ter Haar Romeny, EMBS Berder 2004

Gaussian derivative kernels

x222

2

, x2

22 x

2 3,

x2

22x x

2 5,

x2

22 xx2 3 22 7

, x2

22x4 6 2 x2 3 42 9

The algebraic expressions for the first 4 orders:

ter Haar Romeny, EMBS Berder 2004

4 2 2 40.5

0.5

1

Order4

4 2 2 4

2

1

1

2

Order5

4 2 2 4

6

4

2

2

4Order6

4 2 2 4

10

5

510

Order7

4 2 2 4

0.1

0.2

0.3

0.4Order0

4 2 2 4

0.2

0.1

0.1

0.2

Order1

4 2 2 4

0.40.30.20.1

0.1

Order2

4 2 2 4

0.4

0.2

0.2

0.4

Order3

1, x

2

,x2 2

4, x3 3 x2

6,x4 6 2 x2 3 4

8

The factors in front of the Gaussian are the Hermite polynomials.

ter Haar Romeny, EMBS Berder 2004

They are named after Charles Hermite, a brilliant French mathematician(1822-1901).

nex2

xn

1n Hnxex24 2 2 4

1000

500

500

1000

4 2 2 4

30

20

10

10

20

30

The Hermite polynomials explode, but theGaussian hull suppresses any order.

ter Haar Romeny, EMBS Berder 2004

4 2 2 4

2 108

1 108

1 108

2 108

4 2 2 4

2 108

1 108

1 108

2 108The Gaussian kernelis not the envelope ofthe signal!

The Hermite polynomials are orthogonal functions, the Gaussianderivative functions are not!

ter Haar Romeny, EMBS Berder 2004

12 22

2 ,

12 2 2

2 ,

12 2 22

2

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1

Gaussian derivatives in the Fourier domain:

They act likebandfilters.

ter Haar Romeny, EMBS Berder 2004

Zero crossings of Gaussian derivative functions

5 10 15 20Order

7.5

5

2.5

2.5

5

7.5

Zeros of

HermiteH

10 20 30 40 50Order

2

4

6

8

10

12

14

Widthof Gaussian

derivativein

Range estimation byZernicke (1931)

ter Haar Romeny, EMBS Berder 2004

Correlation between Gaussian derivative kernels

6 4 2 2 4 6

100

50

50

100

Order 20

6 4 2 2 4 6

3000

2000

1000

1000

2000

3000

Order24

4 2 2 4

100

50

50

100

Order 20

4 2 2 4

200

100

100

200

Order 21

correlated

uncorrelated

ter Haar Romeny, EMBS Berder 2004

rn,m 1nm2 1

2 1nm1 mn1

2

12

12n1 2n1

21

2 12m1 2m1

21nm2 mn1

2

2n122m1

2

rn,m

gnxgmxx

gnx2 xgmx2x

One can analytically calculate the formula

5 10 15 20Order

0.8

0.85

0.9

0.95

Correlation

coefficient

ter Haar Romeny, EMBS Berder 2004

Natural limits on observations

0.4 0.6 0.8 1.2

1.05

1.1

1.15

1.2

1.25

1.3

xL

Taking smaller and smallerkernels gives deviatingresults.

First order derivative of a test imagewith slope = 1 image as function of scale

There is a fundamental relation between the order of differentiation, scale of the operator and the accuracy required.

ter Haar Romeny, EMBS Berder 2004

12 2 2

2

12 2 2

2

The Fourier transform of the derivative of a function is – i times the Fourier transform of the function:

20

2

0.51

1.52

0

0.1

0.2

0.3

0.4

fft

20

2

0.51

1.52

20

2x

0.51

1.52

0

0.25

0.5

0.75

gauss

20

2x

0.51

1.52

A smaller kernel in the spatial domain gives rise to a wider kernel in the Fourier domain

ter Haar Romeny, EMBS Berder 2004

The error is defined as the amount of the energy (the square) of the kernel that is 'leaking' relative to the total area under the curve (note the integration ranges)

errorn_, _ 1002n fftgauss , 202n fftgauss , 2

0.1

0.2

0.3

0.4 , .5

0.1

0.2

0.3

0.4 , .5

100 2 n12n 12 E 1

2 n2 2

n 12Solution:

ter Haar Romeny, EMBS Berder 2004

0.5 1 1.5 2

02.5

57.510n

02550

75

error %

0.5 1 1.5 2

02.5

57.510n

0 2 4 6 8 10Order of differentiation

0.25

0.5

0.75

1

1.25

1.5

1.75

2

elacSni

slexip

Fundamental result: