Radial Basis Functions and their Applications to Landmark-based Image Registration

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Radial Basis Functions and their Applications to Landmark-based Image Registration

Hanli Qiao

Department of Mathematics hanli.qiao@unito.it, University of Turin Via Carlo Alberto 10, 10123 Torino, Italy

20th Oct 2016, Seminar in Padova

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Radial Basis Functions (RBFs)

DEFINITION:A function Φ : R^d→ R is called radial if there exsits a univariate function φ : [0, +∞) → R, such that

Φ(x) = φ(r ) here r = kxk, and k.k is Euclidean norm on R^d. PROPERTIES:

Radially or spherically symmetric about the center.

Under all euclidean transformations, (i.e, translations, rotations, reflections) radial function interpolants have nice property of being invariant.

APPLICATIONS:Can be applied in solving scattered data interpolation problem or multivariate approximation problems.

Joint work with A. De Rossi and R. Cavoretto 2 / 64

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RBFs Multivariate Interpolation Scheme

THE MULTIVARIATE INTERPOLATION SCHEMEis defined as follows: given data X = {x₁,x₂, ...,x_N}, xi∈ R^dand the corresponding values is

f_X= {fx₁,fx₂, ...,fx_N}, fx_i∈ R, i = 1, 2, ..., N. where d is the dimension of the working space and N is the number of the data sites, choosing the interpolate kernel Φ : R^d → R such that Φ(x ) = fx.Usually, the RBFs interpolant Φ(x ) is a linear combination that is

Φ(x ) =

N

X

j=1

αjφ(kx − xjk). (1)

then the parameters α can be obtained through solving the linear system Aα = fx,A(i, j) = φ(kxi− xjk).







φ(kx1− x1k) φ(kx1− x2k) · · · φ(kx1− xNk) φ(kx2− x1k) φ(kx2− x2k) · · · φ(kx2− xNk)

.. .

..

. . .. ...

φ(kxN− x1k) φ(kxN− x2k) · · · φ(kxN− xNk)











 α1

α2

.. . αN







=





 fx₁

fx₂

.. . fx_N







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Positive Definite Functions

For obtaining the unique solution of the system Aα = fx, the coefficient matrix should be NON-SINGULAR.

Strictly Positive Definite Functions → Non-singular.

DEFINITION:A real valued continuous function Φ : R^d→ R is called positive definite on R^d if for all pairwise distinct x₁,x₂, ...x_N,and for all α ∈ R^N

N

X

j=1 N

X

k =1

αjαkΦ(xj− xk) ≥0. (2)

function Φ is called strictly positive definite (SPD) when the equality of (2) holds iff α ≡0.

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Illustrations of RBFs-GSRBFs

G

LOBALLY SUPPORTED

RBF

S

(GSRBF

S

)

GSRBFs Φ(r ) Property

Gaussian e^{−r 2 /c2}, c > 0 SPD ∩C^∞(0) Matérn M(r | v ) ^21−v_{Γ(v )}(^r_c)^vKv(^r_c), v > 0 SPD ∩C^{2v −1}(0)

E

XAMPLES OF

GSRBF

S

: images of M

_5/2

with the shape parameter c = 2

(left) and c = 0.2 (right)

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Illustrations of RBFs-CSRBFs

C

OMPACTLY SUPPORTED

RBF

S

(CSRBF

S

)

CSRBFs Φ(_c^r) Property

Wendland ϕ3,1 (1 −^r_c)⁴₊(4^r_c+1),^r_c≤ 1, d ≤ 3 SPD ∩C²(0) Wu ψ1,2 (1 −^r_c)⁴₊(1 + 4^r_c+3_c^r²+³₄^r_c³),^r_c≤ 1, d ≤ 3 SPD ∩C²(0) Gneiting τ2,l (1 −^r_c)^l₊(1 + l_c^r −^(l+1)(l+4)₂ ^r_c²),r ≤ 1, l ≥⁷₂ SPD ∩C²(0)

E

XAMPLES OF

CSRBF

S

: images of τ

2,5

with the shape parameter c = 2 (left) and c = 0.2 (right)

(c)c = 2 (d)c = 0.2

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Conditionally Positive Definite (CPD) Functions

REASON:Not all popular choices of RBFs that are used fit into multivariate interpolant scheme, such as Thin Plate Spline (TPS).

DEFINITION:A continuous function Φ : R^d→ R is called conditionally positive definite of order m on R^dif

N

X

j=1 N

X

k =1

αjαkΦ(xj− xk) ≥0

for any N pairwise distinct points x1, ...,xN∈ R^d, and α = [α1, ..., αN]^T ∈ R^Nsatisfying

N

X

j=1

α_jp(x_j) =0

for any polynomial of degree at most m − 1. The function Φ is called strictly conditionally positive definite (SCPD) of order m on R^dif the quadratic form is zero only for α ≡ 0.

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Advantages of RBFs

T

HIN PLATE SPLINE

: Φ(r ) = r log r .

E

FFICIENT

: RBF is one efficient, frequently used way to solve multivariate approximation problems.

L

ITTLE RESTRICTIONS

: Its applicability in almost any dimension because there are generally little restrictions on the way the data are prescribed.

F

AST CONVERGENCE

: When data become dense, RBFs can produce high accuracy to the approximated target function in many cases.

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I MAGE R EGISTRATION

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Image Registration (IR)

IMAGE REGISTRATIONis the process of overlaying two or more images of the same scene at different times, from different viewpoints, or obtained by different sensors.

APPLICATIONS OFIR:

1 Image fusion:

Combining relevant information from two or more images into a single image.

2 Remote sensing:

Acquisition information of an object or phenomenon without making physical contact with the object–earth science, intelligence, military application.

3 Medicine:

Surgical planning, monitoring of diseases, constructing patient Atlas, computer-aided surgeries.

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Simplified Registration Example

(e)Template image (f)Reference image (g)Transformed template image

Figure 1:Simple example of IR using TPS transformation.

Given two images, which are named by

REFERENCE OR FIXING

image (right) and

TEMPLATE OR MOVING

(left) image. The aim is to determine a

CORRESPONDENCE

(a transformation function) which connects the points of

two images, so that the transformed template image is similar to reference

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Landmark-based Image Registration (LIR)

(a)Template image (b)Reference image

Figure 2:MRI brain images of an anonymous patient taken at different times.

THE PROBLEM:Find the transformation

F: R^d→ R^d,

where d = 2,3, such that each landmark of the template image is mapped to corresponding landmarks of the reference image, that isF(x^T_i )=x^R_i , orFk(x^T_i )=x^R_{k ,i}, i = 1,2,...,n, k = 1,2,...,d .

This problem can be formulated in the context of multidimensional interpolation on scattered data, and solved using the radial basis function(RBF) method.

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Two Linear Systems of LIR

In figure 2, we have 14 template landmarks marked by ◦ and the corresponding reference landmarks marked by ∗.

We denote the template landmarks as x

^T_i

= (x

_i^T

, y

_i^T

) and the reference landmarks denoted by x

^R_i

= (x

_i^R

, y

_i^R

), i = 1, 2, ..., 14.

In this talk, d = 2, therefore, we get two linear systems:

Φ

₁

(x ) = P

N

j=1

α

_1,j

φ(kx − x

j

k) = x

^R

, Φ

₂

(x ) = P

N

j=1

α

_2,j

φ(kx − x

j

k) = y

^R

.

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Two Linear Systems of LIR







φ(kx

^T₁

− x

^T₁

k) · · · φ(kx

^T₁

− x

^T₁₄

k) φ(kx

^T₂

− x

^T₁

k) · · · φ(kx

^T₂

− x

^T₁₄

k)

.. . . .. .. .

φ(kx

^T₁₄

− x

^T₁

k) · · · φ(kx

^T₁₄

− x

^T₁₄

k)











 α

_1,1

α

_1,2

.. . α

_1,14







=





 x

₁^R

x

₂^R

.. . x

₁₄^R





 (3)







φ(kx

^T₁

− x

^T₁

k) · · · φ(kx

^T₁

− x

^T₁₄

k) φ(kx

^T₂

− x

^T₁

k) · · · φ(kx

^T₂

− x

^T₁₄

k)

.. . . .. .. .

φ(kx

^T₁₄

− x

^T₁

k) · · · φ(kx

^T₁₄

− x

^T₁₄

k)











 α

_2,1

α

_2,2

.. . α

_2,14







=





 y

₁^R

y

₂^R

.. . y

₁₄^R





 (4)

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Radial Basis Function Transformations

Generally, applying a radial basis function approach, the general coordinate of the transformation F_k(x ), k = 1,2,...,d , (for simplicity, we write F (x ) instead of it, the same as following) is assumed to have the form

F (x) = ϕ(x) + p(x),

⇓ F (x) =

N

X

i=1

αiΦ(kx − x^T_i k) +

M

X

j=1

a_jπj(x),

where kx − x^T_ik is the Euclidean distance from x to x^T_i,and α_iand a_jare coefficients.

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Specific Functions of Gneiting and Mat´ ern Families

In 2002, Gneiting obtained a family of compactly supported functions, which started with Wendland’s functions, for example

ϕ

_s+2,1

= (1 − r )

^l+1₊

[(l + 1)r + 1]

and using the turning bands operator

τ

s

(r ) = ϕ

s+2

(r ) + r ϕ

s+20

(r ) s

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Two Special Gneiting Functions

Now we set s = 2, the formula of Gneiting function is

τ

_2,l

(r ) = (1 − r )

^l₊

[1 + lr − (l + 1)(l + 4) 2 r

²

] Both of them are in C

²

(R) and SPD when l ≥ 7/2.

A. τ

2,7/2

(

^r_c

) .

= (1 −

_c^r

)

^7/2₊

(1 +

⁷₂_c^r

−

¹³⁵₈

(

^r_c

)

²

) B. τ

_2,5

(

^r_c

) .

= (1 −

_c^r

)

⁵₊

(1 + 5

_c^r

− 27(

_c^r

)

²

)

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Graphs of two Gneiting functions

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Mat´ ern Family

Mat´ern functions have received a great deal of attention recently and they have the following form M(r | v , c) =²_{Γ(v )}^1−v _c^rv

Kv r c , Here Kv is Modified Bessel Functions of the second kind of order v .

Kn(x ) = (_2x^π)^(1/2)e^{(−x )}

1 +⁽⁴ⁿ_{1(8x )}²⁻¹²⁾

1 +⁽⁴ⁿ_{2(8x )}²⁻³²⁾

1 +⁽⁴ⁿ_{3(8x )}²⁻⁵²⁾(...)

,

and c is the coefficient to determine the width or the support of functions. The specific three kinds of Mat´ern functions are listed:

C. M1/2 .

=e^{−r /c} D. M_3/2 .

= (1 +_c^r)e^{−r /c} E. M_5/2= (1 +. _c^r +¹₃^r²

c²)e^{−r /c}

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T OPOLOGY P RESERVATION

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Criteria to Evaluate Topology Preservation Performances

The optimal shape parameter c^∗, which means the minimum value to preserve the topology.

In different cases, the calculations are various.

1

when support size (or shape parameter) c is larger than c

^∗⇒

topology preservation is ensured,

but the locality deformation is extended.

2

when c is less than c

^∗⇒

topology violation occurs in the deformation.

The Jacobian matrix of transformation.⇒if the determinant of Jacobian matrix is closer 1, then the topology is better preserved

One-landmark, Two-landmarks and Four-landmarks matching.

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One-landmark Matching

In this case, necessary conditions to have topology preservation are:

1

Continuity of the function F

2

Positivity of the Jacobian determinant at each point

⇓

The coordinates of transformation are F

1

(x)=x + ∆

x

Φ(||x − p||), F

2

(x)=y + ∆

y

Φ(||x − p||),

⇓

Requiring the determinant of the Jacobian is positive, we obtain det(J(x , y ))=1 + ∆

x∂Φ

∂x

+ ∆

_y^∂Φ_∂y

> 0,

⇓

∆

^∂Φ_∂r

> −

^√¹

2

.

here ∆ = max{∆

x

, ∆

_y

}. c

^∗

can be estimated based on (

^∂Φ_∂r

)

_min

, r = p

x

²

+ y

²

.

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Evaluation of c ^∗

The advantage of having small C is that the influence area of each landmark turns out to be small. This allows us to have a greater local control.

ϕ

_3,1

ψ

_1,2

τ

_2,7/2

τ

_2,5

c > 2.98∆ c > 2.80∆ c > 5.09∆ c > 6.26∆

Table 1:Minimum support size for various CSRBFs, where d = 2.

Gaussian M

_1/2

M

_3/2

M

_5/2

c > 2.42∆ c > 1.10∆ c > 0.52∆ c > 0.3960∆

Table 2:Minimum support size for GSRBFs, where d = 2.

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Numerical Results: One-landmark

The considering grid is 40 × 40 in [0, 1]

²

. Mapping the template landmark (0.5, 0.5) into the corresponding reference landmark (0.6, 0.7) to

transform the grid.

Here we show two examples:

i

topology preservation results with c

^∗

ii

results with a value c which leads topology condition violation In both examples, ∆ = 0.2.

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Numerical Results: Topology Preservation for GSRBFs

(c)Gaussian c = 0.5 (d)Mat´ern,M1/2 c = 0.22

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Numerical Results: Topology Preservation for CSRBFs

(g)Wendland ϕ3,1, c = 0.6 (h)Wu ψ1,2, c = 0.58

(i)Gneiting τ2,7/2, c = 1.02 (j)Gneiting τ2,5, c = 1.26

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Numerical Results: Topology Violation for GSRBFs

(k)Gaussian c = 0.34 (l)Mat´ern,M1/2 c = 0.09

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Numerical Results: Topology Violation for CSRBFs

(o)Wendland ϕ3,1, c = 0.25 (p)Wu ψ1,2, c = 0.25

(q)Gneiting τ2,7/2, c = 0.25 (r)Gneiting τ2,5, c = 0.25

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Two-landmarks Matching

In this model, two landmarks are given as P = {(0, 0), (d , l)} and Q = {(0, ∆), (d , l − ∆)}.

In this case, the displacements along x- and y-coordinate are same but with opposite directions. ∆ < max{d , l}.

1

the locality parameter c is chosen large enough to ensure the influence regions of the two landmarks intersect each other

2

small locality parameters result in a non-preserving topology similar to the one-landmark matching case

Let us now consider components of a generic transformation F : R²→ R²obtained by a transformation of two points, i.e,

1

F

1

(x) =x + α

1,1

Φ(||x − x

^T1

||) + α

1,2

Φ(||x − x

^T2

||),

2

F

2

(x)=y + α

2,1

Φ(||x − x

^T1

||) + α

2,2

Φ(||x − x

^T2

||).

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Determinant of Jacobian Matrix

For obtaining α

1,1

,α

1,2

,α

2,1

and α

2,2

, we require that

F₁((0, 0)) = 0, F₁((d , l)) = d ,

F₂((0, 0)) = ∆, F₂((d , l)) = l − ∆.

Solving these two systems of two equations in two unknowns, we get

α1,1=0, α1,2=0, α2,1= ^∆

1−Φ√

d²+l², α2,2= −α2,1.

It follows that the determinant of the Jacobian matrix is

det (J(x , y )) = 1 + α_2,1^∂Φ

√

x²+y²

∂y + α_2,2^∂Φ

√

(x −d )²+(y −l)²

∂y ,

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Minimum Value of Jacobian Matrix Determinant

The minimum value is occurred at the midpoint between P

1

and P

2

, i.e.,

^d₂

,

₂^l

, when

∆ > 0 and the intersection of the influence regions of two landmarks does not turn out to be negligible. We thus obtain the optimal locality parameter when

det J

^d₂

,

₂^l

= 0.

Obviously, one can observe that

∂Φ

√

x²+y²/c

∂y

x =^d₂,y =₂^l

= −

^∂Φ

√

(x −d )²+(y −l)²/c

∂y

x =^d₂,y =₂^l

,

so we get

det J

^d₂

,

₂^l

= 1 + 2α

2,1

∂Φ

√

x²+y²/c

∂y

x =^d₂,y =₂^l

.

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det J ^d ₂ , ₂ ^l of Various RBFs

Radial Basis Functions det

J

d 2,₂^l

Gneiting τ2,7/2 det J ^d₂,₂^l = 1 − ⁹⁹

16c2

∆l 1−

√

d 2 +l2 2c

!5/2 8−15

√

d 2 +l2 2c

!

1− 1−

√

d 2 +l2 c

!7/2 1+ 72

√

d 2 +l2 c − 135

8 d 2 +l2

c2

!,

Gneiting τ2,5 det J ^d₂,₂^l = 1 −^21∆l

4− 72z1/2

c −3 z c2+ 194 z3/2

c3 −2 z2 c4+ 92z5/2

c5

42z−175 z3/2 c +315 z2

c2−294 z5/2 c3 +140 z3

c4−27 z7/2 c5

,

Mat´ern M1/2 det J ^d₂,₂^l = 1 − ^2l∆e−

√

d 2 +l2 /2c c

√

d 2 +l2 1−e−

√

d 2 +l2 /c

!

Mat´ern M3/2 det J ^d₂,₂^l = 1 − ^l∆e−

√

d 2 +l2 /2c c2 1− 1+

√

d 2 +l2 c

!!

e−

√

d 2 +l2 /c

Mat´ern M5/2 det J ^d₂,^l₂ = 1 −

l∆e−

√

d 2 +l2 /2c 1+

√

d 2 +l2 2c

!

3c2 1− 1+

√

d 2 +l2 c + d2 +l2

3c2

!!

e−

√

d 2 +l2 /c

.

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Evaluation of c ^∗

(s)d = l = 32 (t) d = l = 32

Figure 3:Comparing support size c with fixed d ,l and different ∆

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(a)d = 32, l = 64 (b)d = 32, l = 64

Figure 4:Comparing support size c with fixed d ,l and different ∆

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Evaluation of the Jacobian Determinant

By varying the displacement of such points, we can compare results by two criteria:

1 The number of points where the determinant of the Jacobian is negative

⇒Such number indicates the size of the region with violated topology preservation.

2 The average of the negative Jacobian determinants

⇒This parameter represents the severity of topology violation,The more the value is negative, the more the transformation might be bent or broken compared to the original structure.

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Evaluation of the Jacobian Determinant–Result 1

(a)d = l = 32,∆ = 28 (b)d = l = 32,∆ = 28

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(c)d = l = 32,∆ = 28 (d)d = l = 32,∆ = 28

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Evaluation of the Jacobian Determinant–Result 2

(e)d = 32,l = 64,∆ = 28 (f)d = 32,l = 64,∆ = 28

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(g)d = 32,l = 64,∆ = 28 (h)d = 32,l = 64,∆ = 28

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Evaluation of the Jacobian Determinant–Result 3

(i)d = 8,l = 0,∆ = −8 (j)d = 8,l = 0,∆ = −8

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(k)d = 8,l = 0,∆ = −8 (l)d = 8,l = 0,∆ = −8

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Evaluation of the Jacobian Determinant–Result 4

(m)d = 8,l = 0,∆ = −16 (n)d = 8,l = 0,∆ = −16

Figure 5:Comparison of the negative number and the average of Jacobian determinant

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(a)d = 8,l = 0,∆ = −16 (b)d = 8,l = 0,∆ = −16

Figure 6:Comparison of the negative number and the average of Jacobian determinant

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Numerical Results: Two-landmarks

Considering 40 × 40 grid in [0, 1]

²

and then compare results obtained by the grid distortion in the shift case of two landmarks

{(0.375, 0.350), (0.625, 0.55)} in {(0.375, 0.5), (0.625, 0.4)}

respectively.

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Numerical results: Two-landmarks for GSRBFs

(a)Gaussian, c = 0.5 (b)Mat´ern,M1/2, c = 0.25

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Numerical Results: Two-landmarks for CSRBFs

(e)Wendland ϕ3,1, c = 0.5 (f)Wu ψ1,2, c = 0.5

(g)Gneiting τ2,7/2, c = 0.5 (h)Gneiting τ2,5, c = 0.5

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Bad Results: Two-landmarks for the Whole Deformation

(i)M1/2, c = 0.5 (j)M3/2, c = 0.5 (k)M5/2, c = 0.5

Figure 7:Bad results of Mat´ern transformations

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Four-landmarks Matching

1 In the following we compare topology preservation properties for globally supported transformations.

2 Considering four inner landmarks in a grid, located so as to form a rhombus at the center of the figure, and we suppose that only the lower vertex is downward shifted of ∆

3 The landmarks of template and reference images are P = {(0, 1), (−1, 0), (0, −1), (1, 0)}

and Q = {(0, 1), (−1, 0), (0, −1 − ∆), (1, 0)}, respectively, with ∆ > 0.

4 Let us now consider components of a generic transformationF : R²→ R²obtained by a transformation of four points P1, P2, P3and P4, namely

F₁(x) = x +

4

X

i=1

α1,iΦ(||x − P_i||),

F2(x) = y +

4

X

i=1

α2,iΦ(||x − Pi||).

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Optimal Locality Parameter c ^∗ in Four-landmarks

The coefficients α

1,i

and α

2,i

are obtained so that the transformation sends P

i

to Q

i

, with i = 1, . . . , 4. To do that, we need to solve two systems of four equations in four unknowns, whose solutions are

α

1,1

= 0, α

1,2

= 0, α

1,3

= 0, α

1,4

= 0, and

α

2,1

=

(1−β)[(1+β)^β²^+β−2α²−4α² ²]

∆, α

2,2

= α

(1 + β)

²

− 4α

²

∆, α

2,3

= −

^1+β−2α²

(1−β)[(1+β)²−4α²]

∆, α

2,4

= α

2,2

,

where α = Φ

^√

2 c

and β = Φ

²_c

. For simplicity, we denote

Φ

1

= Φ(||(x , y ) − P

1

||/c), Φ

2

= Φ(||(x , y ) − P

2

||/c), Φ

3

= Φ(||(x , y ) − P

3

||/c)

and Φ

4

= Φ(||(x , y ) − P

4

||/c).

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Four-landmarks: the Determinant of Jacobian matrix

The determinant of the Jacobian matrix is det (J(x , y )) = 1 + P

4

i=1

α

2,i∂Φ_i

∂y

.

The minimum Jacobian determinant is obtained at position (0, y ), with y > 1. In the following, we analyse the value of the Jacobian determinant at (0, y ), with y > 1, for different RBFs. Since the support c is very large, in order to have a global

transformation, we consider || · ||/c to be infinitesimal and omit terms of higher order.

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det (J(0, y )) of Various RBFs

Radial Basis Functions det (J(0, y )) Wendland ϕ3,1 det (J(0, y )) ≈ 1 − 0.6402∆

y²+1 − yp y²+1

, Wu ψ1,2 det (J(0, y )) ≈ 1 − 0.6402∆

y²+1 − yp y²+1

Gaussian det (J(0, y )) ≈ 1 −^y₂∆,

Gneiting τ2,7/2 det (J(0, y )) ≈ 1 − 0.6402∆

y²+1 − yp y²+1

, Gneiting τ2,5 det (J(0, y )) ≈ 1 − 0.6402∆

y²+1 − yp y²+1

, Mat´ern M1/2 det (J(0, y )) ≈ 1 − 2.4142∆

−1 +√^y

y 2 +1

. Mat´ern M3/2 det (J(0, y )) ≈ 1 − 1.7071∆

y²+1 − yp y²+1

. Mat´ern M5/2 det (J(0, y )) ≈ 1 − 2.5607∆

y²+1 − yp y²+1

.

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Evaluation of det (J(0, y ))

(a)CSRBFs (b)RBFs

Figure 8:Value of det(J(0, y )), with y > 1, by varying RBFs.

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Numerical Results: Four-landmarks

Considering the same 40 × 40 grid in [0, 1]

²

and compare results obtained by its distortion, which is created by the shift of one of the four landmarks distributed in rhomboidal position.

The template landmarks are

{(0.5, 0.65), (0.35, 0.5), (0.65, 0.5), (0.5, 0.35)}

and are respectively transformed in the following reference landmarks

{(0.5, 0.65), (0.35, 0.5), (0.65, 0.5), (0.5, 0.25)}

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Numerical Results: Four-landmarks for GSRBFs

(a)Gaussian, c = 100 (b)Mat´ern,M1/2, c = 100

(c)Mat´ern,M3/2, c = 100 (d)Mat´ern,M5/2, c = 100

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Numerical Results: Four-landmarks for CSRBFs

(e)Wendland ϕ3,1, c = 100 (f)Wu ψ1,2, c = 100

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Numerical Results of Medical Brain Images

(i)Gaussian, c = 4 (j)Mat´ern,M1/2, c = 4

(k)Mat´ern,M3/2, c = 4 (l)Mat´ern,M5/2, c = 4

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Numerical Results of Medical Brain Images

(m)Wendland ϕ3,1, c = 4 (n)Gneiting τ2,7/2, c = 4 (o)Gneiting τ2,5, c = 4

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R EAL I MAGE S OFTWARE

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FAIR-Software to Implement LIR

The software based onFAIRis used to implement LIR, which was created by J. Modersitzki in 2009.

The process for satisfying LIR is described as the following steps:

1

Read into images–setupIMMAGINI.m

2

Choose landmarks manually–getLandmarks.m

3

Calcuate the RBF interpolant–getRBFcoefficients.m

4

Register the template image and view the results–evalRBF.m,

viewImage2D.m

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How to Implement the Software on MATLAB

(s) E5-2D-Gneiting

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The Deformed Grid and the Intuitive Result: OPPOSITE

(t) E5-2D-TPS

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Comparison of CPU Time for Various RBFs in Hand Case

RBFs Gaussian M1/2 M3/2 M5/2

CPU Time/S

0.3559 0.3593 0.3287 0.3509

RBFs ψ1,2 ϕ3,1 τ2,7/2 τ2,5

CPU Time/S

1.9173 1.8389 1.8708 1.9067

Table 3:

The CPU time is the average value of 5 experiments. The hand image has 310 × 310 pixels.

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References

[1]J. MODERSITZKI, 2009, FAIR: flexible algorithms for image registration. SIAM Publisher.

[2]G.E. FASSHAUER, 2007, Meshfree Approximation Methods with MATLAB, World Scientific Publishers, Singapore.

[3]X. YANG ANDZ. XUE ANDX. LIU ANDD.R. XIONG, 2011, Topology preservation evaluation of compact-support radial basis functions for image registration, Pattern Recognition Letters, 32, 1162–1177.

[4]H. WENDLAND, 2005, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge.

[5]R. CAVORETTO,ETC, 2014, Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration, LNCS: curves and surfaces 9213: 96-108.

[6]T. GNEITING, W. KLEIBER,ANDM. SCHLATHER, 2010, Mat´ern Cross-Covariance Functions for Multivariate Random Fields, American Statistical Association105(491): 1167–1177.

[7]J. GNEITING, 2002, Compactly Supported Correlation Functions, Journal of Multivariate Analysis - J MULTIVARIATE ANAL,83(2): 493–508.

[8]B. MAT´ERN, 1986, Stochastic models and their application to some problems in forest surveys and other sampling investigations, Spatial Variation, Lecture Notes in Statistics, vol. 36, University of California.

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Thank you for your attention