What are inverse problems?
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1 What are inverse problems? 6th Forum of Polish Mathematicians Warsaw University of Technology Joonas Ilmavirta University of Jyväskylä 8 September 2015 Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
2 Outline 1 Indirect measurements Need for measurements Medical imaging Earth Industrial testing 2 Direct and inverse problems Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
3 Need for measurements Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
4 Need for measurements It is important to be able to measure several things Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
5 Need for measurements It is important to be able to measure several things accurately, reliably and quickly, if possible. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
6 Need for measurements It is important to be able to measure several things accurately, reliably and quickly, if possible. Sometimes direct measurements are impossible. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
7 Need for measurements. It is important to be able to measure several things accurately, reliably and quickly, if possible. Sometimes direct measurements are impossible. Sometimes we want to see inside something that we don t want to break. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
8 Medical imaging Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
9 Medical imaging An important example is the human body: We want to know accurately what is inside but we do not want to tear it apart. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
10 Medical imaging. An important example is the human body: We want to know accurately what is inside but we do not want to tear it apart. Nowadays it is possible for a doctor to see (indirectly) what is inside a patient, and this has revolutionized medicine. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
11 Medical imaging. Ultrasound image (Wikimedia Commons: Wolfgang Moroder) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
12 Medical imaging. Computed tomography (CT) scanner (Wikimedia Commons) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
13 Medical imaging. CT image of a human head (with ameloblastoma) (Wikimedia Commons) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
14 Medical imaging. Magnetic resonance imaging (MRI) scanner (Wikimedia Commons: Jan Ainali) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
15 Medical imaging. MRI image of a human head (Wikimedia Commons) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
16 Earth Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
17 Earth We would like to know what is inside our planet but we cannot see directly. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
18 Earth We would like to know what is inside our planet but we cannot see directly. The deepest manmade holes on Earth are about 12 kilometers deep and the radius of Earth is about 6400 kilometers. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
19 Earth We would like to know what is inside our planet but we cannot see directly. The deepest manmade holes on Earth are about 12 kilometers deep and the radius of Earth is about 6400 kilometers. To measure what is deep inside our planet, one can study for example travel times of earthquakes Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
20 Earth. We would like to know what is inside our planet but we cannot see directly. The deepest manmade holes on Earth are about 12 kilometers deep and the radius of Earth is about 6400 kilometers. To measure what is deep inside our planet, one can study for example travel times of earthquakes or frequencies at which Earth oscillates after a big earthquake. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
21 Earth. Earthquake wave paths (Wikimedia Commons) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
22 Earth. The shadow zone tells us about the core (Wikimedia Commons) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
23 Industrial testing Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
24 Industrial testing Industries want to produce durable products and sometimes even tiny cracks can be dangerous. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
25 Industrial testing Industries want to produce durable products and sometimes even tiny cracks can be dangerous. Think what a crack can do in the concrete structures of a building Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
26 Industrial testing Industries want to produce durable products and sometimes even tiny cracks can be dangerous. Think what a crack can do in the concrete structures of a buildingor the windshield of an airplane. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
27 Industrial testing. Industries want to produce durable products and sometimes even tiny cracks can be dangerous. Think what a crack can do in the concrete structures of a buildingor the windshield of an airplane. We need to be able to see if products have small defects without breaking the products. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
28 Outline 1 Indirect measurements 2 Direct and inverse problems Direct problem Inverse problem X-ray tomography Electric imaging Spectral geometry Boundary rigidity Summary Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
29 Direct problem Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
30 Direct problem A direct problem is to find out how an object behaves, given the object. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
31 Direct problem A direct problem is to find out how an object behaves, given the object. Example: Given an instrument, figure out what sound it makes. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
32 Direct problem A direct problem is to find out how an object behaves, given the object. Example: Given an instrument, figure out what sound it makes. Mathematical example: Given a polynomial, find its roots. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
33 Direct problem. A direct problem is to find out how an object behaves, given the object. Example: Given an instrument, figure out what sound it makes. Mathematical example: Given a polynomial, find its roots. These problems are often straightforward to solve, but not always trivial. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
34 Inverse problem Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
35 Inverse problem An inverse problem asks the opposite question. Given the behaviour, find out what the object is. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
36 Inverse problem An inverse problem asks the opposite question. Given the behaviour, find out what the object is. Example: Given the sound of an instrument, figure out what the instrument is. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
37 Inverse problem An inverse problem asks the opposite question. Given the behaviour, find out what the object is. Example: Given the sound of an instrument, figure out what the instrument is. Mathematical example: Given the roots of a polynomial, find the polynomial. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
38 Inverse problem An inverse problem asks the opposite question. Given the behaviour, find out what the object is. Example: Given the sound of an instrument, figure out what the instrument is. Mathematical example: Given the roots of a polynomial, find the polynomial. Inverse problems are harder than the corresponding direct problem. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
39 Inverse problem. An inverse problem asks the opposite question. Given the behaviour, find out what the object is. Example: Given the sound of an instrument, figure out what the instrument is. Mathematical example: Given the roots of a polynomial, find the polynomial. Inverse problems are harder than the corresponding direct problem. Sometimes the inverse problem is ill-posed: we cannot know the object by its behaviour. But how much can we know? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
40 X-ray tomography Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
41 X-ray tomography Consider an X-ray fired along a line (the real axis). Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
42 X-ray tomography Consider an X-ray fired along a line (the real axis). If the intensity at x R is denoted I(x), then the Beer Lambert law gives us the differential equation I (x) = f(x)i(x), where f(x) is the attenuation coefficient which may depend on position. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
43 X-ray tomography Consider an X-ray fired along a line (the real axis). If the intensity at x R is denoted I(x), then the Beer Lambert law gives us the differential equation I (x) = f(x)i(x), where f(x) is the attenuation coefficient which may depend on position. This can be solved: I(L) = I(0) exp ( L 0 ) f(x)dx. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
44 X-ray tomography. Consider an X-ray fired along a line (the real axis). If the intensity at x R is denoted I(x), then the Beer Lambert law gives us the differential equation I (x) = f(x)i(x), where f(x) is the attenuation coefficient which may depend on position. This can be solved: I(L) = I(0) exp ( L 0 ) f(x)dx. If we measure the initial and final intensities I(0) and I(L), we in fact measure the integral L 0 f(x)dx. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
45 X-ray tomography If we take X-ray images of an object from all directions, we measure the integrals of the attenuation coefficient over all lines through the object. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
46 X-ray tomography If we take X-ray images of an object from all directions, we measure the integrals of the attenuation coefficient over all lines through the object. Direct problem: Integrate a given function over all lines. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
47 X-ray tomography If we take X-ray images of an object from all directions, we measure the integrals of the attenuation coefficient over all lines through the object. Direct problem: Integrate a given function over all lines. Inverse problem: Given the integrals of a function f : R n R over all lines, find the function f. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
48 X-ray tomography If we take X-ray images of an object from all directions, we measure the integrals of the attenuation coefficient over all lines through the object. Direct problem: Integrate a given function over all lines. Inverse problem: Given the integrals of a function f : R n R over all lines, find the function f. This problem was first solved by Johann Radon in 1917 but it was forgotten. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
49 X-ray tomography. If we take X-ray images of an object from all directions, we measure the integrals of the attenuation coefficient over all lines through the object. Direct problem: Integrate a given function over all lines. Inverse problem: Given the integrals of a function f : R n R over all lines, find the function f. This problem was first solved by Johann Radon in 1917 but it was forgotten. It was solved again by Allan Cormack in 1963 in order to do X-ray tomography. Godfrey Hounsfield build a machine to do the measurements and the calculations. In 1979 Cormack and Hounsfield got the Nobel Prize in medicine for developing the CT scan. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
50 X-ray tomography Some modern problems: Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
51 X-ray tomography Some modern problems: How to get a good image from few line integrals (low radiation dose)? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
52 X-ray tomography Some modern problems: How to get a good image from few line integrals (low radiation dose)? What if the Euclidean space is replaced with a manifold? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
53 X-ray tomography Some modern problems: How to get a good image from few line integrals (low radiation dose)? What if the Euclidean space is replaced with a manifold? What if the function f is replaced with a vector field? (Applications in Doppler tomography.) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
54 X-ray tomography. Some modern problems: How to get a good image from few line integrals (low radiation dose)? What if the Euclidean space is replaced with a manifold? What if the function f is replaced with a vector field? (Applications in Doppler tomography.) What if the X-rays are allowed to reflect on some surface? (Broken ray tomography.) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
55 Electric imaging Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
56 Electric imaging Can we determine the electric properties of an object by making measurements at the surface? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
57 Electric imaging Can we determine the electric properties of an object by making measurements at the surface? Suppose we can set any voltage at the surface and measure the resulting current at the boundary (or vice versa). Can we determine the conductivity everywhere inside the object? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
58 Electric imaging Can we determine the electric properties of an object by making measurements at the surface? Suppose we can set any voltage at the surface and measure the resulting current at the boundary (or vice versa). Can we determine the conductivity everywhere inside the object? This is known as Calderón s inverse conductivity problem. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
59 Electric imaging. Can we determine the electric properties of an object by making measurements at the surface? Suppose we can set any voltage at the surface and measure the resulting current at the boundary (or vice versa). Can we determine the conductivity everywhere inside the object? This is known as Calderón s inverse conductivity problem. Electrical impedance tomography (EIT) is based on solving this problem. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
60 Electric imaging. EIT measurement around the chest. (IP CoE) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
61 Electric imaging. A tank with heart and lungs with reconstruction from real measurements. (Isaacson Mueller Newell Siltanen) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
62 Electric imaging Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
63 Electric imaging Consider a domain Ω R n. The conductivity is a function γ : Ω (0, ). Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
64 Electric imaging Consider a domain Ω R n. The conductivity is a function γ : Ω (0, ). Let u: Ω R be the voltage (electric potential). Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
65 Electric imaging Consider a domain Ω R n. The conductivity is a function γ : Ω (0, ). Let u: Ω R be the voltage (electric potential). By Ohm s law the current density is J(x) = γ u(x). Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
66 Electric imaging Consider a domain Ω R n. The conductivity is a function γ : Ω (0, ). Let u: Ω R be the voltage (electric potential). By Ohm s law the current density is J(x) = γ u(x). By Kirchhoff s law div(j) = 0. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
67 Electric imaging Consider a domain Ω R n. The conductivity is a function γ : Ω (0, ). Let u: Ω R be the voltage (electric potential). By Ohm s law the current density is J(x) = γ u(x). By Kirchhoff s law div(j) = 0. The voltage satisfies the conductivity equation div(γ u) = 0. This is a second order partial differential equation. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
68 Electric imaging. Consider a domain Ω R n. The conductivity is a function γ : Ω (0, ). Let u: Ω R be the voltage (electric potential). By Ohm s law the current density is J(x) = γ u(x). By Kirchhoff s law div(j) = 0. The voltage satisfies the conductivity equation div(γ u) = 0. This is a second order partial differential equation. The voltage at the boundary is simply u Ω. The current at the boundary is γν u Ω (the outward component of J Ω ). Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
69 Electric imaging The problem is now this: We can set the value of u at the boundary freely. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
70 Electric imaging The problem is now this: We can set the value of u at the boundary freely. The values of u in Ω are determined by the boundary values and the conductivity equation div(γ u) = 0 but we do not know them. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
71 Electric imaging The problem is now this: We can set the value of u at the boundary freely. The values of u in Ω are determined by the boundary values and the conductivity equation div(γ u) = 0 but we do not know them. We know (measure) the current γν u at the boundary. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
72 Electric imaging The problem is now this: We can set the value of u at the boundary freely. The values of u in Ω are determined by the boundary values and the conductivity equation div(γ u) = 0 but we do not know them. We know (measure) the current γν u at the boundary. We can do this measurement for any choice of u Ω. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
73 Electric imaging The problem is now this: We can set the value of u at the boundary freely. The values of u in Ω are determined by the boundary values and the conductivity equation div(γ u) = 0 but we do not know them. We know (measure) the current γν u at the boundary. We can do this measurement for any choice of u Ω. Is this information enough to find the function γ? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
74 Electric imaging. The problem is now this: Yes! We can set the value of u at the boundary freely. The values of u in Ω are determined by the boundary values and the conductivity equation div(γ u) = 0 but we do not know them. We know (measure) the current γν u at the boundary. We can do this measurement for any choice of u Ω. Is this information enough to find the function γ? In two dimensions if γ is L. [Astala Päivärinta 2006] In higher dimensions if γ is Lipschitz. [Caro Rogers 2014] Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
75 Spectral geometry Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
76 Spectral geometry Kac in 1966: Can One Hear the Shape of a Drum? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
77 Spectral geometry Kac in 1966: Can One Hear the Shape of a Drum? Direct problem: Given the shape of a drum, find the frequencies it produces. (Find the Dirichlet eigenvalues of the Laplacian of a given domain.) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
78 Spectral geometry Kac in 1966: Can One Hear the Shape of a Drum? Direct problem: Given the shape of a drum, find the frequencies it produces. (Find the Dirichlet eigenvalues of the Laplacian of a given domain.) Inverse problem: Given the spectrum (eigenvalues), find the shape of the drum. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
79 Spectral geometry. Kac in 1966: Can One Hear the Shape of a Drum? Direct problem: Given the shape of a drum, find the frequencies it produces. (Find the Dirichlet eigenvalues of the Laplacian of a given domain.) Inverse problem: Given the spectrum (eigenvalues), find the shape of the drum. Sometimes spectra are easier to measure than other things. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
80 Spectral geometry. If the drum sounds like a circle, it is a circle. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
81 Spectral geometry. These two drums sound exactly alike. (Wikimedia Commons) Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
82 Boundary rigidity Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
83 Boundary rigidity Physical problem: Given the travel times of earthquakes between any two points on the surface, find the sound speed everywhere inside the planet. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
84 Boundary rigidity Physical problem: Given the travel times of earthquakes between any two points on the surface, find the sound speed everywhere inside the planet. Mathematical problem: There is a Riemannian manifold with boundary. You know the boundary and the distance between any two boundary points. Find the manifold and the metric. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
85 Boundary rigidity Physical problem: Given the travel times of earthquakes between any two points on the surface, find the sound speed everywhere inside the planet. Mathematical problem: There is a Riemannian manifold with boundary. You know the boundary and the distance between any two boundary points. Find the manifold and the metric. The corresponding direct problem: Given a Riemannian manifold with boundary, find the distance between any two boundary points. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
86 Boundary rigidity Physical problem: Given the travel times of earthquakes between any two points on the surface, find the sound speed everywhere inside the planet. Mathematical problem: There is a Riemannian manifold with boundary. You know the boundary and the distance between any two boundary points. Find the manifold and the metric. The corresponding direct problem: Given a Riemannian manifold with boundary, find the distance between any two boundary points. This inverse problem is very difficult, but it can be solved in some cases. The metric can only be found up to isometries. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
87 Boundary rigidity. Physical problem: Given the travel times of earthquakes between any two points on the surface, find the sound speed everywhere inside the planet. Mathematical problem: There is a Riemannian manifold with boundary. You know the boundary and the distance between any two boundary points. Find the manifold and the metric. The corresponding direct problem: Given a Riemannian manifold with boundary, find the distance between any two boundary points. This inverse problem is very difficult, but it can be solved in some cases. The metric can only be found up to isometries. We do not completely know the interior structure of our planet. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
88 Summary Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
89 Summary In inverse problems we often want to know what is inside by making measurements at the boundary. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
90 Summary In inverse problems we often want to know what is inside by making measurements at the boundary. This is a common feature of all indirect measurements: We cannot go inside but we want to know what there is. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
91 Summary In inverse problems we often want to know what is inside by making measurements at the boundary. This is a common feature of all indirect measurements: We cannot go inside but we want to know what there is. There are unexpected connections between different inverse problems. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
92 Summary In inverse problems we often want to know what is inside by making measurements at the boundary. This is a common feature of all indirect measurements: We cannot go inside but we want to know what there is. There are unexpected connections between different inverse problems. The examples discussed above may look different, but each of them has a strong connection to the X-ray tomography problem. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
93 Summary. In inverse problems we often want to know what is inside by making measurements at the boundary. This is a common feature of all indirect measurements: We cannot go inside but we want to know what there is. There are unexpected connections between different inverse problems. The examples discussed above may look different, but each of them has a strong connection to the X-ray tomography problem. These connections are mathematical, not always physical. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
94 Summary Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
95 Summary What we want to know about inverse problems: Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
96 Summary What we want to know about inverse problems: Uniqueness: Does the data uniquely determine the object? If not, what exactly does the data determine? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
97 Summary What we want to know about inverse problems: Uniqueness: Does the data uniquely determine the object? If not, what exactly does the data determine? Reconstruction: How to reconstruct the object from the data? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
98 Summary What we want to know about inverse problems: Uniqueness: Does the data uniquely determine the object? If not, what exactly does the data determine? Reconstruction: How to reconstruct the object from the data? Stability: If we make a small error in measurements, can we make a big mistake in reconstruction? Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
99 Summary. What we want to know about inverse problems: Uniqueness: Does the data uniquely determine the object? If not, what exactly does the data determine? Reconstruction: How to reconstruct the object from the data? Stability: If we make a small error in measurements, can we make a big mistake in reconstruction? These are hard mathematical problems. Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September /
100 End. Thank you. Slides are available at Joonas Ilmavirta (Jyväskylä) Inverse problems 8 September 2015 /
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