Curve shortening and its applications

Contents v2.1 Curve evolution and Level set representation of curve evolution.. frame-The last chapter in this thesis will give an application of the curve shorteningprocess called aniso

Curve Shortening and its Application

Mohamed Faizal

(B.Science (Hons), NUS)

A thesis submitted for the degree in Master of Science

Associate Professor Xu Xingwang

Department of Mathematics

National University of Singapore

2006-2007

First and foremost, I would like to thank my mummy for being my support forundertaking this postgraduate course and also for being with me even when I wasbeing so unreasonable To my sister and brother in law, thank you for alwaysinviting me over for dinner

To my amazing supervisor, Associate Professor Xu Xingwang He showed metruly what it meant to do research in mathematics Sir I would like to thank you forguiding me and always being patient with me even when I really do not understandsome ideas Sir, you always saying “It’s okay” whenever I face a problem amazesme

I would also like to thank Jiaxuan for being by my side throughout my Master’syears and for bringing me laughter and the strength to carry on Thank you formaking me smile when my days have no more cheer Remember this phrase?

“Things may be tough now but I am sure that the fruits will be good.”

ii

Acknowledgements iii

To Ghazali, for helping me with technical issues that I seem to frequently faceduring my years in NUS

Now for all my dear friends who have been with me all these years Firstly

to my University friends, like Adeline, Eugene, Joan, Kelly, Liyun and Weiliang.Thanks for all the dinners where we laughed and joked about anything under thesun

To Kayjin, Meng Huat and Xiong Dan for helping me whenever I found myself

in dire straits with Mathematics To my old buddy, Terence Thank you forconsistently bugging me to continue running those long runs with you To myyounger “sisters” Weini, Han Ping and Xing En, never give up trying!

To my dear brothers from FCBC, like Eugene, Joshua, John, Vincent, Kelvinand Harold Thank you for listening to me grumble about my life

Finally to Him who is Almighty Thank You for always guiding me in my lifeand always giving me light during my darkest times

1.1 Partial Differential Equations 11.2 Differential Geometry 41.3 Robust Statistics 6

iv

Contents v

2.1 Curve evolution and Level set representation of curve evolution 92.2 Curve shortening 172.3 Geometric Heat Flow and Preservation 42

3.1 Anisotropic Diffusion 473.2 A Robust Statistical View of Anisotropic Diffusion 62

frame-The last chapter in this thesis will give an application of the curve shorteningprocess called anisotropic diffusion The first section will cover on the basic theory

of anisotropic diffusion and there will be examples that describe the functionality

of it The last section of this chapter will give reasons to why the anisotropicdiffusions works and how certain types work better than others

vi

Author’s Contribution

The author expanded on the proofs of the various lemmas and theorems found

in Chapter 2 The proof of Lemma 2.2.1 was extended from the original paper.Details were shown completely In Section 2.3, the proofs were given in detail

In Chapter 3, the author had independently tested the various functions, c(x)using Matlab programming Furthermore, the author independently came up with

a new function of c(x) with the properties needed

The author also verified that the properties were met Finally the author rantests using the new function

vii

List of Figures

2.1 Graphical representation of the inequality and the secant line 30

2.2 Reflection of C1 through the origin 36

3.1 Image with Gaussian filter of variance = 1 48

3.2 Image with Gaussian filter of variance = 25 49

3.3 Image with Gaussian filter of variance = 100 49

3.4 Notation of discretisation for anisotropic diffusion 51

3.5 The image I0 and its edge representation 52

3.6 After applying anisotropic diffusion using c(x) = e−x2k2 53

3.7 Enlarged edge images 54

3.8 The difference after applying anisotropic diffusion using c(x) = 1

1+x2

k2

55

viii

List of Figures ix

3.9 The difference after applying anisotropic diffusion using the Tukey

Biweight 57

3.10 Enlarged edge images 58

3.11 The difference after applying anisotropic diffusion using c(x) defined in (3.9) 60

3.12 The differences after applying anisotropic diffusion with Tukey Bi-weight 62

3.13 Local neighbourhood of pixels at a boundary 63

3.14 c(x) = e−x2k2 68

3.15 ψ(x) = xe−x2k2 68

3.16 ρ(x) = k22e−x2k2 69

3.17 c(x) = 1 1+ x2 2σ2 70

3.18 ψ(x) = 2x 2+x2 σ2 70

3.19 ρ(x) = σ2log(1 + 2σx22) 71

3.20 c(x) 72

3.21 ψ(x) 73

3.22 ρ(x) 73

3.23 c(x) = |erfc(|x|) cos(x)| 75

3.24 ψ(x) = x|erfc(|x|) cos(x)| 75

3.25 c(x) in a restricted domain of 0,3π 4  76

List of Figures x

3.26 The edge representation after applying anisotropic diffusion with

different c(x) for k = 100 773.27 The edge representation after applying anisotropic diffusion with

different c(x) for k = 1000 78

Chapter 1

Basics in PDEs, Differential

Geometry and Robust Statistics

In this chapter, we will give some basic results on partial differential equations,differential geometry and robust statistics The results in this chapter will be statedwithout proof and are taken from [Strauss], [Hampel & Stahel] and [Nasraoui]

We first define the solution of the heat equation

Definition 1.1.1 A function u(x, t) on (x, t) ∈ R2× R+ is a solution of the heatequation if for some c > 0, u(x, t) satisfies

∂u

∂t = c4u(x, t),

Secondly, we define Gaussian filters

1

1.1 Partial Differential Equations 2

Definition 1.1.2 A gaussian filter is a filter such that for any x ∈ R2,

G(x, t) = e

−|x|2

2σ2t

(4πσ2t),where σ, a constant, is the standard deviation of the normal distribution

Note that the Gaussian filter satisfies the heat equation

Theorem 1.1.3 (Green’s Theorem) Over a simply connected region D withboundary ∂D, Green’s Theorem states

where f and g are two functions of (x, y)

In Chapter 3, we would be using a numerical scheme that is derived from astandard numerical scheme for solving differential equations This is called thegradient descent method

Theorem 1.1.4 (Gradient Descent Method) If a vector-valued function F (x)

is defined and differentiable in a neighbourhood of a point a, then F (x) increasesfastest if one goes from a in the direction of the gradient of F at a, i.e., ∇F (a).Thus it follows that if

b = a + γ∇F (a),for γ > 0 small enough then, F (a) 6 F (b) Thus the numerical scheme would be

by starting at an initial point x0 and consider the sequence x0, x1, x2, we have

xn+1 = xn+ γn∇F (xn), n > 0

Thus the sequence xn will eventually converge to a local maximum

1.1 Partial Differential Equations 3

Now we will give results such as the Blaschke Selection theorem and Hausdorffmetric

Definition 1.1.5 Let S be a non-empty convex subset of R2 Then for any given

δ > 0, the parallel body Sδ is defined as

Sδ := [

s∈S

K(s, δ),where K(s, δ) = {x : d(x, s) 6 δ} and d(·, ·) is the ordinary Euclidean planedistance

Definition 1.1.6 Let S and R be non-empty compact convex subsets of R2 Thenthe Hausdorff distance between S and R is defined as

D(S, R) := inf{δ : S ⊂ Rδ and R ⊂ Sδ}

We now list some standard definitions that would be needed later

Definition 1.1.7 1 A sequence {Si} of compact convex subset of R2 is said

to converge (in the Hausdorff metric) to the set S if

lim

i→∞D(S, Si) = 0

2 Let C be the collection of non-empty compact subsets of R2 A subcollection

M of C is uniformly bounded if there exists a disk, k in R2 such that every

m ∈ M, m ⊂ k

3 A collection of subsets {Si} is decreasing if St⊆ Sτ, for all t > τ In the case

of curve evolution, the collection of laminae H(t) associated with the curvesC(t) is called decreasing if H(t) ⊆ H(τ ) for all t > τ

We now state the well-known Blaschke Theorem

Theorem 1.1.8 (Blaschke Theorem) Let M be a uniformly bounded infinitesubcollection of C Then M contains a sequence that converges to a member of C

1.2 Differential Geometry 4

In this section we will cover some basics on planar differential geometry

Definition 1.2.1 A parameterised planar curve is a map given as

C : I → R2,where I = [a, b] ∈ R Thus for each value p ∈ I, we obtain a point C(p) on theplanar curve

Definition 1.2.2 We define the classical Euclidean length as k · k = ph·, ·i, whereh·, ·i is the usual inner product

There are two important objects when we talk about differential geometry Thefirst is the concept of arc length

Definition 1.2.3 Given any p ∈ I, we define the arc length of a parameterisedplanar curve from a point p0 as

2

+ dydp

1.2 Differential Geometry 5

dC

 dCdp

  dpds

,

2

+ dydp

2

Now that we have discussed on the two main properties for differential geometry,

it would be natural to discuss on the derivatives of the curve, C(s)

Definition 1.2.5 We define the unit tangent of a curve C(s), ~T as

dC

ds = ~T Definition 1.2.6 We define the unit normal of a curve C(s) as

hCs, Cssi = 0

1.3 Robust Statistics 6

In this section, we will study some aspects of robust statistics and also the keyidea of robust statistics But first we will loosely define what robust statistics is.Robust statistics is the study of statistics that relates to the deviation fromidealised assumptions that encompasses the rejection of outliers

We give the aim of robust statistics:

• To describe the structure best fitting the bulk of the data

• To identify deviating data points (outliers) or deviating substructures forfurther treatment if desired

• To identify and give warning about highly influential data points

We define deviations from statistical data as gross errors Gross errors are errorswhich occur occasionally but have a large impact to the data They are the mostfrequent reasons for outliers, i.e., data which are far away from the bulk of thedata

There are a few approaches to robust statistics and we will discuss those based

1.3 Robust Statistics 7

where we have k variables observed on n cases and βj for j = 0, , k, are theregression coefficients that measure the average effect on the regression of the unitchange in xij and i ∼ N (0, σ)

Note that βj are unknown and need to be estimated Classically, the elements

of β are estimated by using the method of least squares given as

1 ρ(x) > 0 for all x and has a minimum at 0

1.3 Robust Statistics 8

Definition 1.3.2 The influence function, ψ(x) is the derivative of ρ(x) i.e.,

ψ(x) = ρ0(x),and is bounded

Definition 1.3.3 The robust error norm in robust statistics is denoted as ρ(x)with the above mentioned properties

Chapter 2

Geometric Curves

In this chapter, we will give a theoretical explanation towards geometric curvesand its evolution over time We will then show the relationship between geometriccurve evolution and anisotropic diffusion in the next chapter For this chapter, weuse sources [Gage1], [Gage2] and [Sapiro & Tannenbaum]

2.1 Curve evolution and Level set representation of curve evolution 10

Definition 2.1.1 We say that C(p, t) is a curve that evolves if it satisfies thefollowing PDE,

∂C(p, t)

with the initial condition C(p, 0) = C0(p), where ~T represents the unit tangentdirection of the curve and ~N represents the normal unit direction α(p, t) is thespeed in the tangent direction and β(p, t) is the speed in the normal direction

Lemma 2.1.2 The curve evolution after reparametrization, also satisfies

∂C(p, t)

∂t = β(p, t) ~N (p, t).

Proof We will prove this lemma by first trying to modify the tangential velocityα(p, t) with a new one If this alteration does not affect the curve motion, we aredone

We begin by reparameterizing the curve to ˜C(˜p, τ ) = C(p, t) such that ˜p =

We rewrite the first term on the right hand side, with respect to the Euclideanarc length, s, and we get

2.1 Curve evolution and Level set representation of curve evolution 11

We can also prove this locally

Corollary 2.1.3 (Local property) Let y = γ(x, t) be a local representation ofthe curve C(p, t) then

2.1 Curve evolution and Level set representation of curve evolution 12





+

x

"

α

2.1 Curve evolution and Level set representation of curve evolution 13



+

We can also define this curve as a level set of a Lipschitz function, u(x, y, t) :

R2× [0, T ) → R, in the form

Lc := {(x, y, t) ∈ R2× [0, T ) : u(x, y, t) = c}, (2.6)where c ∈ R We define the initial curve of the level set representation as u0(x, y)

2.1 Curve evolution and Level set representation of curve evolution 14

Thus we have to find the evolution of u(x, y, t) such that C(x, y, t) = Lc(x, y, t)

We do this to ensure that the curve C(p, t) evolves with the level sets of u(x, y, t)

By differentiating the equation u(x, y, t) = c with respect to t, we get

Thus ∂u∂t = βk∇uk

We shall now show that as a closed curve evolves, depending on what β is, willbecome more circular Before that we shall have the following preceding state-ments

2.1 Curve evolution and Level set representation of curve evolution 15

Definition 2.1.4 The support function of any closed convex curve with a chosenpoint as an origin that is contained in the area bounded by the curve, denoted asr(s), where s is the Euclidean arc length, is given as

r(s) =DC(s), − ~N (s)E

We also use L, A to represent the length of C(p, t) and the area enclosed byC(p, t) respectively Furthermore, κ(s) is the curvature of C(p, t) with respect tothe arc length, s

The next few lemmas states that the length and the area of C(p, t) can beexpressed in terms of r(s)

Lemma 2.1.5 The area enclosed by C is given by

2.1 Curve evolution and Level set representation of curve evolution 16

Lemma 2.1.6 The length of C is given as

L 0

By squaring both sides we get

∂p

as seen inthe Chapter 1

We begin by computing

∂C

∂p

∂C

∂p

∂C

∂p

∂C

∂p

... data-page="29">

2.2 Curve shortening 19

Thus we get

∂C

∂p

∂C

∂p

2

On the other hand, we also have