Optical transfer function
Encyclopedia
The optical transfer function (OTF) of an imaging
system (camera
, video
system, microscope
etc.) is the true measure of resolution
(image sharpness) that the system is capable of. The common practice of defining resolution in terms of pixel count is not meaningful, as it is the overall OTF of the complete system, including lens and anti-aliasing filter as well as other factors, that defines true performance. In the most common applications (cameras and video systems) it is the Modulation Transfer Function (the magnitude of the OTF), that is most relevant, although the phase component can have a secondary effect. While resolution
, as commonly used with reference to camera systems, describes only the number of pixels in an image, and hence the potential to show fine detail, the transfer function describes the ability of adjacent pixels to change from black to white in response to patterns of varying spatial frequency, and hence the actual capability to show fine detail, whether with full or reduced contrast. An image reproduced with an optical transfer function that 'rolls off' at high spatial frequencies will appear 'blurred' in everyday language. Modulation Transfer Function or MTF (the OTF magnitude with phase ignored) is roughly the equivalent of frequency response in an audio system, and can be represented by a graph of light amplitude (brightness) versus spatial frequency (cycles per picture width).
theorem says that it should be possible, in a perfect system, to resolve fully (with true black to white transitions) nearly 1920 alternate black and white lines, otherwise referred to as a spatial frequency of 960 line pairs per picture width, or 960 cycles per picture width, (definitions in terms of cycles per unit angle or per mm are also possible but generally less clear when dealing with cameras and more appropriate to telescopes etc.). In practice this is far from the case, and spatial frequencies that approach the Nyquist rate will generally be reproduced with decreasing amplitude, so that fine detail, though it can be seen, is greatly reduced in contrast. This gives rise to the interesting observation that, for example, a standard definition television picture derived from a film scanner that uses oversampling
, as described later, may appear sharper than a high definition picture shot on a camera with a poor Modulation Transfer Function. The two picture show an interesting difference that is often missed, the former having full contrast on detail up to a certain point but then no really fine detail, while the latter does contain finer detail, but with such reduced contrast as to appear inferior overall.
by eliminating spatial frequencies above the Nyquist rate
, but in practice it will have a response that 'rolls off' seriously before the Nyquist frequency is reached.
in the final image, and 'downconvert' or 'interpolate' using special digital processing which cuts off high frequencies above the Nyquist rate to avoid aliasing whilst maintaining a reasonably flat MTF up to that frequency. This approach was first taken in the 1970s when flying spot scanners, and later CCD
line scanners, were developed which sampled more pixels than were needed and then 'downconverted', which is why movies have always looked sharper on television than other material shot with a video camera. The only theoretically correct way to interpolate or downconvert is by use of a steep low-pass spatial filter, realised by convolution
with a two-dimensional sinx/x weighting
function which requires powerful processing. In practice, various mathematical approximations to this are used to reduce the processing requirement. These approximations are now implemented widely in video editing systems and in image processing programs such as Photoshop.
Just as standard definition video with a flat MTF is only possible with oversampling, so HD television with full theoretical sharpness is only possible by starting with a camera that has at least twice as many pixels, and then digitally filtering. With movies now being shot in 4k
and even 8k video for the cinema, using cameras like the Red
, we can expect to see the best pictures on HDTV only from movies or material shot at the higher standard. However much we raise the number of pixels used in cameras, this will always remain true (unless a perfect optical spatial filter can be devised), and the same problem exists of course with stills cameras, where a better image can be expected when, say, a 10 megapixel image is converted to a 5 megapixel image, than could ever be obtained from a even the best 5 megapixel camera. Because of this problem of maintaining a flat MTF, broadcasters like the BBC
did for a long time consider maintaining standard definition television, but improving its quality by shooting and viewing with many more pixels (though as previously mentioned, such a system, though impressive, does ultimately lack the very fine detail which, though attenuated, enhances the effect of true HD viewing).
Another factor in digital cameras and camcorders is lens resolution. A lens may be said to 'resolve' 1920 horizontal lines, but this does not mean that it does so with full modulation from black to white. The 'Modulation Transfer Function' (just a term for the magnitude of the optical transfer function with phase ignored) gives the true measure of lens performance, and is represented by a graph of amplitude against spatial frequency.
Lens aperture diffraction also limits MTF. Whilst reducing the aperture of a lens usually reduces aberrations and hence improves the flatness of the MTF, there is an optimum aperture for any lens and image sensor size beyond which smaller apertures reduce resolution because of diffraction, which spreads light across the image sensor. This was hardly a problem in the days of plate cameras and even 35mm film, but has become an insurmountable limitation with the very small format sensors used in digital cameras and especially video cameras. First generation HD consumer camcorders used 1/4 inch sensors, for which apertures smaller than about f4 begin to limit resolution. Even professional video cameras mostly use 2/3 inch sensors, prohibiting the use of apertures around f16 that would have been considered normal for film formats. Certain cameras (such as the Pentax K10D
) feature an "MTF autoexposure" mode, where the choice of aperture is optimised for maximum sharpness. Typically this means somewhere in the middle of the aperture range.
effects. This has led to such cameras becoming preferred by some film and television programme makers over even professional HD video cameras, because of their 'filmic' potential. In theory the use of cameras with 16 and 21 megapixel sensors offers the possibility of almost perfect sharpness by downconversion within the camera, with digital filtering to eliminate aliasing. In practise such cameras currently fail in this respect and they do not have the processing power to do what is required. The Canon EOS 5D Mark II
is believed to use only every third line, and hence suffers bad aliasing, as its optical filter is optimised for stills use. The Panasonic Lumix DMC-GH2
may do some processing across pixels, producing very sharp images, but with some aliasing. Nevertheless, such cameras produce very impressive results, and appear to be leading the way in video production towards large-format downconversion with digital filtering becoming the standard approach to the realisation of a flat MTF with true freedom from aliasing.
where
and
are spatial frequency in the x- and y-plane, respectively.
Phase is critically important to adaptive optics
and holographic systems.
The OTF is the Fourier transform
of the incoherent Point Spread Function
.
The modulation transfer function represents the Bode plot
of an imaging system (such as a microscope or the human eye), and thus depicts the filtering characteristic of the imaging system. The human eye, for instance, acts as a low-pass filter
, in that very high-frequency components (sharp edges) cannot be perfectly perceived.
Imaging
Imaging is the representation or reproduction of an object's outward form; especially a visual representation .- Imaging methodologies and technologies :...
system (camera
Camera
A camera is a device that records and stores images. These images may be still photographs or moving images such as videos or movies. The term camera comes from the camera obscura , an early mechanism for projecting images...
, video
Video
Video is the technology of electronically capturing, recording, processing, storing, transmitting, and reconstructing a sequence of still images representing scenes in motion.- History :...
system, microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...
etc.) is the true measure of resolution
Image resolution
Image resolution is an umbrella term that describes the detail an image holds. The term applies to raster digital images, film images, and other types of images. Higher resolution means more image detail....
(image sharpness) that the system is capable of. The common practice of defining resolution in terms of pixel count is not meaningful, as it is the overall OTF of the complete system, including lens and anti-aliasing filter as well as other factors, that defines true performance. In the most common applications (cameras and video systems) it is the Modulation Transfer Function (the magnitude of the OTF), that is most relevant, although the phase component can have a secondary effect. While resolution
Optical resolution
Optical resolution describes the ability of an imaging system to resolve detail in the object that is being imaged.An imaging system may have many individual components including a lens and recording and display components...
, as commonly used with reference to camera systems, describes only the number of pixels in an image, and hence the potential to show fine detail, the transfer function describes the ability of adjacent pixels to change from black to white in response to patterns of varying spatial frequency, and hence the actual capability to show fine detail, whether with full or reduced contrast. An image reproduced with an optical transfer function that 'rolls off' at high spatial frequencies will appear 'blurred' in everyday language. Modulation Transfer Function or MTF (the OTF magnitude with phase ignored) is roughly the equivalent of frequency response in an audio system, and can be represented by a graph of light amplitude (brightness) versus spatial frequency (cycles per picture width).
Example
Taking the example of a current High Definition video system, with 1920 by 1080 pixels, the NyquistNyquist rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...
theorem says that it should be possible, in a perfect system, to resolve fully (with true black to white transitions) nearly 1920 alternate black and white lines, otherwise referred to as a spatial frequency of 960 line pairs per picture width, or 960 cycles per picture width, (definitions in terms of cycles per unit angle or per mm are also possible but generally less clear when dealing with cameras and more appropriate to telescopes etc.). In practice this is far from the case, and spatial frequencies that approach the Nyquist rate will generally be reproduced with decreasing amplitude, so that fine detail, though it can be seen, is greatly reduced in contrast. This gives rise to the interesting observation that, for example, a standard definition television picture derived from a film scanner that uses oversampling
Oversampling
In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...
, as described later, may appear sharper than a high definition picture shot on a camera with a poor Modulation Transfer Function. The two picture show an interesting difference that is often missed, the former having full contrast on detail up to a certain point but then no really fine detail, while the latter does contain finer detail, but with such reduced contrast as to appear inferior overall.
Factors affecting MTF in typical camera systems
In practice, many factors result in considerable blurring of a reproduced image, such that patterns with spatial frequency just below the Nyquist rate may not even be visible, and the finest patterns that can be seen appear 'washed out' as shades of grey, not black and white. A major factor is usually the impossibility of making the perfect 'brick wall' optical filter (often realised as a 'phase plate' or a lens with specific blurring properties in digital cameras and video camcorders). Such a filter is necessary to reduce aliasingAliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
by eliminating spatial frequencies above the Nyquist rate
Nyquist rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...
, but in practice it will have a response that 'rolls off' seriously before the Nyquist frequency is reached.
Oversampling and downconversion to maintain MTF
For this reason, the only way in practice to approach the theoretical sharpness possible in a digital imaging system such as a camera is to use more pixels in the camera sensor than samplesSampling (signal processing)
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....
in the final image, and 'downconvert' or 'interpolate' using special digital processing which cuts off high frequencies above the Nyquist rate to avoid aliasing whilst maintaining a reasonably flat MTF up to that frequency. This approach was first taken in the 1970s when flying spot scanners, and later CCD
Charge-coupled device
A charge-coupled device is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time...
line scanners, were developed which sampled more pixels than were needed and then 'downconverted', which is why movies have always looked sharper on television than other material shot with a video camera. The only theoretically correct way to interpolate or downconvert is by use of a steep low-pass spatial filter, realised by convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
with a two-dimensional sinx/x weighting
Weighting
The process of weighting involves emphasizing the contribution of some aspects of a phenomenon to a final effect or result — giving them 'more weight' in the analysis. That is, rather than each variable in the data contributing equally to the final result, some data are adjusted to contribute...
function which requires powerful processing. In practice, various mathematical approximations to this are used to reduce the processing requirement. These approximations are now implemented widely in video editing systems and in image processing programs such as Photoshop.
Just as standard definition video with a flat MTF is only possible with oversampling, so HD television with full theoretical sharpness is only possible by starting with a camera that has at least twice as many pixels, and then digitally filtering. With movies now being shot in 4k
4K
4K or 4-K may refer to:* 4K resolution, a digital film resolution standard* A temperature of 4 kelvins* 4000 * 4096 = 4 × 1024* As a category in demoscene compos* 4K disk sector size* Askari Aviation IATA code...
and even 8k video for the cinema, using cameras like the Red
Red
Red is any of a number of similar colors evoked by light consisting predominantly of the longest wavelengths of light discernible by the human eye, in the wavelength range of roughly 630–740 nm. Longer wavelengths than this are called infrared , and cannot be seen by the naked eye...
, we can expect to see the best pictures on HDTV only from movies or material shot at the higher standard. However much we raise the number of pixels used in cameras, this will always remain true (unless a perfect optical spatial filter can be devised), and the same problem exists of course with stills cameras, where a better image can be expected when, say, a 10 megapixel image is converted to a 5 megapixel image, than could ever be obtained from a even the best 5 megapixel camera. Because of this problem of maintaining a flat MTF, broadcasters like the BBC
BBC
The British Broadcasting Corporation is a British public service broadcaster. Its headquarters is at Broadcasting House in the City of Westminster, London. It is the largest broadcaster in the world, with about 23,000 staff...
did for a long time consider maintaining standard definition television, but improving its quality by shooting and viewing with many more pixels (though as previously mentioned, such a system, though impressive, does ultimately lack the very fine detail which, though attenuated, enhances the effect of true HD viewing).
Another factor in digital cameras and camcorders is lens resolution. A lens may be said to 'resolve' 1920 horizontal lines, but this does not mean that it does so with full modulation from black to white. The 'Modulation Transfer Function' (just a term for the magnitude of the optical transfer function with phase ignored) gives the true measure of lens performance, and is represented by a graph of amplitude against spatial frequency.
Lens aperture diffraction also limits MTF. Whilst reducing the aperture of a lens usually reduces aberrations and hence improves the flatness of the MTF, there is an optimum aperture for any lens and image sensor size beyond which smaller apertures reduce resolution because of diffraction, which spreads light across the image sensor. This was hardly a problem in the days of plate cameras and even 35mm film, but has become an insurmountable limitation with the very small format sensors used in digital cameras and especially video cameras. First generation HD consumer camcorders used 1/4 inch sensors, for which apertures smaller than about f4 begin to limit resolution. Even professional video cameras mostly use 2/3 inch sensors, prohibiting the use of apertures around f16 that would have been considered normal for film formats. Certain cameras (such as the Pentax K10D
Pentax K10D
The Pentax K10D and similar Samsung GX-10 are 10.2 megapixel digital single-lens reflex cameras launched in late 2006. They were developed in a collaboration between Pentax of Japan and Samsung of Korea....
) feature an "MTF autoexposure" mode, where the choice of aperture is optimised for maximum sharpness. Typically this means somewhere in the middle of the aperture range.
The Trend to Digital Large-Format SLRs and improved MTF Potential
There has recently been a shift towards the use of large image format digital single lens reflex cameras driven by the need for low-light sensitivity and narrow depth of fieldDepth of field
In optics, particularly as it relates to film and photography, depth of field is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image...
effects. This has led to such cameras becoming preferred by some film and television programme makers over even professional HD video cameras, because of their 'filmic' potential. In theory the use of cameras with 16 and 21 megapixel sensors offers the possibility of almost perfect sharpness by downconversion within the camera, with digital filtering to eliminate aliasing. In practise such cameras currently fail in this respect and they do not have the processing power to do what is required. The Canon EOS 5D Mark II
Canon EOS 5D Mark II
The Canon EOS 5D Mark II is a 21.1-megapixel full-frame CMOS digital single-lens reflex camera made by Canon. It succeeds the EOS 5D and was announced on September 17, 2008.-Improvements compared to original EOS 5D:...
is believed to use only every third line, and hence suffers bad aliasing, as its optical filter is optimised for stills use. The Panasonic Lumix DMC-GH2
Panasonic Lumix DMC-GH2
The Panasonic Lumix DMC-GH2 is a digital camera with HD video recording capability that uses the Micro Four Thirds System. Though commonly referred to as a DSLR camera, it has no mirror or optical viewfinder, but has instead both a fold-out LCD screen and a electronic viewfinder...
may do some processing across pixels, producing very sharp images, but with some aliasing. Nevertheless, such cameras produce very impressive results, and appear to be leading the way in video production towards large-format downconversion with digital filtering becoming the standard approach to the realisation of a flat MTF with true freedom from aliasing.
Measuring Modulation Transfer Function
Although 'sharpness' is often judged on grid patterns of alternate black and white lines, it should strictly be measured using a sine-wave variation from black to white (a blurred version of the usual pattern). Where a square wave pattern is used (simple black and white lines) not only is there more risk of aliasing, but account must be taken of the fact that the fundamental component of a square wave is higher than the amplitude of the square wave itself (the harmonic components reduce the peak amplitude). A square wave test chart will therefore show optimistic results (better resolution of high spatial frequencies than is actually achieved). The square wave result is sometimes referred to as the 'contrast transfer function' (CTF).More Advanced Details
OTF may be broken down into the magnitude transfer function and phase transfer function components as follows:where
and
are spatial frequency in the x- and y-plane, respectively.Phase is critically important to adaptive optics
Adaptive optics
Adaptive optics is a technology used to improve the performance of optical systems by reducing the effect of wavefront distortions. It is used in astronomical telescopes and laser communication systems to remove the effects of atmospheric distortion, and in retinal imaging systems to reduce the...
and holographic systems.
The OTF is the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the incoherent Point Spread Function
Point spread function
The point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
.
The modulation transfer function represents the Bode plot
Bode plot
A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response...
of an imaging system (such as a microscope or the human eye), and thus depicts the filtering characteristic of the imaging system. The human eye, for instance, acts as a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
, in that very high-frequency components (sharp edges) cannot be perfectly perceived.
See also
- Strehl ratioStrehl ratioThe Strehl ratio, named after the German physicist and mathematician Karl Strehl , is a measure for the optical quality of telescopes and other imaging instruments...
- Optical resolutionOptical resolutionOptical resolution describes the ability of an imaging system to resolve detail in the object that is being imaged.An imaging system may have many individual components including a lens and recording and display components...
- Transfer functionTransfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
- Wavefront codingWavefront codingIn optics and signal processing, wavefront coding is a method for creating optical transfer functions of lenses with specially designed phase masks, encoding, to produce point spread functions of visible light, images, with manipulatable information such as depth of field and distance.Wavefront...
- Modulation transfer function (infrared imaging)Modulation transfer function (infrared imaging)The Modulation Transfer Function is used to approximate the position of best focus of an infrared imaging system. In an imaging system, best focus is typically achieved when the MTF is between 0.4 and 0.6; most often at 0.5 MTF is inversely related to the minimum resolvable temperature difference...
- Signal to noise ratio (image processing)Signal to noise ratio (image processing)The Signal to Noise Ratio is used in imaging as a physical measure of the sensitivity of a imaging system. Industry standards measure SNR in decibels of power and therefore apply the 20 log rule to the "pure" SNR ratio...
- Signal transfer functionSignal transfer functionThe signal transfer function is a measure of the signal output versus the signal input of a system such as an infrared system or sensor. There are many general applications of the SiTF...