Solving ill-posed problems with mollification and an application in biometrics

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Solving ill-posed problems with mollification and

an application in biometrics

Department of Mathematics, Linköping University Emma Lindgren

LiTH-MAT-EX–2018/02–SE

Credits: 16 hp Level: G2

Supervisor: Fredrik Berntsson,

Department of Mathematics, Linköping University Examiner: Bengt-Ove Turesson,

Department of Mathematics, Linköping University Linköping: February 2018

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Abstract

This is a thesis about how mollification can be used as a regularization method to reduce noise in ill-posed problems in order to make them well-posed. Ill-posed problems are problems where noise get magnified during the solution process. An example of this is how measurement errors increases with differentiation. To correct this we use mollification.

Mollification is a regularization method that uses integration or weighted average to even out a noisy function. The different types of error that occurs when mollifying are the truncation error and the propagated data error. We are going to calculate these errors and see what affects them. An other thing worth investigating is the ability to differentiate a mollified function even if the function itself can not be differentiated.

An application to mollification is a blood vessel problem in biometrics where the goal is to calculate the elasticity of the blood vessel’s wall. To do this measurements from the blood and the blood vessel are required, as well as equations for the calculations. The model used for the calculations is ill-posed with regard to specific variables which is why we want to apply mollification. Here we are also going to take a look at how the noise level affects the final result as well as the mollification radius.

Keywords:

Mollification, regularization, ill-posed problems, biometrics, blood vessel URL for electronic version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145539

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Sammanfattning

Den här uppsatsen handlar om hur mollifiering kan användas som en regula-riseringsmetod för att redusera brus i illa-ställda problem med syftet att göra problemet välställt. Illa-ställda problem är problem där brus förstoras genom funktionen. Ett exempel på detta är hur mätfel ökar genom derivering. För att rätta till detta använder vi mollifiering.

Mollifiering är en regulariseringsmetod som använder integration eller viktat medelvärde för att jämna ut den brusiga funktionen. De olika typer av fel som uppstår när man mollifierar är trunkeringsfel och fortplantningsfel. Vi kommer att beräkna dessa fel och se vad det är som påverkar dem. En annan sak vi vill undersöka är möjligheten att derivera ett mollifierat problem även om problemet i sig inte är deriverbart.

Ett användningsområde för mollifiering är blodkärlsproblem inom biometrin där syftet är att beräkna elasticiteten i blodkärlet. För att göra detta behöver man mätvärden från blodet och blodkärlet samt en modell för beräkningarna. Den modell som används är illa-ställd med hänsyn till specifika variabler, vilket är varför vi använder mollifiering. Här tänker vi även studera hur brusnivån och mollifieringsradien påverkar det slutgilitga resultatet.

Nyckelord:

Mollifiering, regularisering, illa-ställda problem, biometri, blodkärl URL för elektronisk version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145539

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Acknowledgments

First of all I want to acknowledge my supervisor Fredrik Berntsson, without whom this paper would not have been possible. Thank you for all the help and for having answers to all my questions. I am also very grateful for your patience while I was working on other projects and taking care of my health. It has truly been an enjoyable experience working with you.

I also want to thank my friends and family for always supporting and encour-aging me. You have helped me through my darkest moments with my burnout and my struggle to get a normal life back. Thanks to you I have learned not to sweat the small stuff and to always have a positive attitude. An extra big thanks goes out to my dad that have helped me throughout this journey with every-thing from keeping me focused on whats matter to helping me with proofreading this paper.

Lastly I want to thank my examiner Bengt-Ove Turesson and Sofia Svensson for a good opposition.

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Introduction

In this work our main purpose is to study mollification[11]. Mollification is a way to smoothen out unwanted noise, for example measurement errors in functions, and hinder the noise from expanding and ruin the accuracy of the result.

In order to understand mollification and why it is important we first have to look into well-posed and ill-posed problems [11]. To make an ill-posed problem well-posed, which is what we want the problem to be, we need some kind of regularization methods. There are many types of regularization methods but the result of it is a stable function. One type of regularization is mollification, which is the one we are interested in here. We are going to look into how mollification works as well as investigate it’s accuracy.

Finally, we are also going to apply the mollification technique to biometrics. Here we will take a closer look into problems with blood vessels, or more specif-ically investigate the elasticity of the blood vessels wall. To do this we need to use mollification to reduce the measurement error in order to get a reliable result.

1.1 Ill-posed problems

In applications measurement error is unavoidable thus data is only available to a certain degree of accuracy. Due to this it is important to study how sensitive the solution of the problem is with respect to changes in the data used. It is desirable that the solution of the problem is stable with respect to changes in the data, however for some important applications this is not the case. In this section we aim to introduce the basic theoretical concept needed to discuss a problems sensitivity with respect to errors in measurements and parameters.

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How the measurement error propagates through different operations can be determined by studying whether an operation is well-posed or ill-posed. The ill-posed problems are sensitive to changes in the data. However before we look at ill-posed problems we first need to start by defining a well-posed problem. The definition we use can be found in [6].

Definition 1.1 A problem iswell-posed if the following holds

1. For all admissible data, a solution exist. 2. For all admissible data, the solution is unique. 3. The solution depends continuously on the data.

For further explanation of Definition 1.1 we have to introduce two linear spaces

G and F , and a linear operator T : G→ F . For a specific f ∈ F we can consider

the operator equation

T g = f,

that is the problem of finding g ∈ G such that T g = f, when f is known. The first condition in the definition is met if range(T ) is the whole space F . Then for every f ∈ F there is an g ∈ G such that T g = f. The second condition

is met if and only if null(T ) = {0}. Since if both g1 and g2 are solutions then

0 = f_{− f = T g}1− T g2= T (g1− g2) and g1− g2∈ null(T ) = {0} which gives

g1 = g2. For concrete problems both the first and the second condition are

usually met.

The third condition on the other hand is not usually met for a concrete problem and may therefor need more explanation. First we introduce the norms

k · kG and k · kF and give a definition of continuity [8].

Definition 1.2 The operatorT : G_{→ F is continuous if}

kT g − T fkG≤ ckg − fkF, ∀g, f ∈ G,

wherec > 0 is a constant that is independent of g and f .

We can also choose to study the operator norm to find out how strong the operator is and even determinate if it is continuous or not.

Definition 1.3 The norm of a linear operator T is defined as

kT k = sup

g6=0

kT gkF

kgkG

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1.1. Ill-posed problems 3

Remark 1.4 This means that if the operator T is bounded in norm then T is

continuous and if T is unbounded then T is not continuous.

To further illustrate the concept we are going to give an example of a well-posed problem.

Example 1.5 We take an operator

T : C0([0, 1])→ C1_{([0, 1])} given by (T f )(t) = Z t 0 f (τ )dτ.

The problem we consider is to evaluate T for a given f ∈ C0_{([0, 1]). By [4]}

we know that there exists a solution and that the solution is unique. In order

to investigate the continuity of T we need to equip the spaces C0_{([0, 1]) and}

C1_{([0, 1]) with norms. In this example we use}

kfk∞= max

0≤t≤1|f(t)|.

We can show that T is continuous by taking f1, f2∈ C0([0, 1]), and estimating

kT f1− T f2k∞= Z t 0 f1(τ )dτ− Z t 0 f2(τ )dτ ∞ = Z t 0 (f1(τ )− f2(τ ))dτ ∞ = max 0≤t≤1 Z t 0 (f1(τ )− f2(τ ))dτ ≤ max0≤t≤1 Z t 0 |f 1(τ )− f2(τ )|dτ ≤ max_0≤t≤1 Z t 0 kf 1− f2k∞dτ ≤ max 0≤t≤1tkf1− f2k∞ =_kf1− f2k∞.

Since kT f1− T f2k∞ ≤ kf1 − f2k∞ we see that T is continuous. Thus all

conditions of Definition 1.1 are satisfied and the problem of evaluating T is well-posed.

Not all problems satisfy the conditions for a problem to be well-posed. Thus we introduce the following definition.

Definition 1.6 A problem is ill-posed if one, or more, of the conditions for

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Example 1.7 We introduce an operator D : C1

→ C0_{, where}

(Df )(t) = f′(t),

and consider the problem of evaluating D for a specific, differentiable, function

f (t). Let fε(t) = f (t) + ε cos(ωt). We will show that the operator D is

un-bounded and thus evaluating D is an ill-posed problem. We know the derivative

is unique and due to the choice of the domain of D equal to C1_{both the first and}

second condition of Definition 1.1 are met. Now we investigate the continuity by calculating.

kDf − Dfεk∞=kωε sin(ωt)k∞= ωε

and

kf − fεk∞=kε cos(ωt)k∞= ε.

Since we are free to choose any value of ω we get that kDf − Dfεk∞ can be

arbitrary large even though kf − fεk∞ is small, which contradicts the third

criterion for a well-posed problem and thereby it is an ill-posed problem.

Remark 1.8 Note that different norms can give different results. If we instead

used the norm kgk1=kgk∞+kg′k∞our operator D would be well-posed.

Remark 1.9 Observe that the operators from Example 1.7 and Example 1.5

are related in a way such that T−1_{= D. In other words, the operator D from}

Example 1.7 can solve the inverse problem T−1_{f = g from Example 1.5 with}

Df = g where f is known.

Since we know that differentiation is an ill-posed problem we expect difficulties in numerical calculations. In order to investigate we give an example.

Example 1.10 For a function f (t) we can approximate the derivative by

Dhf (t) = f (t + h)− f(t − h)

2h .

Thus the operator Dh is an approximation of the derivative d/dt. We then get

the truncation error

Dhf (t)− f′(t) =

h2_f(3)_(t)

12 +O(h

4_).

We can see that the smaller h is the smaller the truncation error Dhf (t)− f′(t)

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1.1. Ill-posed problems 5 10-15 ₁₀-10 ₁₀-5 ₁₀0 h 10-15 10-10 10-5 100 |f (2)-D h f(2)| 10-4 ₁₀-3 ₁₀-2 ₁₀-1 ₁₀0 h 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 |f (2)-D h f (2)|

Figure 1.1: In these graphs we look at how the total error is depending on h. In the left graph there are no additional noise, i.e only round-off errors bounded by machine precision. Compare this to the right graph where we have added some additional noise.

Let the noisy data be expressed as fε(t) = f (t) + e(t) where|e(t)| ≤ ε. We then

get the measurement error

|Dhfε(t)− Dhf (t)| ≤ ε

h

If we instead select h such that h = ε1/3 _{we get}

|Dhfε(t)− f′(t)| ≤ f

(3)_(t)

12 ε

2/3_{+ ε}2/3

which is now getting smaller with a smaller ε which is what we want.

Remark 1.11 Note that the optimal stepsize h is dependent on the noise level

ε.

Example 1.12 To further examine the error in numerically calculated

differ-entiation we take the function

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and the perturbed function

fε(t) = f (t) + e(t)

where |e(t)| ≤ ε and t ∈]0, 1[. As previously we get the total error by calculating

|Dhf ε(t)− f′(t)| where Dhfε(t) = fε(t + h)− fε(t− h) 2h = cos(t + h)√t + h + ε1− cos(t − h)√t− h − ε2 2h and f′_{(t) = cos(t)/2}√_t − sin(t)√t.

In Figure 1.1 we show how the total error depends on the stepsize h. We have

the same functions f (t) and fε(t) as in previous example and do the calculations

for t = 2. We can see that the total error |f′₍₂₎_{− D}

hfε(2)| is smallest when

h ≈ 10−2_{. From the figure we can also see that the error, |f}′₍₂₎_{− D}

hf (2)|,

of the function without added noise is smallest when h ≈ 10−5_{. In this case}

round-off errors gives an error ε ≈ µ · |f(2)| where µ ≈ 1.1 · 10−16_{is the machine}

precision.

For more information on ill-posed problems see [6], [7] and [5].

1.2 Regularization

Recall that we previous considered the problem of solving an operator equation

T g = f , where T : G→ F for a given f ∈ range(T ) = F .

Lemma 1.13 If null(T ) ={0} then T−1 _exists.

If T−1 _{exists then the first and second condition in Definition 1.1 are satisfied.}

From Definition 1.3 we can see that if kT−1

k is bounded, then the problem

g = T−1_{f is well posed. Thus we want to consider the case where}

kT−1

k is

unbounded and the problem ill-posed. We approximate Rα ≈ T−1, where Rα

is bounded and thereby continuous, and where α is a parameter that controls the accuracy of the approximation.

Definition 1.14 Let T : G _{→ F be a linear operator then R}α, α ≥ 0, is a

regularization ofT−1 _{if there exists a parameter}_{α = α(ε, f}

ε) such that lim sup ε→0 {kRα(ε,fε) fε− T−1fk : fε∈ F, kfε− fk ≤ ε} = 0 when lim sup ε→0 {α(ε, fε) : fε∈ F, kfε− fk ≤ ε} = 0

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1.2. Regularization 7

Remark 1.15 Our definition of regularization is slightly different from [6] since

we assume that T−1_{exists. Thus we do not need to consider the pseudo inverse}

T†_.

Now we have a definition so we can solve T g = f , where f is known and T−1

is unbounded. Compare this with (Df )(t) = f′_{(x), from Example 1.7, where}

Dh≈ d/dx and where h has the same function as α. Furthermore we have the

choice rule h = ε1/3 _{that fulfills the definition of a regularization. For more}

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Chapter 2

Mollification

As we have seen in previous chapter, differentiation is a sensitive operation because the noise may be magnified. One way to improve this is to smooth out the function with different kinds of filters. This can be done with a regularization method called mollification [11]. It uses integration or a weighted average in order to smooth out the function and cancel out the noise. In the next section we will investigate the basic principles of mollification before taking a closer look at one of the more commonly used methods.

2.1 Introduction to mollification

To introduce mollification we come up with a simpler version based on the more commonly used method and define it as follows:

Definition 2.1 Let f (t)∈ C0_{(I), I = [a, b]. The moving average mollifier J}

ω is given by Jωf (t) = 1 2ω Z t+ω t−ω f (τ )dτ.

In the definition above f (t) is the function we want to smooth out and ω is the mollification radius.

Remark 2.2 In order to stay within the boundaries [a,b] during the hole

mol-lification process Jωf is defined for a + ω≤ t ≤ b − ω. Because of this we also

need a new norm.

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Definition 2.3 Let I = [a, b] and ω > 0, the norm for the moving average mollifier is then

kfk∞,I= max

a+w≤t≤b−w|f(t)|.

Let us now look closer at the different types of errors that occur when using mollification. The total error in the mollified result is the sum of two different types of errors, the truncation error and the propagated data error. This gives us the following error equation

Jwfε(t)− f(t) = Jw(fε(t)− f(t)) + Jwf (t)− f(t).

Let us first take a closer look at the size of the truncation error.

Lemma 2.4 Let f (t)_{∈ C}1_{(I), I = [a, b]. The truncation error is then}

kJwf− fk∞,I ≤ wkf′k∞,I.

Proof The truncation error can be calculated by

|Jwf (t)− f(t)| = Z t+w t−w 1 2wf (τ )dτ− 1 2w Z t+w t−w 1dτ f (t) = 1 2w Z t+w t−w (f (τ )_{− f(t))dτ} = 1 2w Z t+w t−w f′(ξ(τ ))(τ_{− t)dτ} . Since |f′_{(ξ(τ ))}_{| ≤ kf}′_k

∞,I and |(τ − t)| ≤ ω we get

1 2w Z t+w t−w f′_{(ξ(τ ))(τ} − t)dτ ≤ 1 2w Z t+w t−w kf ′ k∞,Iwdτ = wkf′k∞,I, which gives us kJwf− fk∞≤ wkf′k∞,I.

Let us now investigate what happens if we perturb a function f (t) and use mollification.

Lemma 2.5 Let f (t) ∈ C1_{(I), I = [a, b],} _{kf − f}

εk∞,I ≤ ε . The propagated

data error is then

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2.1. Introduction to mollification 11

Proof The error caused by noise can be calculated by

kJωf − Jωfεk∞,I= max a≤t≤b 1 2ω Z t+ω t−ω (f (τ )− fε(τ ))dτ ≤_2ω1 Z t+ω t−ω kf(τ) − f ε(τ )k∞,Idτ ≤ ε.

In order to investigate how mollification works together with differentiation we first prove the following lemma.

Lemma 2.6 Let f _{∈ C}0_{(I), I = [a, b], then J}

ωf is differentiable on Iω =

[a + ω, b_{− ω].}

Proof We need to show that the limit exist when h → 0.

1 h(Jwf (t + h)− Jwf (t)) = 1 h 1 2ω Z t+ω+h t−ω+h f (τ )dτ−_2ω1 Z t+ω t−ω f (τ )dτ ! = 1 2ωh Z t+ω+h t+ω f (τ )dτ₋ Z t−ω−h t−ω f (τ )dτ ! = (_∗)

Now we use the mean value theorem [4] to show

Z t+ω+h

t+ω

f (τ )dτ = f (ξ1(h))h

where ξ1(h) ∈ [t + ω, t + ω + h] and f(ξ1(h)) → f(t + ω) as h → 0 since f

is assumed to be continuous. Calculations on the second integral in the same manner gives

(_{∗) =} 1

2ωh(f (ξ1)h− f(ξ2)h)→

1

2ω(f (t + ω)− f(t − ω))

as h → 0. That proves that the limit exists as h → 0.

Corollary 2.7 Let f _{∈ C}0_{(I), I = [a, b]. From the proof of Lemma 2.6 we get}

(Jωf )′=

1

2ω(f (t + ω)− f(t − ω)).

We now want to examine the errors that occurs when differentiating a mollified noisy function.

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Lemma 2.8 Let f _{∈ C}0_{(I), I = [a, b] and}

kf − fεk∞,I ≤ ε. Then

k(Jωfε)′− (Jωf )′k∞,I ≤

ε ω.

Proof Here we are going to use Corollary 2.7 to calculate the differentiation.

k(Jωfε)′− (Jωf )′k∞,I≤

ε

ω.

Lemma 2.9 Let f _{∈ C}3_{(I), I}

∈ [a, b] and M = max

a≤t≤b|f (3)_(t) |. Then k(Jωf )′− f′k∞,I ≤ ω2_M 6 .

Proof From Corollary 2.7 and Taylor expansions we get

(Jωf )′(t)− f′(t) = 1 2ω(f (t + ω)− f(t − ω)) − f ′_(t) = 1 2ω((f (t) + f ′_{(t)ω +}f′′(t)ω2 2! + f(3)_(ξ 1)ω3 3! ) − (f(t) − f′_{(t)ω +}f′′(t)ω2 2! − f(3)_(ξ 2)ω3 3! ))− f ′_(t) = 1 2ω f(3)_(ξ 1)ω3 3! + f(3)_(ξ 2)ω3 3! ,

where ξ1∈ [t, t + ω] and ξ2∈ [t − ω, t]. Then we get

|(Jωf )′− f′| = 1 2ω f(3)_(ξ 1)ω3 3! + f(3)_(ξ 2)ω3 3! ≤ ω2_M 6 .

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2.1. Introduction to mollification 13 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 f(t) and f (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 f(t) and J f (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 f(t) and J f (t)

Figure 2.1: To the left we have the function f (t) = t2_{cos(t), without noise, in}

blue and the noisy function fε(t) in red. In the middle we have the function

with noise, fε(t), that has been smoothed out with Jωfε(t) when ω = 3∆t in

red compared with f (t) in blue. To the right we have the same as in the middle graph but with ω = 9∆t instead of ω = 3∆t.

Lemma 2.10 Let f _{∈ C}3_{(I) and f}

ε∈ C3(I), I ∈ [a, b] where kf − fεk∞,I ≤ ε

andM = max a≤t≤b|f (3)_(t) |. Then k(Jωfε)′− f′k∞,I ≤ ε ω + ω2_M 6 .

Proof From Lemma 2.8 and 2.9 we get

k(Jωfε)′− f′k∞,I≤k(Jωfε)′− (Jωf )′k∞,I+k(Jωf )′− f′k∞,I

≤ _ωε +ω

2_M

6 .

We see that the errors that occurs are small and depends on the noise level, ε, the mollification radius, ω, and the character of the function.

Remark 2.11 If we define a kernel by

ρω(t) =

1

2w,

for −ω ≤ t ≤ ω and ρω(t) = 0 for|t| > ω, then

Jωf (t) = (ρω∗ f)(t) = Z ∞ −∞ ρω(t− τ)f(τ)dτ = 1 2ω Z t+ω t−ω f (τ )dτ.

Example 2.12 To examine the moving average mollifier we look at some graphs

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[4]. In Figure 2.1 we have a function f (t) = t2_{cos(t) that we compare with the}

same function but with some added noise fε(t) = t2cos(t) + e(t) where e(t) is

uniformly distributed in the range |e(t)| ≤ ε = 0.05. We have also applied the

moving average mollifier on fε(t) with ω = 3∆t and ω = 9∆t respectively. In

our example we look at t ∈ [0, 1] with ∆t = 1/100. We can see in the figure that the mollified noisy function is close to the original function without noise.

Remark 2.13 Larger ω gives shorter Jωfε(t). This is because if f is defined

for a ≤ t ≤ b, then Jωf is only defined for a + ω/2≤ t ≤ b − ω/2 due to the

integration taking place over [t − ω, t + ω] for a given t.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h -0.5 0 0.5 1 1.5 Diffrentiation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h -0.5 0 0.5 1 1.5 Diffrentiation

Figure 2.2: To the left we have differentiation of Jωfε with ω = 3∆t in red

(dashed) compared with the differentiation of f (t) in blue (solid). To the right we use ω = 9∆t instead of ω = 3∆t.

Example 2.14 Now we want to look at how a mollified function behaves after

derivation. We use the same function, noise level and mollification radius as in Example 2.12. In Figure 2.2 we are looking at how the noise increase after differentiation by comparing with the differentiation of the non noisy function f (t), using the same functions as in previous example. We can see in that the

mollified function Jωfεwith ω = 3∆t still differs a lot from the original function.

With the mollification with ω = 9∆t we get a result which is much closer to the original function.

In this case our function f (t) is smooth and a large value for ω can be used since we do not loose any features from f (t) during the mollification. If we instead had a function with features, these feature risk getting lost during the mollification and we might loose accuracy by using a too large value for ω.

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2.1. Introduction to mollification 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0 0.2 0.4 0.6 0.8 1 1.2 f(t) and f (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0 0.2 0.4 0.6 0.8 1 1.2 f(t) and J f (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0 0.2 0.4 0.6 0.8 1 1.2 f(t) and J f (t)

Figure 2.3: To the left we have the function f (t) without noise in blue and the

noisy function fε(t) in red. In the middle we have the function with noise, fε(t),

that has been smoothed out with Jωfε(t) when ω = 3∆t in red compared with

f (t) in blue. To the right we have the same as in the middle graph but with ω = 12∆t instead of ω = 3∆t. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h -6 -4 -2 0 2 4 6 Diffrentiation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h -6 -4 -2 0 2 4 6 Diffrentiation

Figure 2.4: To the left we have differentiation of Jωfε with ω = 3∆t in red

(dashed) compared with the differentiation of f (t) in blue (solid). To the right we use ω = 12∆t instead of ω = 3∆t and as before we have the exact derivate,

f′_{(t), in blue (solid) with the calculated approximation (J}

ωfε)′ in red (dashed).

Example 2.15 Let us do the same experiment as in Example 2.12 and 2.14

but with the function

f (t) = t2_{cos(t) + e}−30(t−0.3)2

.

In Figure 2.3 we see the noisy function fε(t) compared with the non noisy

function f (t). The figure also shows the mollified noisy function Jωfε(t) with

ω = 3∆t and ω = 12∆t. As we can se this time a bigger ω does not give a better result. That is because a bigger ω not only flattens out the noise

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but also the function itself. This also leaves its marks when differentiating the mollified function as seen in Figure 2.4. When comparing the differentiated non noisy function f (t) with the differentiated mollified functions we see that with ω = 12∆t it eliminates the noise, but also chance the characteristics of the function. When ω = 3∆t the function still has both some noise and the shape of the function left.

2.2 δ-mollification

In this section we present information from [11] where proofs and additional information can be found. Let us now look at a more commonly used mollifi-cation method that we are going to call δ-mollifimollifi-cation. We start off with the definition.

Definition 2.16 Let the kernel be

ρδ(t) = 1 δ√πexp −t2 δ2 ,

and the δ-mollification as

Jδf (t) = (ρδ∗ f)(t) =

Z ∞

−∞

(ρδ(t− τ)f(τ))dτ

whereδ is the mollification radius.

As in previous section we also want to know the different types of error that occurs. Let us first look at the consistency.

Lemma 2.17 If _kf′′_k

∞,I ≤ M, then

k(ρδ∗ f)′− f′k∞,I ≤ 3δM.

This can be compared with Lemma 2.9. Let us now look at the stability.

Lemma 2.18 If f (t)_{∈ C}0_{(I) and}

kfε− fk∞,I≤ ε, then

k(ρδ∗ fε)′− (ρδ∗ f)′k∞,I≤ 2ε

δ√π.

Compare this to Lemma 2.8. We can see that the accuracy of mollification is dependent on the problem, noise level and mollification radius. The error

increases with larger noise levels and with large kf′′_k

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2.2. δ-mollification 17

We can also show that this mollification is differentiable even if the function f (t) is not by (Jδf )′(t) = lim h→0 1 h Z ∞ −∞ ρδ(t + h− τ)f(t)dτ − Z ∞ −∞ ρδ(t− τ)f(t)dτ = Z ∞ −∞ lim h→0 ρδ(t + h− τ) − ρδ(t− τ) h f (τ dτ ) = Z ∞ −∞ ρ′δ(t− τ)f(τ)dτ.

This is true only if we can change the order of the limit and the integral. We are not going to investigate when the limit and the integral can change places but with our kernel,

ρδ(t) = 1 δ√πexp −t2 δ2 ,

we can assume it is possible if f ∈ C0_(R).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 f(t) and f (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 f(t) and J f (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 f(t) and J f (t)

Figure 2.5: To the left we have the function f (t) = t2_{cos(t) in blue compared}

with its noisy function fε(t) in red. In the middle and right graph we have

mollified the noisy function with δ-mollification, Jδfε(t), using δ = 0.02 and

δ = 0.1 respectively.

Example 2.19 We use the same function and do the same experiment as in

Example 2.12, only this time we use δ-mollification. When changing mollifica-tion method we also change the mollificamollifica-tion radius. In this example we are looking at when δ = 0.02 and δ = 0.1 as showed in Figure 2.5.

Example 2.20 Let us now look at how mollification behaves after

differentia-tion when δ = 0.02 and δ = 0.1, and compare it to the differentiated funcdifferentia-tion without noise. We use the same function and noise level as in Example 2.14. The result is shown in Figure 2.6.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h -0.5 0 0.5 1 1.5 Diffrentiation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h -0.5 0 0.5 1 1.5 Diffrentiation

Figure 2.6: Here we compare the differentiation of the non noisy function f (t) in

blue with differentiation of the mollified noisy function Jδfε(t) in red. We have

used δ = 0.02 on the left graph and δ = 0.1 on the right. With a mollification radius of δ = 0.02 we still get a lot of noise but with δ = 0.1 we get a good result.

We have seen in this section that δ-mollification works well when choosing a suitable radius of mollification. More information om mollification can be found at [10], [11], [3], [9] and [13].

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Chapter 3

Application in biometrics

How can we use what we have just learned? An example of application is the calculation of blood flow in patients in order to discover how elastic the blood vessels are. It is the combination of measured data and an ill-posed problem that makes the mollification necessary in order to get an accurate calculation. New measurement equipment also made the delicate measurements of the blood vessel more reliable.

In this chapter we are going to describe how to calculate the elasticity of the blood vessel with the help of δ-mollification. We are also going to use simulations to show how the noise level and mollification radius impact the result.

3.1 A model of blood flow in an elastic vessel

When modeling the blood flow in a vessel we are interested in the pressure,

p(z, t). We have φ1, that is the pressure at the starting point z = −L, and

φ2, that is the pressure at the end point z = L. We are also looking at the

velocity component, vz(z, t), along the length of the vessel and the blood vessels

expansion, ur(z, t).

Let us now take a look at the equations. We are going to use the following two equations: hR−1kϕϕur(z, t) + hRγw∂t2ur(z, t) = p(z, t), t∈ [0, T ], z ∈ (−L, L) (3.1) and R3 16µ∂ 2 zp(z, t) = ∂tur(z, t), t∈ [0, T ], z ∈ (−L, L). (3.2) Lindgren, 2018. 19

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z L 0 −L r R R(1+h)

Figure 3.1: The geometry of the blood vessel. The thickness of the vessel wall h is small compared to the radius R and the length 2L.

Here the symbol kϕϕ denotes the vessel’s elasticity modulus, R is the average

radius and hR is the thickness of the blood vessels wall according to Figure 3.1.

Lastly we have the effective density of the wall γω and the dynamical viscosity

coefficient of the blood µ. The variables are measured over the time period 0 to T and is periodic. For more information about the measuring of the parameters and how they are connected see [2].

3.2 Simulated measurements

In this section we are going to show the simulated measurements that we later are going to use for the reconstruction experiment. We have used the same values on the different variables as in [2] and they are displayed in Table 3.1.

Table 3.1: The physical parameters

Variable Value Unit

R 4.3 mm hR 0.75 mm γ 1050 kg/m3 µ 0.0051 kg/m/s T 0.917 s L 1 m L0 0.1 m

The variables showed in Table 3.1 are chosen to illustrate real values of a blood vessel where the elasticity of the wall is weak at a specific place. The elasticity of

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3.3. The inverse problem 21 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time: t [s] -20 0 20 40 60 80 100 120 Velocity v z (-L,t) [cm/s] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 Elasticity Modulus: k [kPa]

Figure 3.2: In the graph to the left we have the velocity, v0, of the blood over

time. To the right we see the elasticity of the blood vessel, kϕϕ, over the length

of the blood vessel.

the blood vessel, kϕϕ, and the velocity of the blood, v0= v(−L), are illustrated

in Figure 3.2 along with the pressure, p(z, t), and wall displacement, ur(z, t). In

Figure 3.3 we compare the pressure boundary conditions, φ1and φ2, in order to

see how the pressure differs along the blood vessel. We can see how the pressure varies over time and place in the blood vessel. Among the figures we also see that the displacement of the blood wall is depending on the placement as well. Here we compare the displacement at the beginning where z = −1 m and at

z = −0.9279 m where the blood vessel is less elastic. In Figure 3.3 there is

also a graph where we can better see how the displacement differ over time and placement. As we can see this is the measurements of a blood vessel that has a part with less elasticity.

3.3 The inverse problem

Our problem is to find kϕϕ in an interval a ≤ z ≤ b assuming we have the

measured values for ur(z, t), and the pressure boundary conditions p(a, t) =

φ1(t) and p(b, t) = φ2(t). We also have measured data for the other variables,

except p(z, t), but since they do not vary over time and we do not differentiate them we will treat them as constants. The first step is to solve the following problem with regard to p(z, t):

R3

16µ∂

2

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time: t [s] 11 11.5 12 12.5 13 13.5 14 14.5 15 Pressure profiles 1 (t) and 2 (t) [kPa] 11 0 -0.75 12 -0.8 0.2 -0.85 0.4 Time: t [s] Spatial Coordinate: z [m] 13 Pressure P(z,t) [kPa] -0.9 0.6 14 -0.95 0.8 1 -1 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time: t [s] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Displacement u r (-1,t), u r (-0,9279,t) [mm] 0 -0.75 0 0.1 0.2 -0.8 -0.85 0.4 Spatial Coordinate: z [m] Time: t [s] 0.2 Displacement u r (z,t) [mm] -0.9 0.6 0.3 -0.95 0.8 1 -1 0.4

Figure 3.3: To the top left we have φ1in blue and φ2in black. To the top right

the pressure, p(z, t), is shown. On the bottom left we compare the wall displace-ment over time at different places of the blood vessel, u(z(1), t) and u(z(19), t). To the bottom right we see how the wall displacement u(z, t) depends on the time t and space coordinate z.

with boundary conditions

p(a, t) = φ1(t), and p(b, t) = φ2(t).

We can see that (3.3) is well-posed with respect to the right hand side ∂tur

and maintains the continuity of noise, but since the time variable ur(z, t) is

differentiated it makes the problem ill-posed with respect to ur(z, t). Because

of this we mollify ur(z, t) and we also choose to mollify φ1(t) and φ2(t) even

though it is not strictly necessary. We now solve the following system

R3

16µ∂

2

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3.4. Numerical implementation 23

with boundary conditions

p(a, t) = (Jδφ1)(t), and p(b, t) = (Jδφ2)(t).

After having calculated p(z, t), a ≤ z ≤ b, the final step is to get kϕϕfrom (3.1)

by the formula kϕϕ(z, t) = p(z, t)− hRγw∂ 2 tJδur(z, t) hR−1_J δur(z, t) .

Here we have also mollified the ill-posed parameter ∂2

tur(z, t). Now we have

successfully calculated the elasticity of the wall of the blood vessel, kϕϕ.

3.4 Numerical implementation

All the calculations are made in MatLab and most of the code is retrieved from [2] where the equations are discretized using an equidistant grid. The grid consists of 500 grid points for z and 512 for t, i.e N = 500 and M = 512. We can now see the discretized functions as matrices and we write them as

p = p(zi, tj), ur = ur(zi, tj) and kϕϕ = kϕϕ(zi, tj). The finite differential

equations are solved as in [2], more about finite differential equations can be found in [12].

When doing the reconstruction the first thing that needs to be done is to

add noise to the sensitive variables, in this case ur(z, t), φ1 and φ2. Now the

experiment can begin. As described in Section 3.3 we start by mollify ur(z, t), φ1

and φ2with the mollification function from [2]. The next step is to calculate the

first and second derivative of ur(z, t). When having all the needed information

we can calculate the pressure as presented in Section 3.3. Our final step is to

calculate kϕϕ by

kϕϕ= Rh−1(p− Rhγ∂t2ur)/ur,

where the division for the matrices is understood to be point wise. As we see we

now have the elasticity of the blood vessel as a matrix, kϕϕ, where the elements

represent calculated values at each grid point, i.e. we get a matrix kϕϕ(zi, tj).

Since we assume the elasticity to be independent of t we compute the average, kϕϕ(zi) = 1 M M X j=1 kϕϕ(zi, tj).

to get a vector kϕϕ(zi). This extra opportunity for averaging reduces the

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3.5 Reconstruction example

We are now going to try to reconstruct the values showed in Section 3.2. Since

we assume that ur, φ1 and φ2 are measured we need to add noise for

measure-ment error to these. With these new values for ur, φ1 and φ2 we now want to

reconstruct the pressure, p(z, t), and elasticity of the blood vessel, kϕϕ. How

this is done is described in the previous two sections. In order to show how the result is affected by mollification we are going to display various examples with different noise levels and mollification radiuses.

0 0.15 0 0.1 0.2 0.1 0.4 0.05 Spatial Coordinate: z [m] Time: t [s] 0.2 Displacement u r (z,t) [mm] 0 0.6 0.3 -0.05 0.8 1 -0.1 0.4 -0.03 0 0.15 -0.02 0.2 0.1 -0.01 0.05 0.4 Time: t [s] Spatial Coordinate: z [m] Differantiated displacement u r (z,t) [mm] 0 0.6 0 0.01 -0.05 0.8 1 -0.1 0.02 0 0.15 0 0.2 0.1 0.1 0.05 0.4 Spatial Coordinate: z [m] Time: t [s] Displacement u r (z,t) [mm] 0.6 0 0.2 -0.05 0.8 -0.1 1 0.3 -0.01 0 0.15 -0.005 0.1 0.2 0.4 0.05 Time: t [s] Spatial Coordinate: z [m] 0 Differantiated displacement u r (z,t) [mm] 0 0.6 0.005 -0.05 0.8 -0.1 1 0.01

Figure 3.4: To the top left the displacement of the blood vessels wall, ur, with

noise is displayed. The top right graph illustrates the differentiation of the

displacement, ur, with noise. To the bottom left we have mollified the noisy wall

displacement, Jδur, and to the bottom right the differentiation of the mollified

noisy wall displacement, ∂Jδur.

Example 3.1 To each simulated value ur(zi, tj) we add evenly distributed

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3.5. Reconstruction example 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time: t [s] 11 11.5 12 12.5 13 13.5 14 14.5 15 Pressure profiles 1 (t) and 2

(t) [kPa] with noise

11 0 -0.75 12 0.2 -0.8 0.4 -0.85 Spatial Coordinate: z [m] Time: t [s] 13

Reconstructed pressure P(z,t) [kPa] 0.6 -0.9

14 -0.95 0.8 -1 1 15

Figure 3.5: The left graph displays the pressure boundary conditions φ1and φ2

with noise. To the right we have the reconstructed pressure.

Now we want to reconstruct kϕϕ from our "measured" data as described in

Section 3.3. In this example we use a mollification radius off δ = 0.015.

In Figure 3.4 we see the effect off mollifying the wall displacement ur(z, t).

It is clear that ur(z, t) is smoother, especially when differentiated. In Figure 3.5

we see φ1 and φ2 during 0.917 seconds and the reconstructed pressure. From

Figure 3.6 we see that the result of calculating the elasticity of the blood vessel is much smoother when using mollification, at least before taking the mean value. Since we have more data than we actually need we can take the mean value of

kϕϕ(zi, ti) to get an even more accurate result. Because of this we see that even

though there is a big difference between kϕϕ(zi, tj) when using mollification

and kϕϕ(zi, tj) when not using mollification the difference is smaller between

kϕϕ(zi) when using mollification and kϕϕ(zi) when not using mollification.

From this example we can clearly see how effective mollification is.

Example 3.2 Let us now change δ in order to see what roll the mollification

radius plays. We use the same noises level as in Example 3.1.

In Figure 3.7 we see that the mollification radius does impact the result, but since mollification is a stable operation it does not differ much. We can see that smaller δ does not automatically give a better result.

Example 3.3 We have now investigated the effect of the mollification radius,

let us now see what happens if we reduce the noise level. To start with we use the same noise level as in Example 3.1 and 3.2. From Example 3.2 we see that

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10 350 400 5 1 Elasticity Modulus: k [kPa] 0.8 450 Spatial Coordinate: z [cm] 0 _0.6 Time: t [s] 500 0.4 -5 0.2 -10 0 10 350 400 5 1 Elasticity Modulus: k [kPa] 0.8 450 Spatial Coordinate: z [cm] 0 _0.6 Time: t [s] 500 0.4 -5 0.2 -10 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa]

Figure 3.6: To the top left we have the elasticity of the blood vessel, kϕϕ(zi, tj),

with mollification compared with kϕϕ(zi, tj) without mollification to the top

right. On the bottom we have taken the mean value kϕϕ(zi) using mollification

to the left and not using mollification to the right.

-10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa]

Figure 3.7: Here the elasticity of the blood vessels wall, kϕϕ with different δ is

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3.5. Reconstruction example 27 -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa] -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa]

Figure 3.8: Starting to the top left with a noise level of εi,j∼ N (0, 5 · 10−6) for

u_rand εj ∼ N (0, 50) for φ1 and φ2. We use a mollification radius of δ = 0.01

to calculate the elasticity of the blood vessel, kϕϕ. Then the graph to the top

right has half of the noise level to εi,j ∼ N (0, 2.5 · 10−6) and εj ∼ N (0, 25)

respective with the same mollification radius, δ = 0.01. To the bottom left we have chosen a smaller mollification radius, δ = 0.005 and use the same noise

εi,j ∼ N (0, 2.5 · 10−6) and εj ∼ N (0, 25) for ur and φ1, φ2respective. We half

the noise level a final time to get the graph to the bottom right with noise level

εi,j ∼ N (0, 1.25 · 10−6) and εj ∼ N (0, 12.5) and with the same mollification

radius of δ = 0.005.

δ = 0.01 was a good choice for the mollification radius. From Figure 3.8 we now see that the elasticity of the blood vessel get more and more accurate with a lower noise level. Note also that with a lower noise level we can choose a smaller

δ. Compare this to the regularization rule where Jδfε≈ f and δ has the same

function as α. We also have the choice rule where the mollification radius is dependent on the noise level.

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-0.1 0 0.15 0 0.2 0.1 0.1 0.05 0.4 Time: t [s] Spatial Coordinate: z [m] Displacement u r (z,t) [mm] 0.2 0 0.6 0.3 -0.05 0.8 1 -0.1 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time: t [s] 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 Pressure profiles 1 (t) and 2

(t) with noise [kPa]

11 0 -0.75 12 -0.8 0.2 -0.85 0.4 Time: t [s] Spatial Coordinate: z [m] 13

Reconstructed pressure P(z,t) [kPa] 0.6 -0.9

14 -0.95 0.8 1 -1 15 -10 -8 -6 -4 -2 0 2 4 6 8 10 Spatial Coordinate: z [cm] 320 340 360 380 400 420 440 460 Elasticity Modulus: k [kPa]

Figure 3.9: To the top left we have the wall displacement, ur, with noise. The

top right graph is of the pressure boundaries φ1 and φ2. From these we have

calculated the pressure displayed at the bottom left. Finally to the bottom right

we have the elasticity of the blood vessel, kϕϕ.

Example 3.4 Let us now see what happens if we increase the noise level to

εi,j ∼ N (0, 2 · 10−5) for ur and εj ∼ N (0, 100) for φ1 and φ2. We use a

mollification radius of δ = 0.01. In Figure 3.9 we can see that even though the noise level has increased significant we still get a good result.

For more information about this specific blood vessel problem see [2] and for more applications of mollification in biometrics and blood vessel problem see [1].

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Chapter 4

Conclusion

Some problems are ill-posed, i.e noise increases while calculating and can ruin the accuracy of the results. In order to reduce the noise a regularization method can be used to smooth out the function. Mollification is a kind of regularization method that uses integration or weighted average in order to cancel out the unwanted noise. The accuracy of mollification depends on the problem, noise level and the mollification radius. The error increases with larger noise levels

and with large kf′′_k

∞,I. The mollification radius may neither be to big nor to

small. A large radius can smooth out not only the noise but also the function. On the other side a too small radius can not sufficiently cancel out the noise.

In this paper we have studied how mollification can be used in order to calcu-late the elastics in a blood vessels wall given measured data, with measurement noise, about the blood vessel and the blood flow. Since we use the mean value

of the elasticity of the blood vessel, kϕϕ, in our calculation it means that the

final value do not get out of control and the result is good even with the noise.

We have showed that kϕϕwhen using mollification is better compared to when

not using mollification. We have also illustrated how the noise level impact the result in a way that the result gets better with a smaller noise level. The choice of the mollification radius do also matter in the result but, since mollification is a stable procedure, the difference is small.

In conclusion, mollification is an effective method to cancel out unwanted noise in order to get a more accurate result.

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Solving ill-posed problems with mollification and an application in biometrics