# Estimation Of Modulation Transfer Function Information Technology Essay

## Abstract

Modulation Transfer Function(MTF) is one of the accepted ways of describing the optical properties of an Imaging system. In almost every major application of image science, MTF is considered as the important property for measuring the image quality.

The MTF of an optical device can be used to determine its resolution and sharpness as function of spatial frequencies. Testing the performance of a digital cameras is very complicated due to its incorporation with several technologies. But still MTF is considered as the best way to test quality of complex imaging devices like Digital cameras.

The main aim of the project is to determine the MTF of a commercial Digital Camera using two different methods. First one is, MTF determination using Traditional Method and the second one is Histogram Analysis Method. We then compare the results of both Methods to choose the one that gives the better results. The programme coding for the methods will be written in matlab and runned for the results.

## Introduction

Modulation Transfer Function is one of the important factors that are used in the field of Digital image processing to test image quality. MTF is defined as the Spatial frequency response of the an optical system.

## Modulation Transfer Function :

It is considered to be the most widely used scientific method to describe the lens performance. It can be defined as the measure of transfer of modulation from subject to image. In other words, it can be defined as the measurement done to predict the faithfulness of a lens in reproducing the detail from the object to the image[4].

When an image is reproduced by an optical system, it cannot be a perfect blur free image. This is not because of using a bad lens by optical system. Even if a perfect lens is used, still some blur can be found in image. The phenomenon causing this blur, is called ‘Diffraction’. Diffraction is the only factor that can limit the performance of a perfect lens , therefore such lens are called as Diffraction limited lens[4].

MTF can be analytically determined by Fourier transform of Line Spread Function (LSF)

[8,9,10,11]

And LSF can be found by differentiation of Edge Spread Function or Edge Scan Function(ESF)

[8,9,10,11]

## Importance of MTF in Optical designing and Fabrication:

Modulation Transfer function has always been prominent in measuring the performance of Optical systems. The optical system designers and the scientists have been doing research to characterize MTF as a mode to measure the performance of Optical systems. MTF is considered as the understood and well developed concept that bridges the gap between the Metrology engineers, lens designers and optical fabricators.

## Statement of the problem :

The companies that manufacture Digital photo equipment show marketing specifications like Effective pixels(Mega pixels), Maximum resolution, Digital zoom, Price etc.., as the parameters of the performance leaving aside the technical specifications.

Actually, technical specifications like Electronic signal processing, optical elements, number of pixels used, type of electronic sensor, software and compression mode are the factors which are responsible for the performance of an equipment and its operation. However, a customer being unaware of such technical specifications will have to choose the devices according to the shown marketing specifications.

The main aim of this project is to determine the Modulation Transfer Function(MTF) of a Digital camera which is a good measure of its practical performance and also a most widely used parameter to characterise the optical and electro optical equipment.

## Scope of the project

This project has following limitations

Gray scale colour mapping.

Image capturing environment.

One dimensional plot.

Gray scale colour mapping :

The images under test are mapped to gray scale to view the image properly without bright colours which are obtained after scanning operation.

Image capturing environment :

The images under test are captured in bright environment to get the images perfectly and to reduce any corruptions that result due to lack of enough light.

One dimensional plots :

All the plots shown in the project are one dimensional and this was done in order to study, avoid confusion and compare the various parameters of different geometrical shaped images simultaneously.

## Objectives :

The main objectives of the project are

Finding the MTF using Traditional method.

Finding the MTF using Histogram Analysis Method.

Comparing the results of both methods to choose the one that gives best results.

## Software :

The software required for the project was programmed in MATLAB(R2007b) coding. It was very much helpful in studying and analysing the plots of various parameters resulted during the process of MTF estimation.

MATLAB stands for ‘Matrix Laboratory’ a software developed by Math works, Inc. which is re-known as a mathematical computing software developer. This software can be used in various Engineering and Technical fields for some specialised tasks like Data analysis, Signal processing, Image processing, etc....

## Overview of Thesis:

The thesis of the project contains four chapters

Chapter 1 :

It gives information about the project showing its brief introduction, statement of problem, project objectives, its scope & limitations, software used and overview of thesis.

Chapter 2 :

It is about literature review and theory about the information gathered from others works such as Work thesis, Journals, Books, Internet, etc..,

Chapter 3 :

It is about the Methodology of the project. That means methods used, different steps of doing it and other related tasks of the project.

Chapter 4 :

It is the discussion about the obtained results and their comparison.

Chapter 5:

It is about the conclusion and expected future works. This chapter explains which one is the best method among two and why. It also gives brief information about any improvements required and the expected practical usage of project results in other applications and future works.

## Literature review

## Modulation Transfer Function :

Modulation Transfer Function (MTF) is one of the accepted ways of describing optical properties of an imaging system. It is not only used in many major applications of image science to measure the quality of an image but also to determine resolution and sharpness.

Modulation Transfer Function (MTF) can be defined and explained in many ways. The below discussion gives brief information about its definitions and explains about it importance in fields of Image science and Technology.

## Below shown are some of the its definition in different contexts:

In the field of Digital image processing, MTF is defined as the magnitude of spatial frequency response of the optical system.

It is defined as the measure of transfer of modulation from subject to image[4].

It can be defined as the measurement done to predict the faithfulness of a lens in reproducing the detail from the object to the image[4].

It is defined as the magnitude of Optical Transfer Function(OTF) of a spatially invariant imaging system[12,13].

OTF can be broken down in to two components as magnitude and phase.

[13]

Where,

and (ξ,η) are spatial frequencies in x-plane and y-plane respectively. [13]

Finally, MTF can be analytically defined by Fourier transform of Line Spread Function (LSF)

[8,9,10,11]

And LSF can be found by differentiation of Edge Spread Function or Edge Scan Function(ESF)

[8,9,10,11]

In the next coming pages the definitions for PTF, LSF, ESF are given and explained each in detail

Theoretical Background of MTF:

As defined in earlier stages of this thesis, MTF is the magnitude of OTF. That means the real part of OTF.

[13]

Where,

and (ξ,η) are spatial frequencies in x-plane and y-plane respectively. [13]

## Optical Transfer Function :

The full description of image quality can be obtained by Optical Transfer Function(OTF), which is a combination of Modulation Transfer Function(MTF) and Phase Transfer Function(PTF)

Where,

Modulation Transfer Function(MTF) is a plot of ratio of image modulation to object modulation plotted against image spatial frequency in cycles per millimetre.

Phase Transfer Function(PTF) is the plot of displacement of image when compared with geometrically correct position in radian plotted against image spatial frequency.

In terms of Spatial frequency, OTF is defined as the frequency response of an optical system.

[14]

It is defined as the Fourier transform of Point spread Function(PSF)

[15]

Point Spread Function is an another important property which will be explained in coming sections of the thesis.

The Optical Transfer Function will allow us to know about the spatial frequencies which are transmitted by imaging system and to what extent they are attenuated.

[16]

An optical system can have more than one component with corresponding OTF. And aggregate OTF value of such system can be obtained by multiplying all MTFs and adding all PSFs.

## Phase Transfer Function :

From the definition of Optical Transfer Function, the phase of Optical Transfer Function is Phase Transfer Function.

As discussed in the previous sections, the magnitude of OTF is called MTF which is real component of the function. Whereas the imaginary component is called PTF.

For a symmetric impulse response which is centred at an ideal image point, PTF will have a value of zero or π as a function of spatial frequency.

PTF determines the change in phase of the image at each angular frequency at respective axes.

Phase distortion is present in Coma and Astigmatism like asymmetrical aberrations which describe radial linear shift of intensity pattern.

Though phase transfer function is not very much useful, but at higher frequencies, it can determine whether image artefacts can occur or not.

## Phase Distortion(Non-linearity in Phase):

Non-linearity in Phase Transfer Function is called Phase distortion, which make different spatial frequencies of image to recombine with different relative phases. The phase distortion results in change of shape of spatial waveform that describes image. Phase distortion in a severe range can result in image waveforms that are very much different from object wave form. [ 16] [16]

## Point Spread Function(PSF)

In general terms, PSF can be defined as the response of the optical imaging system to a point object or a point source. Or it can also be defined as the impulse response of Optical imaging system.

## PSF definition with Mathematical approach.

Let us assume an Ideal point source as the Object which can be mathematically represented as

## f(x,y) = δ(x,y)

If the assumed point object is a perfect point source, then the impulse response h(x,y) should be identically equal to the two dimensional image distribution g(x,y). This is called as Point Spread Function(PSF).

## h(x,y) = g(x,y) = PSF(x,y)

The above Point Spread Function when undergoes two dimensional Fourier Transform gives two dimensional Optical Transfer Function.

## [PSF(x,y)] = OTF()

The Magnitude of Above function gives Modulation Transfer Function(MTF)

## │ [PSF(x,y)]│ = │OTF()│ = MTF()

The evaluation of two dimensional Fourier transform can be done along any desired profile

For Instant, MTF() or MTF()

## [16]

## Line Spread Function(LSF) : [16]

Line spread Function can be evaluated in two different methods.

Ideal line approximation method using slit test target

Differentiation of ESF(Edge Spread Function)

Importance and evaluation of Edge spread function will be discussed in the next section of the thesis.

Evaluation of LSF using Ideal line approximation method :

Let us assume, the same measurement setup which is used for PSF test has been used here. But instead of point source, a line source object is used. Let it be a delta function in x and let the other variable y be a constant.

The above conditions would result in the equation taking form as follows

## f(x,y) = δ(x)1(y)

Then its two dimensional image irradiance distribution g(x,y) is called as Line Spread Function(LSF).

Functionally, LSF can be defined as function of one of the spatial variables which was responsible for impulsive behaviour of Line source.

In this case, spatial variable along x-direction is considered.

## g(x,y) = LSF(x,y)

## [16]

Let us consider the used line source object as collection of point source objects placed in a straight line. Then each point produces a PSF in image plane. Such PSFs overlap vertically whose sum is the resultant PSF. This concept can be visualised better by the following figure and equation.

[16]

## g(x,y) = LSF(x) = f(x,y) * * h(x,y) = [δ(x)1(y)] * * PSF(x,y)

## The LSF is the two dimensional convolution of line source object with PSF

The previous equation when convoluted in the direction of y is nothing but its integration over the same direction.

It can be verified from the above results, that LSF can be a function only in x. This is due to the object source being independent of y. Being impulsive in single direction, the object can give imformation about only one spatial frequency component of the transfer function. When this line response undergoes one dimensional Fourier transform gives one profile of OTF, whose magnitude gives MTF in one profile.

## │ [LSF(x)]│ = │OTF()│ = MTF()

The reorientation of Line source gives the other profiles of transfer function. For example, the orientation of Line source by 90 degrees gives the following equation.

## f(x,y) = 1(x)δ(y)

This gives LSF in y – direction and the corresponding MTF would be MTF()

It should be noted that PSF and LSF will have different functional forms due to the summation along constant direction in line source image. The x profile of PSF(x,y) is not simply equal to LSF(x).

## LSF(x) ≠ PSF(x,0)

## [16]

when a diffraction limited optical system is considered, the comparison of PSF and LSF gives the following results.

It may noted in the pattern that PSF(x,0) has zeros, whereas LSF(x) does not.

## Comparison of x-directional functional terms of PSF and LSF for a diffraction limited system

Edge Spread Function(ESF) : [16]

Below shown is the measurement set up to evaluate Edge Spread Function(ESF), where the input is a step function, a knife-edge source.

## f(x,y) = Step(x)1(y)

## Measurement setup to evaluate Edge Spread Function

Edge spread function is defined as the resulting function when PSF convoluted with unit step function.

## g(x,y) = ESF(x) = PSF(x,y) * * step(x)1(y)

The PSF when convoluted in y-direction with a constant produces LSF and when convoluted in x-direction with a step function gives cumulative integration as shown below in the following figure.

## The ESF is two dimensional convolution of Edge source object with PSF [16]

ESF is cumulative function that increases monotonically. For a diffraction limited system, the ESF can be as shown in below figure.

## Plot of ESF for a diffraction limited system

ESF can also be explained in the terms of superposition of PSF. In the open part of aperture, each vertical strip produces LSF at corresponding location in image plane. The ESF is formed when displaced LSFs are overlapped in horizontal direction and summed. This process can be shown as follows

The ESF data can be converted to MTF by taking the spatial derivative of ESF data.

If once LSF is evaluated, its one dimensional Fourier transform gives OTF in one profile and MTF with same profile can be obtained by taking the magnitude of achieved OTF. The reorientation of Knife edge gives the corresponding one dimensional profile of MTF.

## Comparison of PSF, LSF and ESF :

A detailed study about PSF, LSF and ESF in previous sections show that PSF is the one which provides two dimensional OTF in one measurement. The main disadvantage of PSF is, the point object source used provides very little flux which makes it difficult to detect. This drawback mainly comes in to existence in the infra red portion of spectrum, where the flux sources being used are black bodies. This draw back may not be effective in the visible region, where the flux sources being used are the hotter sources which have higher flux densities.

The pinhole can be illuminated by using a laser source. Irrespective of illumination source, a laser from a pinhole can produce True PSF data, which will be spatially coherent across it and such properties will not complicate interpretation PSF data. While conducting the tests with LSF and ESF, care must be taken to avoid interference artefacts in to data that are introduced by coherent properties of the laser. The image plane flux provided by LSF test will be more than that of PSF test. And the flux provided by ESF is even more than that of LSF. Slit width issues avoided by knife edge is an another added advantage of LSF test. In ESF test spatial derivative operation is used which results in accentuation in noise of data. Noise can be reduced by convolution using a spatial kernel and this data smoothening operation contributes to Instrumental MTF.

## Histogram of image and its Cumulative Distribution Function:

The histogram of image is defined as the bar plot graphical representation of relative distribution of various tones that form a digital photograph. An image with eight bit per channel gray scale will have a single channel and uses 256 gray shades to describe all the tone levels from black to white. The level value for Black is zero whereas that of for a white is 255. And all the values between 0 and 255 will represent the gray shades gradually going from black to white.

The closed integral of Histogram is called as Cumulative Distribution Function.

## MTF measuring methods

Measuring of the MTF can be done in several methods. Few of them can be done by generation of continous or descrete frequencies, scanning the image, and analysis of wavefront. The current advancement in optical technology and precision mechanics allows to measure MTF with a very high accuracy.

The four popular methods in practice are

Frequency generation.

Scanning the image.

Video methods.

Interferometric methods. [17]

## Frequency Generation method:

This method tests the MTF directly, where the lens under test images the object that has a pattern with single spatial frequency. Using this method, the operator can measure the image contrast directly. This is formally called as Discrete or Single frequency measurement where Eye charts, Bar charts, USAF 1951 resolution charts are used in series of such tests to plot a graph of MTF over a range of Spatial frequencies.

Various methods have been developed for the systems which vary the source frequencies continuously by measuring the contrast constantly. An example of this method would be a rotating radial grating using slit aperture as the object. A detector monitors the light passing through a pinhole which was placed in lens’s focal plane. The white and black bars will sweep through the pin hole when grating is rotate. It has the facility to vary the object’s spatial frequency, by moving grating relative to the pinhole. Synchronizing the rotation to the detector output gives direct measure of its Modulation Transfer Function at the corresponding radial grating spatial frequency and it’s harmonics.

Advantage:

The direct measure is the main advantage of the Frequency generation method.

Disadvantage:

These methods limit the flexibility of instrument due to the requirement of manipulating the sources and detectors simultaneously. [17]

## Scanning method:

The scanning method is most commonly used commercial method for measuring MTF. These kinds of methods are operated on the principle of linear system theory. For example, a known input of infinitely small pin hole is imaged by lens under test and MTF is determined with this information.

If an image is scanned in one-dimensional its spatial profile is called Line spread function(LSF). Whereas when it is done in two-dimensional Point spread function can be obtained. The image of point source is edge-scanned with an obscuration called Knife-edge. The throughput is monitored and differentiated to get Line spread function. Moving the pin hole or slit using a slit source can also produce Line spread function. As per the requirement either the Tangential or Sagittal scanning can be achieved by horizontal or vertical orientation of knife edge respectively. Both tangential and sagittal scanning can be achieved when a diagonally traversed image with a knife-edge at right angle is scanned sequentially in vertical and horizontal directions. The one-dimensional MTF can be obtained from the Fourier transform of LSF.

In order to get a true impulse response function, the source size must be corrected to a finite value. Using linear system theory, this correction can be done by dividing the MTF measured by the source’s Fourier transform. Where the corrected MTF data will be the quotient of measured MTF data when divided by proper correction factor at discrete frequencies.

MTF(f) = CorrectedMTF(f) / Correction(f) [17]

For an aperture of finite size like pin hole and slit the measured data of MTF can be corrected quickly through computer algorithms. Then the data which is corrected fully can be compared to theoretical performance. Re-measuring the MTF data at various focal planes can generate through-focus MTF mapping. These curves help in determining the various effects of Spherical aberration, Chromatic aberration, Field curvature, Astigmatism and Defocus. The best performance of focus can be determined by comparing the MTF at these focal plane whilst choosing a single spatial frequency.

Advantages:

Even without Image magnification, this method allows very high resolution because of using scanning systems which are equipped with leads screws driven by accurate synchronous motors or stepper motors.

Disadvantages:

The duration of the scan is the main disadvantage of this method. The number of data points required for properly sampled image are dictated by the sampling theory and parameters of lens under test. This will lead to insufficient data sampling and can affect the MTF’s accuracy. A thirty seconds of measuring time is often required to scan a long image.

## Video Methods:

This method uses the same theoretical considerations that are used for the previous scanning method. In the focal plane of lens under test, a solid state array is placed. Its digitized output offers PSF directly, when a pinhole source is used. This data when undergoes Fourier transform in two dimensions gives Optical transfer Function. The integration of Point spread function gives Line spread function and Edge spread function. Using a slit source instead of pin hole source in the above case gives Line spread function directly and when it undergoes one-dimensional Fourier transform gives Optical Transfer function. In both the cases, the magnitude of Optical transfer function gives Modulation Transfer function(MTF).

Advantages:

The accomplishing speed of video MTF measuring method is its major advantage. As soon as the solid state array is sampled electronically and Fourier Transform is calculated, its MTF is updated. This can provide a continuously updated spread function and its MTF curve.

These systems are very useful where the MTF data specifies the alignment of optical systems. The effects of the perturbation of the MTF can be monitored by an operator by moving the optical component or assembling.

Disadvantages:

The drawback of this method arises from design of electronic solid state array. The maximum frequency resolvable is 30 to 80 1p/mm approx. Due to sizes of detector elements which are finite and in order of microns. This problem can be compensated by magnifying the image on to the array, by addition of a optical relay system. In addition to that, the relay optics being used should be of high quality. Because while working at angles of high axis, it is required to have high numerical aperture in order to capture the whole output of fast systems and lenses. And the relay optics being used must be diffraction limited ones in order not to impact he measured MTF.

The measured MTF is affected by increase in apparent size of image due to pixel-to-pixel cross talk both electrically and optically. The MTF should be corrected for this effects. Normally 8-bit precision digitisation is done by digitizing board that work with high speed. The blooming effect and pixel saturation should be avoided that can be achieved by controlling the illumination in video camera. The digitizing level limits the computed MTF’s accuracy. When compared to 8-bit video MTF systems, the conventional scanning systems are more accurate. Anyhow, video sampling methods are valuable with right application.

## Interferometric methods:

An interferometer can be used to measure a system’s MTF using one of the two methods.

Auto correlation of the pupil function of lens under test.

Analysing the PSF calculated through Fourier transform of pupil wave front.

This method is convenient for the systems that suit to test with interferometer, without any significant aberrations. This method also suits the systems whose wave front errors do not vary substantially over the wavelength of interest. For full polychromtic testing, spectral filters and wide band sources are used by adjusting wavelength range in video frequency methods and scanning methods. Since monochromatic sources like lasers are used in interferometer, the computed MTF will be available only at these wavelengths. This method requires the correction of wave front, because of having the limited capability of phase measuring interferometers to sample the wave front. MTF measurement with Interferometres will be difficult for the lenses with excessive wave front errors.

## Methodology

The project mainly deals with estimation of Modulation transfer function using two different methods.

Traditional method of MTF determination.

Histogram Analysis method.

Both the methods used in the project fall under the category of scanning methods for MTF estimation which were explained in previous sections of the Thesis.

## Traditional Method of MTF determination

In this method, the process starts with edge scan of an image perpendicular to edge. Parallel to the edge, in vertical direction all the pixels are averaged by the scan whose result can be shown on graphically with reflectance R on y-axis an position x on x-axis as shown in the following figure.

## Edge scan Edge scan function(Edge spread function)

Edge scan function is also called as Edge spread function, which can give a good description of what a line scan looks like. Here in this case, it can be noticed that the edge is not an instantaneous transition. It is a gradual transition from black to white through gray. This gives an idea about the treatment of an imaging system with an edge.

The second step is to do differentiation of Edge scan function (Edge spread function) to get Line spread function.

The resultant function is called Line spread function which gives information about how the original image has been spread out in the process of imaging. This is a probability density function to locate the edge in the output image. The plot of resultant function would like the one as shown below.

## Line spread function

If an imaging system was used with perfect resolution, then LSF of such system would be a delta function of zero width.

The representation of Line spread function in frequency domain is called Modulation Transfer Function, MTF(). The MTF gives the knowledge about the frequencies which were attenuated by the used imaging system. In order to obtain Modulation transfer function from Line spread function, it should undergo a Fourier transform as shown in the following equation.

## MTF() = {

And the plot of resultant function, Modulation Transfer function will be as shown in the following figure.

## Modulation Transfer Function

At each spatial frequency, the fraction of the contrast of the image that was attenuated by image system is given by the value of MTF from the above figure. In this graph it can be noticed that though the low frequencies get through with some attenuation, high frequencies drop off quickly. In this way, MTF is considered as the important tool to describe how an imaging system can preserve or loose resolution of an image.

Now the various images of different geometrical shapes will be used as test images to repeat the same experiment with each image and Modulation transfer function for all images will be measured and noted to compare the results of that of another method of project called as Histogram Analysis method. Those different images being used for experiment are as follows.

## Various images being used for experiment with different geometrical shapes

## Histogram Analysis method:

The scientific and research literature of Digital image processing direct that gray level of an image will not have any spatial information. This may not be true to its full. The histogram alone may not be enough sufficient to give all the knowledge about spatial information. But having in hand some additional information about image’s spatial characteristics can help to obtain the quantitative spatial information from histogram. And the additional information about image’s spatial characteristics, for this experiment is that the image is an edge which means that the gray level in the image increase monotonically in the direction perpendicular to the edge.

The below shown are the histograms of sharp and soft edge images and it can be noticed that from left to right there is an increase in gray level of histogram. It can also be viewed that the spatial motion is monotonic in the direction perpendicular to the edge scan.

## Histogram of sharp Edge Histogram of soft Edge

The rate of change of position x, with respect to change in R can be interpreted as the histogram of an Edge image.

## Histogram(R) =

And the Cumulative distribution of the Histogram can be obtained by doing its integration over a closed interval of R = 0 to R. The plot of image we get for that is as shown below.

## Cumulative Distribution Function

Rotating the plot ninety degrees experimentally, gives the plot of Edge spread function then following the same steps of first method (Traditional method) MTF can be achieved.

This experiment has to be repeated with various images that we used in Traditional method.

Then the comparison of the results of the both methods should be done in order to choose the one that gives the same MTF for various images.

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