0133 0.0596 0.0983 0.0596 0.01330.0219 0.0983 0.1621 0.0983 0.02190.0133 0.0596 0.0983 0.0596 0.01330.0030 0.0133 0.0219 0.0133 0.0030
Let r be odd.
fi,j =r∑
h=1
r∑l=1
kh,r ui+h− r+12 ,j+l− r+1
2
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
How can we discretize a blur?
Image: u ∈ Rn×m.
Blur kernel: k ∈ Rr×r , typically r << minn,m. Assume zerovalues outside. For example
k =
0.0030 0.0133 0.0219 0.0133 0.00300.0133 0.0596 0.0983 0.0596 0.01330.0219 0.0983 0.1621 0.0983 0.02190.0133 0.0596 0.0983 0.0596 0.01330.0030 0.0133 0.0219 0.0133 0.0030
Let r be odd.
fi,j =r∑
h=1
r∑l=1
kh,r ui+h− r+12 ,j+l− r+1
2
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
How can we discretize a Gaussian blur?
Image: u ∈ Rn×m.
Gaussian blur kernel: Separable!
f (x , y) =1
2πσ2
∫Ω
exp(− (s − x)2 + (t − y)2
2σ2
)u(s, t) ds dt
=1√2πσ
∫exp
(− (t − y)2
2σ2
)v(x , t) dt
with
v(x , t) =1√2πσ
∫exp
(− (s − x)2
2σ2
)u(s, t) ds
We only have to do two 1d convolutions!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
How can we discretize a Gaussian blur?
Image: u ∈ Rn×m.
Gaussian blur kernel: Separable!
f (x , y) =1
2πσ2
∫Ω
exp(− (s − x)2 + (t − y)2
2σ2
)u(s, t) ds dt
=1√2πσ
∫exp
(− (t − y)2
2σ2
)v(x , t) dt
with
v(x , t) =1√2πσ
∫exp
(− (s − x)2
2σ2
)u(s, t) ds
We only have to do two 1d convolutions!
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizationsGaussian blur kernel: Separable!
k =(
0.1615 0.2180 0.2409 0.2180 0.1615)
fi,j =r∑
h=1
kh
r∑l=1
kl ui+h− r+12 ,j+l− r+1
2
Or in the pure 1d case
ci =r∑
h=1
khbi+h− r+12
can be written as
~c =
... ... ... ... ...0 k 0 ... 00 0 k ... 00 ... 0 k 0... ... ... ... ...
︸ ︷︷ ︸
A1
~b
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
~c =
... ... ... ... ...0 k 0 ... 00 0 k ... 00 ... 0 k 0... ... ... ... ...
︸ ︷︷ ︸
A1
~b
What happens at the boundary? What are b0,b−1,...?
Most common assumption for image blurring:bh = b1 for h ≤ 1, bh = bn for h ≥ n.
First rows of the matrix A1:k1 + k2 + k3 k4 k5 ... 0
k1 + k2 k3 k4 ... 0k1 k2 k3 ... 00 k 0 ... 0... ... ... ... ...
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
~c =
... ... ... ... ...0 k 0 ... 00 0 k ... 00 ... 0 k 0... ... ... ... ...
︸ ︷︷ ︸
A1
~b
What happens at the boundary? What are b0,b−1,...?
Most common assumption for image blurring:bh = b1 for h ≤ 1, bh = bn for h ≥ n.
First rows of the matrix A1:k1 + k2 + k3 k4 k5 ... 0
k1 + k2 k3 k4 ... 0k1 k2 k3 ... 00 k 0 ... 0... ... ... ... ...
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
~c =
... ... ... ... ...0 k 0 ... 00 0 k ... 00 ... 0 k 0... ... ... ... ...
︸ ︷︷ ︸
A1
~b
What happens at the boundary? What are b0,b−1,...?
Most common assumption for image blurring:bh = b1 for h ≤ 1, bh = bn for h ≥ n.
First rows of the matrix A1:k1 + k2 + k3 k4 k5 ... 0
k1 + k2 k3 k4 ... 0k1 k2 k3 ... 00 k 0 ... 0... ... ... ... ...
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
• We know how to write a 1d blur as c = A1b. For image u:
v = A1u ∈ Rn×m.
• Now we need to blur v in x-direction. Generate matrixA2 ∈ Rm×m with the kernel k appearing in the columns,and the boundaries treated similar to the y -direction case.
• Computef = A1uA2.
• Kronecker product:
A⊗ B =
A1,1B A1,2B ... A1,mBA2,1B A2,2B ... A2,mB... ... ... ...
Am,1B Am,2B ... Am,mB
∈ Rnm×nm
Vectorization
f = A1uA2 ⇔ vec(f ) = (AT2 ⊗ A1)vec(u).
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
• We know how to write a 1d blur as c = A1b. For image u:
v = A1u ∈ Rn×m.
• Now we need to blur v in x-direction. Generate matrixA2 ∈ Rm×m with the kernel k appearing in the columns,and the boundaries treated similar to the y -direction case.
• Computef = A1uA2.
• Kronecker product:
A⊗ B =
A1,1B A1,2B ... A1,mBA2,1B A2,2B ... A2,mB... ... ... ...
Am,1B Am,2B ... Am,mB
∈ Rnm×nm
Vectorization
f = A1uA2 ⇔ vec(f ) = (AT2 ⊗ A1)vec(u).
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring - discretizations
• We know how to write a 1d blur as c = A1b. For image u:
v = A1u ∈ Rn×m.
• Now we need to blur v in x-direction. Generate matrixA2 ∈ Rm×m with the kernel k appearing in the columns,and the boundaries treated similar to the y -direction case.
• Computef = A1uA2.
• Kronecker product:
A⊗ B =
A1,1B A1,2B ... A1,mBA2,1B A2,2B ... A2,mB... ... ... ...
Am,1B Am,2B ... Am,mB
∈ Rnm×nm
Vectorization
f = A1uA2 ⇔ vec(f ) = (AT2 ⊗ A1)vec(u).
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring as a linear equation
• We have discretized f = k ∗ u as
~f = A~u.
• Matlab: A is invertible!• Unique solution with A−1 → Problem not ill-posed?!?
• Give it a try! Use backslash.
How is this possible?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring as a linear equation
• We have discretized f = k ∗ u as
~f = A~u.
• Matlab: A is invertible!• Unique solution with A−1 → Problem not ill-posed?!?• Give it a try! Use backslash.
How is this possible?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring as a linear equation
• We have discretized f = k ∗ u as
~f = A~u.
• Matlab: A is invertible!• Unique solution with A−1 → Problem not ill-posed?!?• Give it a try! Use backslash.
How is this possible?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Deblurring as a linear equation
• We have discretized f = k ∗ u as
~f = A~u.
• Matlab: A is invertible!• Unique solution with A−1 → Problem not ill-posed?!?• Give it a try! Use backslash.
How is this possible?
Examples of Ill-PosedProblems
Michael Moeller
Ill-Posedness
Differentiation
Inverse Diffusion
Image Deblurring
updated 11.10.2014
Implementation
• When writing a convolution as a matrix vectormultiplication, always use sparse matrices!
• A full double matrix (AT2 ⊗ A1) for a 256× 256 image is
over 34GB!
• See “help spdiags“ in Matlab.
• See “help kron” in Matlab.
• See “help reshape” in Matlab.