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103-541: Either transfer function specifies the response to a periodic sine-wave pattern passing through the lens system, as a function of its spatial frequency or period, and its orientation. Formally, the OTF is defined as the Fourier transform of the point spread function (PSF, that is, the impulse response of the optics, the image of a point source). As a Fourier transform, the OTF is complex-valued; but it will be real-valued in

206-414: A knife-edge test target image back-illuminated by a black body . The box area is defined to be approximately 10% of the total frame area. The image pixel data is translated into a two-dimensional array ( pixel intensity and pixel position). The amplitude (pixel intensity) of each line within the array is normalized and averaged. This yields the edge spread function. where The line spread function

309-454: A support that is half of that of the confocal microscope in all three-dimensions, confirming the previously noted lower resolution of the wide-field microscope. Note that along the z -axis, for x  =  y  = 0, the transfer function is zero everywhere except at the origin. This missing cone is a well-known problem that prevents optical sectioning using a wide-field microscope. The two-dimensional optical transfer function at

412-427: A transfer function (also known as system function or network function ) of a system, sub-system, or component is a mathematical function that models the system's output for each possible input. It is widely used in electronic engineering tools like circuit simulators and control systems . In simple cases, this function can be represented as a two-dimensional graph of an independent scalar input versus

515-405: A two-port electronic circuit, such as an amplifier , might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical actuator might be the mechanical displacement of the movable arm as a function of electric current applied to the device; the transfer function of a photodetector might be

618-568: 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. 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

721-431: A high definition picture shot on a camera with a poor modulation transfer function. The two pictures 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. Although one typically thinks of an image as planar, or two-dimensional,

824-406: A large area of the camera, this mainly benefits the accuracy at low spatial frequencies. As with the line spread function, each measurement only determines a single axes of the optical transfer function, repeated measurements are thus necessary if the optical system cannot be assumed to be rotational symmetric. As shown in the right hand figure, an operator defines a box area encompassing the edge of

927-505: A reflex camera will generally demagnify objects at a distance of 5 meter by a factor of 100 to 200. The resolution of a digital imaging device is not only limited by the optics, but also by the number of pixels, more in particular by their separation distance. 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

1030-501: 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

1133-432: A single dimension, by consequence, the optical transfer function can only be determined for a single dimension using a single line-spread function (LSF). If necessary, the two-dimensional optical transfer function can be determined by repeating the measurement with lines at various angles. The line spread function can be found using two different methods. It can be found directly from an ideal line approximation provided by

SECTION 10

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1236-461: A slit test target or it can be derived from the edge spread function, discussed in the next sub section. The two-dimensional Fourier transform of an edge is also only non-zero on a single line, orthogonal to the edge. This function is sometimes referred to as the edge spread function (ESF). However, the values on this line are inversely proportional to the distance from the origin. Although the measurement images obtained with this technique illuminate

1339-413: A system to a sinusoidal input beginning at time t = 0 {\displaystyle t=0} will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady-state response; it corresponds to the homogeneous solution of

1442-413: A system to a sinusoidal input beginning at time t = 0 {\displaystyle t=0} will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady-state response; it corresponds to the homogeneous solution of

1545-506: Is The output is related to the input by the transfer function H ( s ) {\displaystyle H(s)} as and the transfer function itself is If a complex harmonic signal with a sinusoidal component with amplitude | X | {\displaystyle |X|} , angular frequency ω {\displaystyle \omega } and phase arg ⁡ ( X ) {\displaystyle \arg(X)} , where arg

1648-506: Is The output is related to the input by the transfer function H ( s ) {\displaystyle H(s)} as and the transfer function itself is If a complex harmonic signal with a sinusoidal component with amplitude | X | {\displaystyle |X|} , angular frequency ω {\displaystyle \omega } and phase arg ⁡ ( X ) {\displaystyle \arg(X)} , where arg

1751-401: Is actually achieved). The square wave result is sometimes referred to as the 'contrast transfer function' (CTF). 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 appear 'washed out' as shades of grey, not black and white. A major factor

1854-482: Is an acceptable representation of their input-output behavior. Descriptions are given in terms of a complex variable , s = σ + j ⋅ ω {\displaystyle s=\sigma +j\cdot \omega } . In many applications it is sufficient to set σ = 0 {\displaystyle \sigma =0} (thus s = j ⋅ ω {\displaystyle s=j\cdot \omega } ), which reduces

1957-405: Is captured in the image as a function of spatial frequency. The MTF tends to decrease with increasing spatial frequency from 1 to 0 (at the diffraction limit); however, the function is often not monotonic . On the other hand, when also the pattern translation is important, the complex argument of the optical transfer function can be depicted as a second real-valued function, commonly referred to as

2060-415: 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

2163-599: Is identical to the first derivative of the edge spread function, which is differentiated using numerical methods . 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

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2266-468: Is important to consider the vectorial nature of the fields that carry light. By decomposing the waves in three independent components corresponding to the Cartesian axes, a point spread function can be calculated for each component and combined into a vectorial point spread function. Similarly, a vectorial optical transfer function can be determined as shown in () and (). The optical transfer function

2369-438: Is not only useful for the design of optical system, it is also valuable to characterize manufactured systems. The optical transfer function is defined as the Fourier transform of the impulse response of the optical system, also called the point spread function . The optical transfer function is thus readily obtained by first acquiring the image of a point source, and applying the two-dimensional discrete Fourier transform to

2472-545: Is often used exclusively to refer to linear time-invariant (LTI) systems. Most real systems have non-linear input-output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear that LTI system theory is an acceptable representation of their input-output behavior. Descriptions are given in terms of a complex variable , s = σ + j ⋅ ω {\displaystyle s=\sigma +j\cdot \omega } . In many applications it

2575-410: Is provided by the optical transfer function. Optical systems, and in particular optical aberrations are not always rotationally symmetric. Periodic patterns that have a different orientation can thus be imaged with different contrast even if their periodicity is the same. Optical transfer function or modulation transfer functions are thus generally two-dimensional functions. The following figures shows

2678-513: Is sufficient to set σ = 0 {\displaystyle \sigma =0} (thus s = j ⋅ ω {\displaystyle s=j\cdot \omega } ), which reduces the Laplace transforms with complex arguments to Fourier transforms with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often

2781-415: Is superior to that of a wide-field microscope in all three dimensions. A three-dimensional optical transfer function can be calculated as the three-dimensional Fourier transform of the 3D point-spread function. Its color-coded magnitude is plotted in panels (b) and (d), corresponding to the point-spread functions shown in panels (a) and (c), respectively. The transfer function of the wide-field microscope has

2884-452: Is the amplitude of the output as a function of the frequency of the input signal. The transfer function of an electronic filter is the amplitude at the output as a function of the frequency of a constant amplitude sine wave applied to the input. For optical imaging devices, the optical transfer function is the Fourier transform of the point spread function (a function of spatial frequency ). Transfer functions are commonly used in

2987-650: Is the argument is input to a linear time-invariant system, the corresponding component in the output is: In a linear time-invariant system, the input frequency ω {\displaystyle \omega } has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. The frequency response H ( j ω ) {\displaystyle H(j\omega )} describes this change for every frequency ω {\displaystyle \omega } in terms of gain and phase shift The phase delay (the frequency-dependent amount of delay introduced to

3090-650: Is the argument is input to a linear time-invariant system, the corresponding component in the output is: In a linear time-invariant system, the input frequency ω {\displaystyle \omega } has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. The frequency response H ( j ω ) {\displaystyle H(j\omega )} describes this change for every frequency ω {\displaystyle \omega } in terms of gain and phase shift The phase delay (the frequency-dependent amount of delay introduced to

3193-716: Is the circle segment angle. By substituting | ν | = cos ⁡ ( θ / 2 ) {\displaystyle |\nu |=\cos(\theta /2)} , and using the equalities sin ⁡ ( θ ) / 2 = sin ⁡ ( θ / 2 ) cos ⁡ ( θ / 2 ) {\displaystyle \sin(\theta )/2=\sin(\theta /2)\cos(\theta /2)} and 1 = ν 2 + sin ⁡ ( arccos ⁡ ( | ν | ) ) 2 {\displaystyle 1=\nu ^{2}+\sin(\arccos(|\nu |))^{2}} ,

Optical transfer function - Misplaced Pages Continue

3296-485: Is the operator defined on the relevant function space transforms u into r . That kind of equation can be used to constrain the output function u in terms of the forcing function r . The transfer function can be used to define an operator F [ r ] = u {\displaystyle F[r]=u} that serves as a right inverse of L , meaning that L [ F [ r ] ] = r {\displaystyle L[F[r]]=r} . Solutions of

3399-485: Is the operator defined on the relevant function space transforms u into r . That kind of equation can be used to constrain the output function u in terms of the forcing function r . The transfer function can be used to define an operator F [ r ] = u {\displaystyle F[r]=u} that serves as a right inverse of L , meaning that L [ F [ r ] ] = r {\displaystyle L[F[r]]=r} . Solutions of

3502-654: Is the spatial frequency normalized to the highest transmitted frequency. In general the optical transfer function is normalized to a maximum value of one for ν = 0 {\displaystyle \nu =0} , so the resulting area should be divided by π {\displaystyle \pi } . The intersecting area can be calculated as the sum of the areas of two identical circular segments : θ / 2 − sin ⁡ ( θ ) / 2 {\displaystyle \theta /2-\sin(\theta )/2} , where θ {\displaystyle \theta }

3605-402: 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). Such a filter is necessary to reduce aliasing by eliminating spatial frequencies above the Nyquist rate of the display. The only way in practice to approach the theoretical sharpness possible in

3708-426: Is valid for any number of transfer-function poles. If x ( t ) {\displaystyle x(t)} is the input to a general linear time-invariant system , and y ( t ) {\displaystyle y(t)} is the output, and the bilateral Laplace transform of x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)}

3811-426: Is valid for any number of transfer-function poles. If x ( t ) {\displaystyle x(t)} is the input to a general linear time-invariant system , and y ( t ) {\displaystyle y(t)} is the output, and the bilateral Laplace transform of x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)}

3914-406: 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, realized by convolution with a two-dimensional sin( x )/ x weighting function which requires powerful processing. In practice, various mathematical approximations to this are used to reduce

4017-420: Is widely used in electronic engineering tools like circuit simulators and control systems . In simple cases, this function can be represented as a two-dimensional graph of an independent scalar input versus the dependent scalar output (known as a transfer curve or characteristic curve ). Transfer functions for components are used to design and analyze systems assembled from components, particularly using

4120-415: Is zero around 250 cycles/mm, or periods of 4 μm. This explains why the images for the out-of-focus system (e,f) are more blurry than those of the diffraction-limited system (b,c). Note that although the out-of-focus system has very low contrast at spatial frequencies around 250 cycles/mm, the contrast at spatial frequencies near the diffraction limit of 500 cycles/mm is diffraction-limited. Close observation of

4223-525: The Laplace transforms with complex arguments to Fourier transforms with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case in signal processing and communication theory ), not the fleeting turn-on and turn-off transient response or stability issues. For continuous-time input signal x ( t ) {\displaystyle x(t)} and output y ( t ) {\displaystyle y(t)} , dividing

Optical transfer function - Misplaced Pages Continue

4326-462: The block diagram technique, in electronics and control theory . Dimensions and units of the transfer function model the output response of the device for a range of possible inputs. The transfer function of a two-port electronic circuit, such as an amplifier , might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical actuator might be

4429-417: The contrast transfer function (CTF) . A perfect lens system will provide a high contrast projection without shifting the periodic pattern, hence the optical transfer function is identical to the modulation transfer function. Typically the contrast will reduce gradually towards zero at a point defined by the resolution of the optics. For example, a perfect, non-aberrated , f/4 optical imaging system used, at

4532-507: The convolution theorem it can be seen that the optical transfer function is in fact the autocorrelation of the pupil function . The pupil function of an ideal optical system with a circular aperture is a disk of unit radius. 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 2 ν {\displaystyle 2\nu } , where ν {\displaystyle \nu }

4635-569: The differential equation . The transfer function for an LTI system may be written as the product: where s P i are the N roots of the characteristic polynomial and will be the poles of the transfer function. In a transfer function with a single pole H ( s ) = 1 s − s P {\displaystyle H(s)={\frac {1}{s-s_{P}}}} where s P = σ P + j ω P {\displaystyle s_{P}=\sigma _{P}+j\omega _{P}} ,

4738-569: The differential equation . The transfer function for an LTI system may be written as the product: where s P i are the N roots of the characteristic polynomial and will be the poles of the transfer function. In a transfer function with a single pole H ( s ) = 1 s − s P {\displaystyle H(s)={\frac {1}{s-s_{P}}}} where s P = σ P + j ω P {\displaystyle s_{P}=\sigma _{P}+j\omega _{P}} ,

4841-489: The discrete Fourier transform of the line spread function. This data is graphed against the spatial frequency data. In this case, a sixth order polynomial is fitted to the MTF vs. spatial frequency curve to show the trend. The 50% cutoff frequency is determined to yield the corresponding spatial frequency. Thus, the approximate position of best focus of the unit under test is determined from this data. The Fourier transform of

4944-400: The phase transfer function ( PhTF ). The complex-valued optical transfer function can be seen as a combination of these two real-valued functions: where and a r g ( ⋅ ) {\displaystyle \mathrm {arg} (\cdot )} represents the complex argument function, while ν {\displaystyle \nu } is the spatial frequency of

5047-631: The z -axis of the 3D optical transfer function correspond to the Dirac delta function . Most optical design software has functionality to compute the optical or modulation transfer function of a lens design. 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 can be calculated following two approaches: Mathematically both approaches are equivalent. Numeric calculations are typically most efficiently done via

5150-408: The 3D point-spread function in object space of a wide-field microscope (a) alongside that of a confocal microscope (c). Although the same microscope objective with a numerical aperture of 1.49 is used, it is clear that the confocal point spread function is more compact both in the lateral dimensions (x,y) and the axial dimension (z). One could rightly conclude that the resolution of a confocal microscope

5253-463: The Fourier transform; however, analytic calculation may be more tractable using the auto-correlation approach. Since the optical transfer function is the Fourier transform of the point spread function , and the point spread function is the square absolute of the inverse Fourier transformed pupil function , the optical transfer function can also be calculated directly from the pupil function . From

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5356-879: The Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the z-transform (notated with a corresponding capital letter, like X ( z ) {\displaystyle X(z)} and Y ( z ) {\displaystyle Y(z)} ), so a discrete-time system's transfer function can be written as: H ( z ) = Y ( z ) X ( z ) = Z { y [ n ] } Z { x [ n ] } . {\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.} A linear differential equation with constant coefficients where u and r are suitably smooth functions of t , and L

5459-824: The Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the z-transform (notated with a corresponding capital letter, like X ( z ) {\displaystyle X(z)} and Y ( z ) {\displaystyle Y(z)} ), so a discrete-time system's transfer function can be written as: H ( z ) = Y ( z ) X ( z ) = Z { y [ n ] } Z { x [ n ] } . {\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.} A linear differential equation with constant coefficients where u and r are suitably smooth functions of t , and L

5562-407: The Laplace transform of a general sinusoid of unit amplitude will be 1 s − j ω i {\displaystyle {\frac {1}{s-j\omega _{i}}}} . The Laplace transform of the output will be H ( s ) s − j ω 0 {\displaystyle {\frac {H(s)}{s-j\omega _{0}}}} , and

5665-407: The Laplace transform of a general sinusoid of unit amplitude will be 1 s − j ω i {\displaystyle {\frac {1}{s-j\omega _{i}}}} . The Laplace transform of the output will be H ( s ) s − j ω 0 {\displaystyle {\frac {H(s)}{s-j\omega _{0}}}} , and

5768-622: The Laplace transform of the input, X ( s ) = L { x ( t ) } {\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}} , yields the system's transfer function H ( s ) {\displaystyle H(s)} : which can be rearranged as: Discrete-time signals may be notated as arrays indexed by an integer n {\displaystyle n} (e.g. x [ n ] {\displaystyle x[n]} for input and y [ n ] {\displaystyle y[n]} for output). Instead of using

5871-817: The Laplace transform of the output, Y ( s ) = L { y ( t ) } {\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}} , by the Laplace transform of the input, X ( s ) = L { x ( t ) } {\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}} , yields the system's transfer function H ( s ) {\displaystyle H(s)} : which can be rearranged as: Discrete-time signals may be notated as arrays indexed by an integer n {\displaystyle n} (e.g. x [ n ] {\displaystyle x[n]} for input and y [ n ] {\displaystyle y[n]} for output). Instead of using

5974-423: The analysis of systems such as single-input single-output filters in signal processing , communication theory , and control theory . The term is often used exclusively to refer to linear time-invariant (LTI) systems. Most real systems have non-linear input-output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear that LTI system theory

6077-521: The case in signal processing and communication theory ), not the fleeting turn-on and turn-off transient response or stability issues. For continuous-time input signal x ( t ) {\displaystyle x(t)} and output y ( t ) {\displaystyle y(t)} , dividing the Laplace transform of the output, Y ( s ) = L { y ( t ) } {\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}} , by

6180-411: The center of the image, and deteriorate toward the edges of the field-of-view. When significant variation occurs, the optical transfer function may be calculated for a set of representative positions or colors. Sometimes it is more practical to define the transfer functions based on a binary black-white stripe pattern. The transfer function for an equal-width black-white periodic pattern is referred to as

6283-457: The common case of a PSF that is symmetric about its center. The MTF is formally defined as the magnitude (absolute value) of the complex OTF. The image on the right shows the optical transfer functions for two different optical systems in panels (a) and (d). The former corresponds to the ideal, diffraction-limited , imaging system with a circular pupil . Its transfer function decreases approximately gradually with spatial frequency until it reaches

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6386-410: The definition of transfer function , O T F ( 0 ) = M T F ( 0 ) {\displaystyle \mathrm {OTF} (0)=\mathrm {MTF} (0)} should indicate the fraction of light that was detected from the point source object. However, typically the contrast relative to the total amount of detected light is most important. It is thus common practice to normalize

6489-416: The dependent scalar output (known as a transfer curve or characteristic curve ). Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory . Dimensions and units of the transfer function model the output response of the device for a range of possible inputs. The transfer function of

6592-410: The detector axially. By consequence, the three-dimensional optical transfer function can be defined as the three-dimensional Fourier transform of the impulse response. Although typically only a one-dimensional, or sometimes a two-dimensional section is used, the three-dimensional optical transfer function can improve the understanding of microscopes such as the structured illumination microscope. True to

6695-401: The diffraction-limit, in this case at 500 cycles per millimeter or a period of 2 μm. Since periodic features as small as this period are captured by this imaging system, it could be said that its resolution is 2 μm. Panel (d) shows an optical system that is out of focus. This leads to a sharp reduction in contrast compared to the diffraction-limited imaging system. It can be seen that the contrast

6798-468: The equation for the area can be rewritten as arccos ⁡ ( | ν | ) − | ν | 1 − ν 2 {\displaystyle \arccos(|\nu |)-|\nu |{\sqrt {1-\nu ^{2}}}} . Hence the normalized optical transfer function is given by: A more detailed discussion can be found in and. The one-dimensional optical transfer function can be calculated as

6901-587: 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

7004-477: The focal plane can be calculated by integration of the 3D optical transfer function along the z -axis. Although the 3D transfer function of the wide-field microscope (b) is zero on the z -axis for z  ≠ 0; its integral, the 2D optical transfer, reaching a maximum at x  =  y  = 0. This is only possible because the 3D optical transfer function diverges at the origin x  =  y  =  z  = 0. The function values along

7107-404: The gray circular bands in the spoke image shown in the above figure. In between the gray bands, the spokes appear to invert from black to white and vice versa , this is referred to as contrast inversion, directly related to the sign reversal in the real part of the optical transfer function, and represents itself as a shift by half a period for some periodic patterns. While it could be argued that

7210-809: The homogeneous constant-coefficient differential equation L [ u ] = 0 {\displaystyle L[u]=0} can be found by trying u = e λ t {\displaystyle u=e^{\lambda t}} . That substitution yields the characteristic polynomial The inhomogeneous case can be easily solved if the input function r is also of the form r ( t ) = e s t {\displaystyle r(t)=e^{st}} . By substituting u = H ( s ) e s t {\displaystyle u=H(s)e^{st}} , L [ H ( s ) e s t ] = e s t {\displaystyle L[H(s)e^{st}]=e^{st}} if we define Other definitions of

7313-809: The homogeneous constant-coefficient differential equation L [ u ] = 0 {\displaystyle L[u]=0} can be found by trying u = e λ t {\displaystyle u=e^{\lambda t}} . That substitution yields the characteristic polynomial The inhomogeneous case can be easily solved if the input function r is also of the form r ( t ) = e s t {\displaystyle r(t)=e^{st}} . By substituting u = H ( s ) e s t {\displaystyle u=H(s)e^{st}} , L [ H ( s ) e s t ] = e s t {\displaystyle L[H(s)e^{st}]=e^{st}} if we define Other definitions of

7416-464: The image in panel (f) shows that the image of the large spoke densities near the center of the spoke target is relatively sharp. Since the optical transfer function (OTF) is defined as the Fourier transform of the point-spread function (PSF), it is generally speaking a complex-valued function of spatial frequency . The projection of a specific periodic pattern is represented by a complex number with absolute value and complex argument proportional to

7519-414: The imaging system will produce a three-dimensional intensity distribution in image space that in principle can be measured. e.g. a two-dimensional sensor could be translated to capture a three-dimensional intensity distribution. The image of a point source is also a three dimensional (3D) intensity distribution which can be represented by a 3D point-spread function. As an example, the figure on the right shows

7622-412: 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. Typically either a line or a black-white edge is used. The two-dimensional Fourier transform of a line through the origin, is a line orthogonal to it and through the origin. The divisor is thus zero for all but

7725-547: The line spread function (LSF) can not be determined analytically by the following equations: Therefore, the Fourier Transform is numerically approximated using the discrete Fourier transform D F T {\displaystyle {\mathcal {DFT}}} . where The MTF is then plotted against spatial frequency and all relevant data concerning this test can be determined from that graph. At high numerical apertures such as those found in microscopy, it

7828-410: The mechanical displacement of the movable arm as a function of electric current applied to the device; the transfer function of a photodetector might be the output voltage as a function of the luminous intensity of incident light of a given wavelength . The term "transfer function" is also used in the frequency domain analysis of systems using transform methods, such as the Laplace transform ; it

7931-469: 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. Taking

8034-439: The optical transfer function depicted in the following figure. As the ideal lens system, the contrast reaches zero at the spatial frequency of 500 cycles per millimeter. However, at lower spatial frequencies the contrast is considerably lower than that of the perfect system in the previous example. In fact, the contrast becomes zero on several occasions even for spatial frequencies lower than 500 cycles per millimeter. This explains

8137-403: The optical transfer function to the detected intensity, hence M T F ( 0 ) ≡ 1 {\displaystyle \mathrm {MTF} (0)\equiv 1} . 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. E.g. the image contrast and resolution are typically optimal at

8240-447: The output as a function of the frequency of a constant amplitude sine wave applied to the input. For optical imaging devices, the optical transfer function is the Fourier transform of the point spread function (a function of spatial frequency ). Transfer functions are commonly used in the analysis of systems such as single-input single-output filters in signal processing , communication theory , and control theory . The term

8343-409: The output voltage as a function of the luminous intensity of incident light of a given wavelength . The term "transfer function" is also used in the frequency domain analysis of systems using transform methods, such as the Laplace transform ; it is the amplitude of the output as a function of the frequency of the input signal. The transfer function of an electronic filter is the amplitude at

8446-406: The period of 500 cycles per millimeter. A higher number of pixels on the same sensor size will not allow the resolution of finer detail. On the other hand, when the pixel spacing is larger than 1 micrometer, the resolution will be limited by the separation between pixels; moreover, aliasing may lead to a further reduction of the image fidelity. An imperfect, aberrated imaging system could possess

8549-420: The periodic pattern. In general ν {\displaystyle \nu } is a vector with a spatial frequency for each dimension, i.e. it indicates also the direction of the periodic pattern. 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

8652-415: The point source. 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. The corresponding transfer functions are referred to as the line-spread function and the edge-spread function, respectively. Such extended objects illuminate more pixels in the image, and can improve the measurement accuracy due to

8755-413: The processing requirement. These approximations are now implemented widely in video editing systems and in image processing programs such as Photoshop . Transfer function In engineering , a transfer function (also known as system function or network function ) of a system, sub-system, or component is a mathematical function that models the system's output for each possible input. It

8858-402: The relative contrast and translation of the projected projection, respectively. Often the contrast reduction is of most interest and the translation of the pattern can be ignored. The relative contrast is given by the absolute value of the optical transfer function, a function commonly referred to as the modulation transfer function ( MTF ). Its values indicate how much of the object's contrast

8961-435: The resolution of both the ideal and the imperfect system is 2 μm, or 500 LP/mm, it is clear that the images of the latter example are less sharp. A definition of resolution that is more in line with the perceived quality would instead use the spatial frequency at which the first zero occurs, 10 μm, or 100 LP/mm. Definitions of resolution, even for perfect imaging systems, vary widely. A more complete, unambiguous picture

9064-431: The sampled image. Such a point-source can, for example, be a bright light behind a screen with a pin hole, a fluorescent or metallic microsphere , or simply a dot painted on a screen. Calculation of the optical transfer function via the point spread function is versatile as it can fully characterize optics with spatial varying and chromatic aberrations by repeating the procedure for various positions and wavelength spectra of

9167-707: The sinusoid by the transfer function) is The group delay (the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency ω {\displaystyle \omega } , The transfer function can also be shown using the Fourier transform , a special case of bilateral Laplace transform where s = j ω {\displaystyle s=j\omega } . Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used: In control engineering and control theory ,

9270-707: The sinusoid by the transfer function) is The group delay (the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency ω {\displaystyle \omega } , The transfer function can also be shown using the Fourier transform , a special case of bilateral Laplace transform where s = j ω {\displaystyle s=j\omega } . Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used: In control engineering and control theory ,

9373-429: The spokes become more and more blurred towards the center until they merge into a gray, unresolved, disc. 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. Conversion between the two is typically a matter of a multiplication or division. E.g. a microscope typically magnifies everything 10 to 100-fold, and

9476-419: The surface plots in the above figure indicate phase. It can be seen that, while for the rotational symmetric aberrations the phase is either 0 or π and thus the transfer function is real valued, for the non-rotational symmetric aberration the transfer function has an imaginary component and the phase varies continuously. While optical resolution , as commonly used with reference to camera systems, describes only

9579-417: The temporal output will be the inverse Laplace transform of that function: The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σ P is positive. For a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative and

9682-417: The temporal output will be the inverse Laplace transform of that function: The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σ P is positive. For a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative and

9785-486: The transfer function are used, for example 1 / p L ( i k ) . {\displaystyle 1/p_{L}(ik).} A general sinusoidal input to a system of frequency ω 0 / ( 2 π ) {\displaystyle \omega _{0}/(2\pi )} may be written exp ⁡ ( j ω 0 t ) {\displaystyle \exp(j\omega _{0}t)} . The response of

9888-486: The transfer function are used, for example 1 / p L ( i k ) . {\displaystyle 1/p_{L}(ik).} A general sinusoidal input to a system of frequency ω 0 / ( 2 π ) {\displaystyle \omega _{0}/(2\pi )} may be written exp ⁡ ( j ω 0 t ) {\displaystyle \exp(j\omega _{0}t)} . The response of

9991-435: The transfer function is derived with the Laplace transform . The transfer function was the primary tool used in classical control engineering. A transfer matrix can be obtained for any linear system to analyze its dynamics and other properties; each element of a transfer matrix is a transfer function relating a particular input variable to an output variable. A representation bridging state space and transfer function methods

10094-435: The transfer function is derived with the Laplace transform . The transfer function was the primary tool used in classical control engineering. A transfer matrix can be obtained for any linear system to analyze its dynamics and other properties; each element of a transfer matrix is a transfer function relating a particular input variable to an output variable. A representation bridging state space and transfer function methods

10197-496: The transient behavior will tend to zero in the limit of infinite time. The steady-state output will be: The frequency response (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude: which is the absolute value of the transfer function H ( s ) {\displaystyle H(s)} evaluated at j ω i {\displaystyle j\omega _{i}} . This result

10300-496: The transient behavior will tend to zero in the limit of infinite time. The steady-state output will be: The frequency response (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude: which is the absolute value of the transfer function H ( s ) {\displaystyle H(s)} evaluated at j ω i {\displaystyle j\omega _{i}} . This result

10403-405: The two-dimensional equivalent of the ideal and the imperfect system discussed earlier, for an optical system with trefoil , a non-rotational-symmetric aberration. Optical transfer functions are not always real-valued. Period patterns can be shifted by any amount, depending on the aberration in the system. This is generally the case with non-rotational-symmetric aberrations. The hue of the colors of

10506-411: The visible wavelength of 500 nm, would have the optical transfer function depicted in the right hand figure. It can be read from the plot that the contrast gradually reduces and reaches zero at the spatial frequency of 500 cycles per millimeter, in other words the optical resolution of the image projection is 1/500 of a millimeter, or 2 micrometer. Correspondingly, for this particular imaging device,

10609-540: Was proposed by Howard H. Rosenbrock , and is known as the Rosenbrock system matrix . In imaging , transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light. Transfer functions do not exist for many non-linear systems , such as relaxation oscillators ; however, describing functions can sometimes be used to approximate such nonlinear time-invariant systems. Transfer function In engineering ,

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