Optical transfer function

As a Fourier transform, the OTF is generally complex-valued; however, it is real-valued in the common case of a PSF that is symmetric about its center.

Its transfer function decreases approximately gradually with spatial frequency until it reaches the diffraction-limit, in this case at 500 cycles per millimeter or a period of 2 μm.

The impulse response of a well-focused optical system is a three-dimensional intensity distribution with a maximum at the focal plane, and could thus be measured by recording a stack of images while displacing the detector axially.

Generally, the optical transfer function depends on factors such as the spectrum and polarization of the emitted light and the position of the point source.

When significant variation occurs, the optical transfer function may be calculated for a set of representative positions or colors.

Note that sometimes the optical transfer function is given in units of the object or sample space, observation angle, film width, or normalized to the theoretical maximum.

As explained by the Nyquist–Shannon sampling theorem, to match the optical resolution of the given example, the pixels of each color channel should be separated by 1 micrometer, half the period of 500 cycles per millimeter.

While optical 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.

Taking the example of a current high definition (HD) video system, with 1920 by 1080 pixels, the Nyquist theorem states that it should be possible, in a perfect system, to resolve fully (with true black to white transitions) a total of 1920 black and white alternating lines combined, otherwise referred to as a spatial frequency of 1920/2=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.

Ideal systems such as in the examples here are readily calculated numerically using software such as Julia, GNU Octave or Matlab, and in some specific cases even analytically.

The optical transfer function of such a system can thus be calculated geometrically from the intersecting area between two identical disks at a distance of

At high numerical apertures such as those found in microscopy, it is important to consider the vectorial nature of the fields that carry light.

When the aberrations can be assumed to be spatially invariant, alternative patterns can be used to determine the optical transfer function such as lines and edges.

Such extended objects illuminate more pixels in the image, and can improve the measurement accuracy due to the larger signal-to-noise ratio.

The optical transfer function is in this case calculated as the two-dimensional discrete Fourier transform of the image and divided by that of the extended object.

It can be found directly from an ideal line approximation provided by a slit test target or it can be derived from the edge spread function, discussed in the next sub section.

Although the measurement images obtained with this technique illuminate a large area of the camera, this mainly benefits the accuracy at low spatial frequencies.

As shown in the right hand figure, an operator defines a box area encompassing the edge of a knife-edge test target image back-illuminated by a black body.

In case it is more practical to measure the edge spread function, one can determine the line spread function as follows: Typically the ESF is only known at discrete points, so the LSF is numerically approximated using the finite difference: where: 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).

A square wave test chart will therefore show optimistic results (better resolution of high spatial frequencies than is actually achieved).

A major factor is usually the impossibility of making the perfect 'brick wall' optical filter (often realized as a 'phase plate' or a lens with specific blurring properties in digital cameras and video camcorders).

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 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.

The only theoretically correct way to interpolate or downconvert is by use of a steep low-pass spatial filter, realized by convolution with a two-dimensional sin(x)/x weighting function which requires powerful processing.

However much we raise the number of pixels used in cameras, this will always remain true in absence of a perfect optical spatial filter.

Because of the problem of maintaining a high contrast 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).

[12] 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 field effects.

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 realization of a flat MTF with true freedom from aliasing.

The optical contrast reduction can be partially reversed by digitally amplifying spatial frequencies selectively before display or further processing.

Although more advanced digital image restoration procedures exist, the Wiener deconvolution algorithm is often used for its simplicity and efficiency.

Illustration of the optical transfer function (OTF) and its relation to image quality. The optical transfer function of a well-focused (a), and an out-of-focus optical imaging system without aberrations (d). As the optical transfer function of these systems is real and non-negative, the optical transfer function is by definition equal to the modulation transfer function (MTF). Images of a point source and a spoke target with high spatial frequency are shown in (b,e) and (c,f), respectively. Note that the scale of the point source images (b,e) is four times smaller than the spoke target images.
Various closely related characterizations of an optical system exhibiting coma, a typical aberration that occurs off-axis. (a) The point-spread function (PSF) is the image of a point source. (b) The image of a line is referred to as the line-spread function, in this case a vertical line. The line-spread function is directly proportional to the vertical integration of the point-spread image. The optical-transfer function (OTF) is defined as the Fourier transform of the point-spread function and is thus generally a two-dimensional complex function. Typically only a one-dimensional slice is shown (c), corresponding to the Fourier transform of the line-spread function. The thick green line indicates the real part of the function, and the thin red line the imaginary part. Often only the absolute value of the complex function is shown, this allows visualization of the two-dimensional function (d); however, more commonly only the one-dimensional function is shown (e). The latter is typically normalized at the spatial frequency zero and referred to as the modulation transfer function (MTF). For completeness, the complex argument is sometimes provided as the phase transfer function (PhTF), shown in panel (f).
When viewed through an optical system with trefoil aberration, the image of a point object will look as a three-pointed star (a). As the point-spread function is not rotational symmetric, only a two-dimensional optical transfer function can describe it well (b). The height of the surface plot indicates the absolute value and the hue indicates the complex argument of the function. A spoke target imaged by such an imaging device is shown by the simulation in (c).
The three-dimensional point spread functions (a,c) and corresponding modulation transfer functions (b,d) of a wide-field microscope (a,b) and confocal microscope (c,d). In both cases the numerical aperture of the objective is 1.49 and the refractive index of the medium 1.52. The wavelength of the emitted light is assumed to be 600 nm and, in case of the confocal microscope, that of the excitation light 500 nm with circular polarization. A section is cut to visualize the internal intensity distribution. The colors as shown on the logarithmic color scale indicate the irradiance (a,c) and spectral density (b,d) normalized to the maximum value.
The MTF data versus spatial frequency is normalized by fitting a sixth order polynomial to it, making a smooth curve. The 50% cut-off frequency is determined and the corresponding spatial frequency is found, yielding the approximate position of best focus .
In evaluating the ESF , an operator defines a box area equivalent to 10% [ citation needed ] of the total frame area of a knife-edge test target back-illuminated by a black body . The area is defined to encompass the edge of the target image.