Image Processing Reference

In-Depth Information

Performance

63

The curve
A
represents the “blurring” of the image due to diffraction of the tele-

scope only. Curve
B
displays image motion due to seeing, calculated using Eq. 6.1.

Curve
C
, the blurring function, has been obtained as an interpolation between the

two extreme cases; i.e., diffraction-limited or seeing-limited. It is noteworthy that

different interpolation expressions do not modify the position of the minimum sig-

nificantly. The minimum of the curve is roughly at a value of
D
/
r
0
~3. The total

spread angle, curve
D
,isan
ad hoc
expression of curves
B
and
C
. Since the total

spread angle is not defined as a standard deviation, the equality
D
2
=
B
2
+
C
2
is not

valid.

The ratio
D
/
r
0
becomes a particularly important term in describing the effect of

image motion on an image. Image motion is produced by the global tilt in the atmo-

spheric turbulence, which is dominated by disturbances larger than the telescope

aperture. This allows the angular variation,

σ

, to be written as (Hardy 1998)

5

3

2

D

r

λ

2

σ

=

0 182

.

,

(6.1)

α

D

0

where
D
is the aperture,
r
0
is Fried's parameter, and

λ

is the wavelength.

6.3 Strehl Ratio

A diffraction-limited imaging system produces an image of an unresolved object

whose shape is defined by the Fourier transform of the entrance aperture. For circu-

lar apertures, the resulting image is an Airy function. Figure 6.3 shows two super-

imposed Airy functions, one a high-fidelity image and the other corresponding to a

slightly aberrated image. Notice that the minima of both functions are at the same

place, but the height of the point spread function (PSF) has changed. Clearly, using

the term diffraction-limited to refer to the location and visibility of the minima is

not a sufficient measure of performance in the presence of small aberrations.

A more sensitive measure of the performance of an optical system in the pres-

ence of small aberrations is to compare the height of the PSF to the ideal case. This

comparison, the Strehl ratio, is an image plane measure of the performance of an

optical system. The most common approach is to compare the ratio of the intensity

at the center of the PSF to that of an optimum or ideal system.

The effect of small wavefront aberrations on the final image is to move light out

of the focused point, reducing the peak height. This can be quantified, resulting in the

development of a relationship between the PSF height and the wavefront error. The

intensity of the light in the PSF can be determined using the Fresnel-Kirchoff diffrac-

tion integral, which was derived in Born and Wolf (1999). Hardy (1998) shows how

to quantize the reduction of the peak height, which is given as

2

2

Aa

R

*

2

I

=

π

,

(6.2)

2

λ