Name

TwoDTestFunc- Create a 2D test function or quantities related to it, where these quantities

• have analytic expressions

Usage

output = TwoDTestFunc(Size, p, q, a, b, flag=0, alpha=0)

Input

Size

Is the size of the image. MUST be odd.

p

the x-frequency

q

the y-frequency

a

the x-falloff

b

the y-falloff

flag

takes on one of 5 possible values. (0) 2D function, (1) 2D FT , (2) projection , (3) FT of projection , (4) FH

alpha

the projection angle (when needed)

Output

output

depends on flag=0,1,2,3 or 4

• ImBW, which is a sampling of side Size of the test function.
• ImBWFFT, which is a sampling of side Size of the Fourier Transform of the test function
• pofalpha, which is a sampling of the analytical projection of the test function in Real space.
• pofalphaFT, which is the analytical projection of the test function in Fourier space.
• rhoOfkm, which is a sampling of the Fourier Harmonic of the test function.

Description

Purpose: Create a nicely bounded function from which one can calculate projections analytically. The Function (flag=0) is

cos(px) cos(qy) exp(-x² /a² /2) exp(-y²/b² /2)

There is an additional normalization, such that the total integral is unity. The normalization constant is

(1/ 2/ $\pi$) exp(a² p² /2) exp(b² q² /2)

The Fourier transform (flag=1) is

• cosh(p r a²) cosh(q s b²) exp(-a² r² /2) exp(-b² s² /2)

The projection in Fourier space (flag=3) is (Fourier variable is v)

• 5* ( cosh(v(P+Q)D) + cosh(v(P-Q)D) ) exp(-v² D² /2)

with

• D = sqrt( sin² (alpha) b² + cos² (alpha) a² )$

P = p cos(alpha) a² /D Q = q sin(alpha) b² /D

The projection (flag=2) is (parametrized by x)

• 5* ( cos(X(P+Q)) exp((P+Q)²/2) + cos(X(P-Q)) exp((P-Q)²/2) ) exp(-X² /2) /

  • sqrt( 2 pi ) D

with X=x/D

The Fourier Harmonic (flag=4) is (parametrized by k,m) when a=b

• exp(-k² b²/2) cos(m arctan (q/p)) I_m(k b² sqrt(p²+q²))

with X=x/D

Author / Maintainer

P. R. Baldwin

Keywords

category 1

UTILITIES

category 2

SPATIAL.

Files

• util_sparx.cpp

See also

• References to relevant other routines.

Maturity

• alpha

Bugs

Only works with odd size image. flag=4 not yet implimented