MTF and derivative parameters

MTF (modulation transfer function) is a function of the contrast of image of a sine pattern at a given spatial frequency generated by the tested camera relative to a contrast of an image of sine pattern at spatial frequency equal to zero. Spatial frequency is typically measured per cycles (or line pairs) per a unit angle or a unit length (in case of thermal cameras MTF in line pairs per millionaires [LP/mrad]

or in inverted millionaires [mrad^-1])² (see Fig.3.25).

θ

sine pattern

distance R cycle angular size θ

pattern cycle Tx

spatial frequency:

ν = 1 / θ

Fig.3.25. Graphical interpretation of spatial frequency.

Images of several sine patterns of different spatial frequency generated by a thermal camera of MTF presented in Fig. 3.26 are shown in Table 3.9.

2 Attention: TV resolution is measured in line widths instead of pairs, where there are two line widths per pair, over the total height of the display

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 2 4 6 8 10

spatial frequency [lp/mrad]

MTF

Fig. 3.26. Exemplary MTF.

Table 3.9. Images of sine patterns generated by a thermal camera of assumed MTF function.

Frequency [LP/mrad]

MTF Original pattern Image

1 0.94

2 0.78

3 0.57

4 0.37

5 0.21

6 0.11

7 0.05

8 0.02

9 0.01

10 0.0

From mathematical point of view MTF is defined as

MTF( ν ^{) = C(}ν ^)/C(ν^{= 0) , (3.13)}

where C(ν= 0) is the contrast of image of the sine pattern at near zero frequency.

The contrast of the image of the sine pattern is defined as

( )

Max is the maximum intensity for a pattern of spatial frequency ν ("white peak") and Min is the minimum intensity for a pattern of spatial frequency ν ("black val-ley").

The relationship between MTF function and contrast of a sine target creates the possibility of determination of MTF by measurement of contrast of series of sine targets of different spatial frequencies.

Measurement of MTF using sine targets is an oldest, classical MTF measure-ment technique. However, this measuremeasure-ment method is time consuming because measurement of contrast must be done for a series of sine targets. At the same time this method is also an expensive one, particularly in case of thermal cameras.

Manufacturing sine targets is difficult and costly even in case of visible targets when a sine target is created by deposition of a layer of spatially variable transmit-tance over a transparent glass. It is technically possible to manufacture sine filters for far infrared range but it is so difficult and costly technology that MTF of thermal imagers is never measured using sine targets method.

Contrast transfer function (CTF) can be treated as a substitute of MTF that can be measured using targets that are simpler to manufacture. CTF is defined in the same way as MTF with an exception that a square wave pattern is used instead of a sine wave pattern. The CTF values are usually higher than the MTF values.

The difference between MTF and CTF is usually not great. Next, it is much easier to measure CTF than to measure MTF. Therefore, several decades ago CTF was typically used instead of MTF to evaluate thermal cameras. However, nowadays CTF is rarely measured because in present era of computer technology and image processing MTF function can be easily measured by capturing images of some standard targets and using mathematical apparatus of Fourier transform.

MTF can also be defined as the magnitude of a complex function Optical Trans-fer Function:

OTF(ν) = MTF(ν) exp (i PTF(ν)) , (3.15)

where OTF is Optical Transfer Function, MTF is Modulation Transfer Function, and PTF is a function called Phase Transfer Function that represent the change in phase position as a function of spatial frequency³.

A perfect optical system would have MTF of unity at all spatial frequencies, and PTF equal to 0 at all spatial frequencies. In case of real imaging systems, MTF always decreases to zero at some spatial frequency. The shape of MTF function gives precise information about imager ability to produce sharp images.

In most imaging systems PTF is not significant and therefore it is usually assumed that OTF equals MTF. Therefore MTF, not OTF, is typically used as a measure of quality of imaging systems.

Modern measurement methods of MTF of thermal imagers are based on rela-tionships between MTF function and two other functions (LSF and ESF):

MTF(ν )=Magnitude {F[LSF(x)]}, (3.16)

MTF(ν )=Magnitude {F[derivative from ESP(x)]} (3.17) where F is the Fourier transform operator, LSF (line spread function) is one direc-tional distribution of the flux in the image of an ideal line-like target, ESF (edge spread function) is one directional distribution of flux in the image of an ideal edge-like target.

Measurement of MTF of thermal imagers is usually carried out on the basis of captured images of two types of targets: narrow slit target or edge target (Fig.

3.27). When an image of one of these targets is captured and digitized then later MTF is calculated using formula (3.16) or formula (3.17). Practically measurement of MTF is not as simple as formulas (3.16) and (3.17) suggest due to necessity to use noise correction algorithm but these formulas present the principle of modern measurement of MTF of thermal cameras.

a) b)

Fig. 3.27. Images captured during MTF measurement a)image of a narrow slit target, b)image of an edge target.

3 If case of a linear PTF, only simple lateral displacement of the image is observed. Non-linear PTF can adversely affect image quality. An extreme case is a phase shift of 180º produces a reversal of image contrast.

MTF function is an excellent criterion of image quality of thermal cameras.

However, interpretation of a curve is more complicated than interpretation of simple numerical parameters. Next, graphical presentation of MTF function was difficult several decades ago when computers were rarely used. Therefore several numeric-al parameters based on MTF function were proposed in the past to characterize thermal cameras. Nowadays, these numerical parameters are rarely used but still it is useful to know these five numerical parameters related to MTF function:

1. Equivalent frequency (equivalent line number or equivalent bandwidth) Ne,.

2. Half frequency HF, 3. Effective resolution ER,

4. Effective instantaneous field of view EIFOV, 5. Limiting resolution LR.

First, the equivalent frequency Ne is defined as

∞

∫

The equivalent frequency Ne concept is based on Shade criterion [24], who stated that perceived image quality can be described using the formula (3.18).

Second, the effective resolution ER is defined as

Ne

ER= ⋅ 2

1 . (3.19)

The effective resolution is presented using angle units (in case of thermal cameras typically millionaires are used).

Third, experiments have shown that perceived image sharpness is closely re-lated to the spatial frequency where MTF is 0.5. This means that the spatial fre-quency at which MTF drops to 0.5 can be a good indicator of imager quality. This frequency is called the half frequency HF. It is expressed in spatial frequency units.

Fourth, the effective instantaneous field of view EIFOV is defined as

EIFOV HF

= ⋅ 2

1 , (3.20)

EIFOV is presented in angle units (typically in millionaires).

Fifth, the limiting resolution is defined as spatial frequency at which MTF equals from about 0.02 to 0.05. The definition is based on the fact that humans usu-ally cannot distinguish high contrast sine pattern at frequencies where MTF drops below the level 0.02−0.05. Exact value of the limiting MTF level depends on the observer.

As we can see, determination of these five numerical parameters related to MTF function is quite easy when the latter function is known. However, nowadays in era

of computer technology when it is easy to measure, present and store MTF function it is better always to use MTF function as original data. The numerical parameters listed earlier should be used only for comparisons of thermal cameras of different MTF functions when we need to have a simple numerical criterion of comparison.