September 1967 L E T T E R S T O T H E E D I T O R 1159

Gaussian and Exponential Approximations of the Modulation Transfer Function

U. V. GOPALA RAO AND V. K. JAIN Department of Radiology, The Johns Hopkins Medical Institutions,

Baltimore, Maryland 21205 (Received 5 May 1967)

INDEX HEADINGS: Modulation transfer; Spread function.

IN 1958, Jones1 published an article on the interrelationships between the point spread function, the line spread function,

and the modulation transfer function (MTF). Using these general relationships, he derived analytical expressions for the spread functions of systems whose MTF can be approximated by an ex­ponential function. From these expressions, he obtained the half width at half maximum of the line spread function and the half diameter at half maximum of the point spread function.

As Jones has pointed out, the MTF of a transducer such as a photographic film can often be approximated by an exponential function. However, in the case of systems where the image blurring is caused by a number of randomly varying factors, the MTF tends to be a gaussian function. A familiar example is a radioiso­tope imaging system.2 The purpose of this communication is to discuss some interesting points concerning the exponential and gaussian approximations of the MTF.

Jones1 has shown that the spread functions can be computed from the MTF by use of the equations

and

where P{r) is the point spread function, L{x) is the line spread function, M{f) is the MTF and J0 is the Bessel function of first kind and zero order.

Gaussian Approximation. For gaussian distributions, we write

It then follows from Eqs. (1) and (2), respectively, that

and

In the form derived above, L {x) and P (r) are normalized so that the total area under each spread function is unity. In other words,

However, the practice most commonly used with spread func­tions is to normalize them so that the maximum value is unity. The maximum values of the spread functions given by Eqs. (3) and (4) are (π)½/k and π/k2, respectively. When normalized in this way, the spread functions are given, respectively, by

and

This means that if the MTF of a system can be approximated by a gaussian function, the point and the line spread functions are identical, when normalized so that the maximum value is unity. In other words, the point and line spread functions are the same for systems in which the spread is the result of a number of ran­domly varying factors.

From Eqs. (6) and (7), it is seen that the half width at half maximum of the line spread function and the half diameter at half maximum of the point spread function are both equal to (0.693k2/π2 or 0.266k.

Exponential Approximation. For exponential approximation, we write

Putting Eq. (8) in Eqs. (1) and (2) and simplifying, we get

and

As before, we can normalize Eqs. (9) and (10) so that the maxi­mum value is unity. We then get

and

Equations (11) and (12), though exact, are inconvenient to handle. For this reason, we tried to fit them with appropriate exponential functions. This was done by plotting Eqs. (11) and (12) on semi­log paper and approximating the curves thus obtained by straight lines. The exponential functions thus obtained are as follows:

and

1160 L E T T E R S T O T H E E D I T O R Vol. 57

The half width and the half diameter at half maximum are then 0.1477τ and 0.110τ, respectively. The half width and the half diameter obtained from Eqs. (11) and (12) are 0.1S9τ and 0.122τ, respectively. The errors involved at the half maxima are thus less than 10%.

We have further found that in applications where it is not ob­jectionable to make L(x) and P(r) equal to something other than unity at x=0 and r = 0, respectively, a very close fit (within 2% at most points) can be obtained over the regions 0.80 <L(x) <0.1 and 0.85 <P( r )<0 .03 by using the following approximations instead of Eqs. (13) and (14):

and

Summarizing, if the MTF of a transducer can be approximated by a gaussian function, the line and point spread functions are also gaussian; when normalized so that the maximum value is unity, the line and point spread functions are identical. If the MTF can be approximcted by an exponential function, the line and point spread functions can also be approximated by expo­nential functions, within the limits of accuracy normally required in experimental work.

1 R. C. Jones, J. Opt. Soc. Am. 48, 934 (1958). 2 H. S. Frey, Investigative Radiology 1, 314 (1966).