Mathematical preliminaries - Intrinsic dimensionality in vision: Nonlinear filter design and ap

Weber reported that two heavy weights have to differ more than two lighter weights to observe a difference in weight [81]. This corresponds to the reported size effect in numerosity estimation. Weber formulated the following law: In order to be able to distinguish two sensory stimuli, they must differ at least by a fraction k of the stimulus intensity I . The difference in stimulus intensity ∆I thus becomes

∆I = kI. (3.1)

The fraction k = ∆I/I is also referred to as the Weber fraction.

In 1860, Fechner extended Weber’s law that it is related to the perceived stimulus intensity explicitly [21]. The larger the stimulus intensity, the larger the difference in stimulus intensity must be to cause equal differences in perceived stimulus intensity. Fechner observed that the perceived stimulus intensity S is a logarithmic function of the stimulus intensity I with a constant factor k, i.e.

S = k ln(I). (3.2)

The relation extended by a constant summand is also referred to as Weber-Fechner law. This law corresponds to the distance effect in numerosity estimation. It states that an exponential increase in stimulus intensity is perceived linearly only.

The reported effects allow two possible explanations for the mental representation of number [22]. On the one hand, the numerosity can be represented linearly. Then the uncertainty in the belief in a specific numerosity increases with the absolute cardinality. On the other hand, the number line is represented logarithmically with constant uncertainty at all numerosities.

Neural findings give rising evidence for the logarithmic representation [55].

In order to avoid repetition the reader is referred to Sections 3.3 and 3.4 as further related

works and background information are considered within the context of the presented articles.

space, we start with the consideration of space curves. A space curve p is a subset of the three-dimensional space which is defined by a parametrization, e.g. p(t) = (x(t), y(t), z(t))

, t

∈

I

⊂R

. A particular case, which is of major interest for later consideration with respect to numerosity estimation, is a space curve with constant height which lies on the surface (x, y, l(x, y))

. All presented theoretical quantities are derived for this case. In each point on a space curve three important characteristic vectors can be determined: the tangent vector, the principal normal vector, and the binormal vector.

Definition 3.1

(Tangent vector). Let X : I

⊂R→R³

be a differentiable space curve. Then the tangent vector T is defined by

T (t) = X

^′

(t)

X

^′

(t)

. (3.3)

Corollary 3.2.

If a space curve is given implicitly by l(x(t), y(t)) = c with the parametrization X(t) = (x(t), y(t), l(x(t), y(t)))

, then the tangent vector is given by

T (t) =





l

−

l

0





(l

²_x

+ l

²_y

)

⁻^1/2

. (3.4)

Proof. Using the constraint l(x(t), y(t)) = c, we get

X

^′

(t) =





x

^′

y

^′

x

^′

l

(x, y) + y

^′

l

(x, y)





. (3.5)

The definition of the curve yields

x

^′

l

(x, y) + y

^′

l

(x, y) = 0. (3.6)

We thus get

T (t) =





x

^′

−^l_l^x_y^(x,y)_(x,y)

x

^′

0





(x

^′²

+ l

(x, y)

l

(x, y)

x

^′²

)

⁻^1/2

=





l

−

l

0





(l

²_x

+ l

_y²

)

⁻^1/2

. (3.7)

Definition 3.3

(Principal normal vector). At any point with T (t)

= 0 the principal normal

vector is defined by P (t) := T

^′

T

^′k

. (3.8)

Corollary 3.4.

For the implicitly given curve (l(x(t), y(t)) = c) the principal normal is given by

P (t) =

−

1 (l

_x²

+ l

²_y

)

^1/2





l

0





. (3.9)

Proof. The tangent vector for a general space curve (x(t), y(t), c)

is given by

T (t) = 1 (x

^′²

+ y

^′²

)

^1/2





x

^′

y

^′

0





. (3.10)

The derivative of the tangent vector thus becomes

T

^′

(t) = 1 (x

^′²

+ y

^′²

)

^1/2





x

^′′

y

^′′

0



−

x

^′

x

^′′

+ y

^′

y

^′′

(x

^′²

+ y

^′²

)

^3/2





x

^′

y

^′

0





= 1

(x

^′²

+ y

^′²

)

^3/2









x

^′′

x

^′²

+ x

^′′

y

^′²

y

^′′

x

^′²

+ y

^′′

y

^′²

0



−





x

^′′

x

^′²

+ y

^′′

x

^′

y

^′

x

^′′

x

^′

y

^′

+ y

^′′

y

^′²

0









= 1

(x

^′²

+ y

^′²

)

^3/2





x

^′′

y

^′²−

y

^′′

x

^′

y

^′

y

^′′

x

^′²−

x

^′′

x

^′

y

^′

0





= x

^′′

y

^′−

y

^′′

x

^′

(x

^′²

+ y

^′²

)

^3/2





y

^′

−

x

^′

0





. (3.11)

With this result the principal normal vector is given by

P (t) = 1 (x

^′²

+ y

^′²

)

^1/2





y

^′

−

x

^′

0





. (3.12)

With x

^′

= l

and y

^′

=

−

l

follows

P (t) =

−

1 (l

_x²

+ l

_y²

)

^1/2





l

0





.

Definition 3.5

(Binormal vector). The binormal vector B is defined by the vector product of th orthogonal vectors T and P , i.e.

B = T

P. (3.13)

Remark 3.6.

For the implicitly defined curve X the binormal vector is

B =

−

1 l

²_x

+ l

²_y









l

−

l

0



×





l

0









=





0 0

−

1





. (3.14)

Definition 3.7.

The triple of vectors (T, P, B) associated to each point of a continuous space curve is called the Frenet frame.

With the knowledge of the Frenet frame specific properties of the curve can be determined.

One example is the curvature of a space curve which is used subsequently to determine the desired operators.

Definition 3.8

(Curvature). Let X

∈

C(I)

be a parametrized curve. The curvature κ : I

→ R⁺

is defined as

κ(t) =

T

^′

(t)

s

^′

(t) (3.15)

where s

^′

(t) =

X

^′

(t)

. By multiplication both sides with T

^′

(t), it follows

T

^′

(t) = s

^′

(t)κ(t)P(t). (3.16)

Corollary 3.9.

If a space curve is given implicitly by l(x(t), y(t)) = c with the parametrization X(t) = (x(t), y(t), l(x(t), y(t)))

, then the curvature is given by

κ(t) = l

²_x

l

_yy

+ l

²_y

l

_xx−

2l

l

_xy

(l

²_x

+ l

²_y

)

^3/2

. (3.17)

Proof. Using T

^′

from Equation (3.11) and using P from Equation (3.12) in Equation (3.16)

yield

x

^′′

y

^′−

y

^′′

x

^′

(x

^′²

+ y

^′²

)

^3/2





y

^′

−

x

^′

0





= (x

^′²

+ y

^′²

)

^1/2

κ 1 (x

^′²

+ y

^′²

)

^1/2





y

^′

−

x

^′

0





.

With x

^′

= l

, y

^′

=

−

l

, x

^′′

= l

_yx

l

_y−

l

_yy

l

, and y

^′′

=

−

(l

_xx

l

_y−

l

_xy

l

) follows κ(t) = x

^′′

y

^′−

y

^′′

x

^′

(x

^′²

+ y

^′²

)

^3/2

= 1

(l

²_x

+ l

²_y

)

^3/2

(x

^′′

y

^′−

y

^′′

x

^′

)

= 1

(l

²_x

+ l

²_y

)

^3/2

(l

_yx

l

_y−

l

_yy

l

)

−

l

(l

_xx

l

_y−

l

_xy

l

))

= l

²_x

l

_yy

+ l

²_y

l

_xx−

2l

l

_xy

(l

²_x

+ l

²_y

)

^3/2

.

The curvature is a property of the space curve but if the the space curve is assumed to lie on a parametrized surface, further quantities can be determined with the aid of the curvature of the space curve. In order to deal with parametrized surfaces, this term is specified in the following definition.

Definition 3.10.

S

⊂R³

is called a parametrized surface if

∀

p

∈

S :

∃

U

⊂R²

, V (p)

⊂R³

, X

∈

C(U,

R³

) : X(U ) = V

∩

S (3.18) where V (p) is an open neighborhood of p. X is called a parametrization.

The parametrization plays an important role as it enables one to determine properties of the described objects, e.g. surfaces or space curves on a surface. The previously described case of interest makes use of a specific kind of parametrization. The described surface is given by the parametrization X(u, v) = (u, v, l(u, v))

. An important space in the study of curved surfaces is the tangent space which is defined with the aid of tangent vectors as follows.

Definition 3.11

(Tangent space). Let S

⊂R³

be a parametrized surface, p

∈

S, and M is the set of tangent vectors to S in p. If dim(M ) = 2, M is called the tangent space and it is denoted by T

S. Then the set

{

p + v

v

∈

T

S

}

is called the tangent plane.

This formal definition of the tangent space does not take a specific parametrization into

account. How the tangent space and the tangent plane is determined by a given

parametriza-tion and more importantly whether the tangent space exists, is considered in the following

proposition.

Proposition 3.12.

Let S

⊂R³

be a parametrized surface with the parametrization X : U

⊂ R² →R³

. Assume X is injective and p = X(u

₀

, v

₀

). Then the set of tangent vectors forms a subspace of

R³

if and only if

X

(u

₀

, v

₀

)

X

(u

₀

, v

₀

)

= 0. (3.19)

The corresponding tangent plane is then defined by

{

x

∈R³|

(x

−

p)

(X

(u

₀

, v

₀

)

X

(u

₀

, v

₀

)) = 0

}

. (3.20) Further considerations require more regularity with respect to the parametrization which is provided by a regular surface as defined in the following.

Definition 3.13.

S

⊂R³

is a regular surface if for all p

∈

S exists an open set U

⊂R²

, an open neighborhood V (p)

⊂R³

, and a surjective continuous function X : U

→

S

∩

V (p) such that

(i) X is differentiable,

(ii) X is a homeomorphism, i.e. X is continuous and X

⁻¹

exists and is also continuous, (iii) X satisfies the regularity condition (ker(DX(u, v)) =

{

0

}

).

The following proposition is an important result and guarantees in the “case of interest”

that the surface is a regular surface.

Proposition 3.14.

Let U

⊂R²

be open. Then if a function f : U

→R

is differentiable, the set

S =

{

(u, v, f (u, v))

∈R³|

(u, v)

∈

U

}

(3.21)

is a regular surface.

Similar to the principal normal vector of a space curve, the unit normal vector of a surface is defined by the vector which is orthogonal to the tangent plane.

Definition 3.15

(Unit normal vector). Let S

⊂ R³

be a regular surface, p

∈

S, X a parametrization of a neighborhood of p, and p = X(q). Then the unit normal vector is defined by

N (q) = X

_u×

X

_u×

X

_vk

(q). (3.22)

The quantities of the previous definitions are determined for the studied case in the following

remark. These findings play an important role in the subsequent derivation of the geodesic

curvature and the Gaussian curvature.

Remark 3.16.

Let l : U

⊂R² →R

be a differentiable function. The surface S

⊂R³

defined by S :=

{

X(x, y) = (x, y, l(x, y))

^T|

(x, y)

∈

U

}

is a regular surface and X : U

→ R³

is the parametrization. The tangent and unit normal vectors in p = X(x

, y

), p

∈

S, then become

X

(x, y) =





1 0 l





, X

(x, y) =





0 1 l





, and N (x, y) = 1 (1 + l

_x²

+ l

²_y

)

^1/2





−

l

1





.

(3.23) The first fundamental form is an important mapping defined with respect to the tangent space. With the aid of the first fundamental form, quantities of the intrinsic geometry of the surface can be computed. This means quantities with respect to the surface itself, e.g. length of curves on the surface or the area of regions on the surface. But more importantly in the context of this work is the relation to the Gaussian curvature.

Definition 3.17

(First fundamental form). Let S be a regular surface. The first fundamental form I

(

,

) is the restriction of the usual dot product in

R³

to the tangent plane T

S . That means I

(x, y) = x

y for all x, y

∈

T

S with respect to the standard basis of

R³

.

The first fundamental form is defined on the tangent space of the surface. But it remains unclear how the parametrization of the surface influences it. The following proposition gives an answer to this question.

Proposition 3.18.

Let I

: T

S

T

S

→R

be the first fundamental form at a point p on a regular surface S. Given a regular parametrization X : U

→ R³

, the matrix associated with the first fundamental form I

with respect to the basis

=

{

X

, X

_v}

is

g = g

₁₁

g

₁₂

g

₂₁

g

₂₂

= X

_u·

X

_u·

X

_v·

X

_v·

X

. (3.24)

The first fundamental form can be expressed by the quadratic form I

(x, y) = x

gy for x and y being points on the tangent plane and expressed in the basis

.

In literature the coefficients of the matrix associated with the first fundamental form are often denoted by E, F , and G such that

g = X

u·

X

u·

X

_v·

X

_v·

X

=: E F

F G

. (3.25)

The following formula by Brioschi relates the first and second derivatives of the coefficients

of the first fundamental form to the Gaussian curvature. In particular it gives an explicit

formula which enables one to determine the Gaussian curvature. The theorem and a proof

can be found in [26].

Theorem 3.19

(Brioschi’s formula). Let S be a regular surface with its parametrization X : U

→R³

. Then the Gaussian curvature of X is given by

K = 1

(EG

−

F

)





−¹₂

E

_vv

+ F

_uv−¹₂

G

_uu ¹₂

E

F

_u−¹₂

E

F

v−¹₂

G

E F

G

F G

−

0

¹₂

E

_v ¹₂

G

E

E F

G

F G





.

(3.26)

Corollary 3.20.

Let l : U

⊂R² →R

be a differentiable function. The surface S

⊂R³

defined by S :=

{

X(x, y) = (x, y, l(x, y))

^T|

(x, y)

∈

U

}

is a regular surface and X : U

→ R³

is the parametrization. Then the Gaussian curvature is given by

K = l

_xx

l

_yy−

l

_xy²

(1 + l

²_x

+ l

_y²

)

. (3.27)

Proof. By using Equation (3.23) the quantities E, F , and G become E = 1 + l

_x²

, F = l

l

, and G = 1 + l

²_y

with

E

= 2l

l

_xx

, E

= 2l

l

_xy

, E

_yy

= 2(l

²_xy

+ l

l

_xyy

)

F

= l

_xx

l

+ l

l

_yx

, F

= l

_xy

l

+ l

l

_yy

, F

_xy

= l

_xxy

l

+ l

_xx

l

_yy

+ l

l

_yxy

+ l

²_xy

G

= 2l

l

_yx

, G

= 2l

l

_yy

, G

_xx

= 2(l

_xy²

+ l

l

_yxx

).

With Equation (3.26) follows

K = 1

(1 + l

²_x

+ l

_y²

)





l

yy−

l

_xy²

l

_yy

1 + l

²_x

l

_yy

l

1 + l

²_y

−

0 l

l

_xy

1 + l

²_x

l

_xy

l

1 + l

_y²





= 1

(1 + l

²_x

+ l

_y²

)

(l

_xx

l

_yy−

l

²_xy

)(1 + l

_x²

)(1 + l

²_y

) + 2l

²_x

l

²_y

l

_xx

l

_yy

−

l

_y²

l

(1 + l

_x²

)

−

l

_x²

l

(1 + l

²_y

)

−

l

_x²

l

_y²

(l

l

yy−

l

²_xy

)

−

2l

²_x

l

²_y

l

²_xy

+ l

_x²

l

_xy²

(1 + l

_y²

) + l

²_y

l

_xy²

(1 + l

_x²

)

= l

_xx

l

_yy−

l

_xy²

(1 + l

²_x

+ l

_y²

)

(1 + l

_x²

)(1 + l

²_y

)

−

l

²_y

(1 + l

²_x

)

−

l

²_x

(1 + l

²_y

) + l

²_x

l

²_y

| {z }

.

Remark 3.21.

The definition of the Gaussian curvature is not given in this section as it

re-quires knowledge about the second fundamental form. The shortcut without this knowledge would not be possible without Brioschi’s formula and Gauss’ prominent Theorema Egregium, which states that the Gaussian curvature can be computed without knowledge of the sec-ond fundamental form, by using first and secsec-ond derivatives of the coefficients of the first fundamental form only. The interested reader is referred to standard differential geometry literature [5, 26, 48]. The following is a constructive description of what is meant by Gaussian curvature. From the definition it can be derived that the Gaussian curvature K is the product of the two principal curvatures k

₁

and k

₂

in each point on a regular surface S, i.e.

K = k

k

. (3.28)

Let T

S denote the tangent plane and N

₁

and N

₂

are two orthogonal planes which are both orthogonal to T

S. The curvature k

_n_i

of the intersection curves in N

is termed the normal curvature. If N

₁

and N

₂

are chosen so that the corresponding k

_n_i

become extreme values, the normal curvatures with respect to N

₁

and N

₂

are the principal curvatures k

₁

and k

₂

. The product of the principal curvatures is then the Gaussian curvature.

The last quantity which is considered describes a specific type of curvature of curves on a surface. This is the geodesic curvature. Given a curved surface and a curve on the surface.

Then the geodesic curvature in one point is the curvature of the curve which results from the projection of the curve to the tangent plane. The following lemma describes how this curvature can be determined.

Lemma 3.22

(Geodesic curvature). Let S be a regular surface with parametrization X : U

⊂ R² →R³

. And let Y : I

⊂R→

S be a space curve on S. Given the Frenet frame (T, P, B) and the unit normal vector N, the geodesic curvature can be calculated by

κ

= κP

U (3.29)

where

U = N

T. (3.30)

Corollary 3.23.

Let l : U

⊂R² →R

be a differentiable function. The surface S

⊂R³

defined by S :=

{

X(x, y) = (x, y, l(x, y))

^T|

(x, y)

∈

U

}

is a regular surface and X : U

→ R³

is the parametrization. Furthermore a space curve on S is given implicitly by l(x(t), y(t)) = c with Y (t) = (x(t), y(t), l(x(t), y(t)))

. Then the geodesic curvature of Y is given by

κ

= l

²_x

l

_yy

+ l

_y²

l

_xx−

2l

l

_xy

(l

²_x

+ l

_y²

)

^3/2

(1 + l

²_x

+ l

_y²

)

^1/2

. (3.31)

Proof. Given the vectors P , N , and T from Equations (3.9), (3.23), and (3.4), we get

U = 1

(l

_x²

+ l

²_y

)

^1/2

(1 + l

²_x

+ l

²_y

)

^1/2





l

_x²−

l

²_y





. (3.32)

With

P

U = 1

(1 + l

²_x

+ l

²_y

)

^1/2

(3.33)

and the curvature κ from Equation (3.17) it follows κ

= l

²_x

l

_yy

+ l

²_y

l

_xx−

2l

l

_xy

(l

²_x

+ l

²_y

)

^3/2

(1 + l

²_x

+ l

²_y

)

^1/2

. (3.34)

We now have the Gaussian curvature and the geodesic curvature, the elementary quantities in the Gauss-Bonnet theorem, which play the key role in the subsequent model for numerosity estimation. But the relation to the concept of intrinsic dimensionality is still unclear. The following theorem makes the relation explicit as it shows that both, the Gaussian curvature and the geodesic curvature, define i2D-operators.

Theorem 3.24.

The operators T

: C

(Ω)

→

C(Ω), i = 1, 2, with compact Ω

⊂R²

defined by

T

₁

(u)(x) :=

∂²

∂x²₁

u

_∂x^∂²₂

u

−

(

_∂x^∂²

1∂x2

u)

(1 + (

_∂x^∂

u)

+ (

_∂x^∂

u)

)

(3.35)

and

T

₂

(u)(x) :=

(

_∂x^∂

u)

²_∂x^∂²2

u + (

_∂x^∂

u)

²_∂x^∂²2

u

−

2

_∂x^∂

u

_∂x^∂

u

_∂x^∂²

1∂x2

u ((

_∂x^∂

u)

+ (

_∂x^∂

u)

)

^3/2

(1 + (

_∂x^∂

u)

+ (

_∂x^∂

u)

)

^1/2

(3.36)

are i2D-operators.

Proof. Let u

∈

C

(Ω) and x

∈

I

₀

(u)

∪

I

₁

(u) with respect to B

_ǫ

=

{

x

∈R²|k

x

< ǫ

}

for an arbitrary ǫ > 0. Then there exists a direction v

∈

S

such that for all t

∈R

with tv

∈

B

_ǫ

the following holds

u(x) = u(x + tv). (3.37)

The directional derivative in the direction of v then becomes D

u(x) = lim

t→0

u(x + tv)

−

u(x)

t = 0. (3.38)

u is continuosly differentable by definition such that it is also totally differentiable. For totally differentiable functions the following relation to the directional derivate holds

D

u =

∇x

u

v = v

₁

∂

∂x

₁

u + v

₂

∂

∂x

₂

u. (3.39)

With Equation (3.38) we obtain the relation

∂

∂x

₁

u = k ∂

∂x

₂

u (3.40)

where k :=

−

v

₂

/v

₁

. In the following it is shown that the nominator of the operators T

₁

and T

becomes zero within this setup. The nominator of T

becomes

∂

∂x

²₁

u ∂

∂x

²₂

u

−

( ∂

∂x

₁

∂x

₂

u)

= k ∂

∂x

₁

∂x

₂

u 1 k

∂

∂x

₂

∂x

₁

u

−

( ∂

∂x

₁

∂x

₂

u)

= 0.

The nominator of T

₂

becomes ( ∂

∂x

u)

∂

∂x

²₂

u + ( ∂

∂x

u)

∂

∂x

²₁

u

−

2 ∂

∂x

u ∂

∂x

u ∂

∂x

u

=(k ∂

∂x

u)

∂

∂x

u + k( ∂

∂x

u)

∂

∂x

u

−

2k ∂

∂x

u ∂

∂x

u ∂

∂x

u = 0.

Both operators thus fulfill the requirements to be an i2D-operator.

3.3 Article: Spatial numerosity: A computational model based on a

Im Dokument Intrinsic dimensionality in vision: Nonlinear filter design and applications (Seite 64-75)