# Optical modulation transfer function of an image and its cutoff

• chis54
In summary, the conversation discusses calculating the cutoff spatial frequency for an image taken with a camera by using the f/#, focal length, CCD characteristics, and approximate distance to the scene. It is suggested to calculate the cutoff frequency directly using the entrance pupil diameter and wavelength, or to measure the OTF directly from an image by imaging a sharp edge and correcting for pixel size. However, this method may be affected by non-stationary measurements and aliasing.

#### chis54

I have an image taken with a camera. I know the f/#, focal length, CCD characteristics, and an approximate distance to my scene. I computed the modulation transfer function but I am trying to map the pixel space to cy/m and having trouble. I essentially want to determine the cutoff spatial frequency. Any tips on how to do this?

The limitation on the spatial frequency will be set either by the diffraction limit of the lens (see the Raleigh criterion) or by the Nyquist frequency of the sensor. You should be able to compute those given the data you have.

voko said:
The limitation on the spatial frequency will be set either by the diffraction limit of the lens (see the Raleigh criterion) or by the Nyquist frequency of the sensor. You should be able to compute those given the data you have.

My image is 1024x1024 and after computing the MTF, integrating it about theta from 0 to 2pi, I have a plot from 0 to 512 in pixels on my x-axis. I think I divide my pixel value by my ccd pixel size and that will have my x-axis in cy/m. Am I thinking of that correctly?

I don't understand what you are trying to accomplish. I understand what spatial frequency and cutoff spatial frequency are, but "mapping pixel space to cy/m" means nothing to me.

voko said:
I don't understand what you are trying to accomplish. I understand what spatial frequency and cutoff spatial frequency are, but "mapping pixel space to cy/m" means nothing to me.

Just converting the x-axis from pixels to cylces/m

The x-axis is the spatial frequency, correct? Then how come it is in pixels? Did you mean cycles/pixel?

voko said:
The x-axis is the spatial frequency, correct? Then how come it is in pixels? Did you mean cycles/pixel?

yes, yes

chis54 said:
I have an image taken with a camera. I know the f/#, focal length, CCD characteristics, and an approximate distance to my scene. I computed the modulation transfer function but I am trying to map the pixel space to cy/m and having trouble. I essentially want to determine the cutoff spatial frequency. Any tips on how to do this?

You can calculate the cutoff frequency directly (assuming diffraction-limited performance, etc.) with the information you already have- the entrance pupil diameter 'd' is given by the focal length and f/#, and the cutoff frequency for coherent imaging is f_c = d/(2*λ*f), where λ is the wavelength and f the focal length of the lens. For incoherent imaging, the cutoff frequency is twice that of coherent imaging.

Measuring the OTF directly from an image requires some care, but is straightforward. The procedure I use is to image a sharp edge (a sharp black-to-white transition) and differentiate the image to obtain the line spread function. The OTF (in one direction) is the Fourier transform of the LSF, but the specific frequency values (in 1/pixels) will scale with how many image pixels you used. Correct for that, and then knowing the pixel size, you can obtain the OTF in image space (in, for example, cycles/mm). If you then want the OTF in object space ('resolution limits'), you need to know the magnification of the lens.

Does this help?

Andy Resnick said:
You can calculate the cutoff frequency directly (assuming diffraction-limited performance, etc.) with the information you already have- the entrance pupil diameter 'd' is given by the focal length and f/#, and the cutoff frequency for coherent imaging is f_c = d/(2*λ*f), where λ is the wavelength and f the focal length of the lens. For incoherent imaging, the cutoff frequency is twice that of coherent imaging.

Measuring the OTF directly from an image requires some care, but is straightforward. The procedure I use is to image a sharp edge (a sharp black-to-white transition) and differentiate the image to obtain the line spread function. The OTF (in one direction) is the Fourier transform of the LSF, but the specific frequency values (in 1/pixels) will scale with how many image pixels you used. Correct for that, and then knowing the pixel size, you can obtain the OTF in image space (in, for example, cycles/mm). If you then want the OTF in object space ('resolution limits'), you need to know the magnification of the lens.

Does this help?

Yes, thank you

Andy Resnick said:
You can calculate the cutoff frequency directly (assuming diffraction-limited performance, etc.) with the information you already have- the entrance pupil diameter 'd' is given by the focal length and f/#, and the cutoff frequency for coherent imaging is f_c = d/(2*λ*f), where λ is the wavelength and f the focal length of the lens. For incoherent imaging, the cutoff frequency is twice that of coherent imaging.

Measuring the OTF directly from an image requires some care, but is straightforward. The procedure I use is to image a sharp edge (a sharp black-to-white transition) and differentiate the image to obtain the line spread function. The OTF (in one direction) is the Fourier transform of the LSF, but the specific frequency values (in 1/pixels) will scale with how many image pixels you used. Correct for that, and then knowing the pixel size, you can obtain the OTF in image space (in, for example, cycles/mm). If you then want the OTF in object space ('resolution limits'), you need to know the magnification of the lens.

Does this help?

Presumably that only works if your OTF dominates over the effects of the pixel size, otherwise you'll be making a non-stationary measurement of the pixel transfer function.

To get the true pre-sampling OTF you need to tilt the edge and interpolate.

mickybob said:
Presumably that only works if your OTF dominates over the effects of the pixel size, otherwise you'll be making a non-stationary measurement of the pixel transfer function.

To get the true pre-sampling OTF you need to tilt the edge and interpolate.

To be sure, there are a lot of subtleties I glossed over- the point spread function may vary over the field of view, there may be aliasing due to the spatial frequencies of the sampling, etc. etc.

## 1. What is the optical modulation transfer function (MTF) of an image?

The optical modulation transfer function (MTF) of an image is a measure of the ability of an imaging system to transfer contrast from the object being imaged to the final image. In simpler terms, it describes how well an image reproduces fine details and sharp edges.

## 2. How is the MTF of an image calculated?

The MTF of an image is typically calculated by measuring the contrast of a series of black and white lines in the image, and then plotting the contrast values against the spatial frequency of the lines. The resulting curve is the MTF, with the cutoff frequency indicating the point at which the image can no longer accurately reproduce high-frequency details.

## 3. What factors affect the MTF of an image?

The MTF of an image is affected by several factors, including the quality of the lens and other optical components in the imaging system, the resolution of the image sensor, and any image processing techniques used. Additionally, factors such as lighting, motion blur, and noise can also impact the MTF of an image.

## 4. Why is the cutoff frequency important in the MTF of an image?

The cutoff frequency in the MTF of an image is important because it indicates the maximum level of detail that can be accurately reproduced in the image. If an image has a low cutoff frequency, it means that it cannot accurately reproduce fine details and will appear blurry or lacking in sharpness.

## 5. How can the MTF of an image be improved?

The MTF of an image can be improved by using higher quality lenses and optical components, increasing the resolution of the image sensor, and reducing any sources of image degradation such as motion blur and noise. Additionally, image processing techniques such as sharpening can also help to improve the MTF of an image.