# Diffraction, Photosite Size, Modulation Transfer Functions

This is a continuation of previous efforts here: http://www.dpreview.com/forums/thread/3447868

How lens diffraction effects, sensor photosite size, and optical low-pass filter (OLPF, AA Filter) effects influence the net composite MTF of a lens/camera system when the product of Wavelength and F-Number is increased (for the purpose of reducing the effects of certain types of lens aberrations, and/or realizing deeper Depth of Field) is a matter of interest to me.

The following analysis is not intended to be a rigorous accounting for all factors involved in determining the net composite Modulation Transfer Function (MTF) of a lens/camera system.

For the sake of simplicity, it assumes photosite aperture equal to photosite pitch (100% fill factor).

It does not account for the effects of micro-lenses, and it does not account for effects of the other optical filtering located within an image-sensor's "filter stack". It appears (from some things that I have read) that effects of lens aberrations - as well as diffusion in epitaxial layers (at least when photosite aperture does equal photosite pitch) - are not directly multiplicative in the spatial frequency domain, thus requiring ray-tracing analysis in order to determine their effects upon the composite system MTF.

While it is indeed true that a number of factors (lens aberrations, focus integrity, and camera movement, in addition to post-capture processing, print/display limitations, etc.) will all act to decrease the composite MTF, the relevant contributions of diffraction, photosite aperture, and optical low-pass filtering can still be compared in order to determine their composite effects.

Because de-mosaicing algorithms are complicated and differing in how many individual photosites are combined in the process of rendering their image-data into pixels in some RGB color-space, they are not considered here. The analytical model used relates to a monochromatic image-sensor.

.

All BLUE colored plots in the diagrams below represent the MTF resulting from lens-system diffraction through a circular aperture (only). The product of the Wavelength multiplied by the F-Number (in spatial units of distance) is equal to the reciprocal of the unit spatial sampling frequency (X-axis).

.

All RED colored plots in the diagrams below represent the composite MTF resulting from lens-system diffraction through a circular aperture combined (convolved together) with reference photosites of dimensions equal to the reciprocal of the unit spatial sampling frequency (appearing along the X-axis).

In all but the first diagram appearing below, AA Filters of varying "strengths" (varying locations of the first zero response spatial frequency as a fraction of the appropriate spatial sampling frequency) are also combined (convolved) together with the lens-system diffraction and photosites in order to calculate the composite MTF response.

Note that for the RED colored plots, the Shannon-Nyquist spatial frequency limit exists in the center of the X-axis (denoted by a small green colored square). In those cases, the area of primary interest exists in the left half of the diagrams, and the comparative analysis describing the diagrams below will relate to the MTF response existing at that Shannon-Nyquist spatial frequency limit.

.

All GREEN colored plots in the diagrams below represent the composite MTF resulting from lens-system diffraction through a circular aperture combined (convolved together) with photosites of dimensions that are equal to one-half the dimensions of the reference photosites represented in the RED plots. The purpose is to compare results when photosite size is halved.

In all but the first diagram appearing below, AA Filters of varying "strengths" (varying locations of the first zero response spatial frequency as a fraction of the appropriate spatial sampling frequency) are also combined (convolved) together with the lens-system diffraction and photosites in order to calculate the composite MTF response.

Note that for the GREEN colored plots, the Shannon-Nyquist spatial frequency limit exists at the right-most end of the X-axis. In those cases, the area of interest extends across the entire diagram. This additional spatial frequency resolution afforded by the higher spatial sampling frequency represents a significant portion of the benefits realized from using the smaller photosites.

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. No AA Filter exists in the model in the case of this first diagram. The use of the smaller photosite results in a moderate (25% > 35%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 3/4 (75.00%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a more significant (13% > 30%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 2/3 (66.67%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a slightly larger (10% > 29%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 5/8 (62.50%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a slightly larger (8% > 28%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 1/2 (50.00%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a more significant (0% > 25%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

Comments:

Degradation of composite system Modulation Transfer Function (MTF) due to lens-system diffraction through a circular aperture, though gradual in nature, is indeed tangible. When spatial frequency equals the product of the Wavelength multiplied by the F-Number, the composite system MTF is reduced to 0% (complete "extinction" of the MTF composite response).

Nothing will make the effects of lens-system diffraction upon the composite system MTF better without exacting some cost in terms of performance. Sharpening techniques may be able to improve matters somewhat by selectively boosting higher frequencies using subtractive high-pass or deconvolution filtering techniques - but at some cost of reduced signal/noise ratio. Attempting to increase the signal/noise ratio using noise-reduction techniques reduces the composite system MTF.

Reducing the physical dimensions of photosites (and any associated AA Filters) will (up to a point) increase the composite system MTF somewhat, and more importantly will extend the spatial frequency response to higher spatial frequencies. The lower limit of photosite size exists where the photosite size equals one-half of the product of the Wavelength multiplied by the F-Number. Further, meaningful resolution also depends upon the existence of a viable system signal/noise ratio.

DM ...

For those who may be interesting in the mathematical identities utilized to calculate the MTFs:

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Lens Diffraction: (2/pi) * ( ArcCosine(f) - (f) * Sqrt ( 1 - (f)^2 ) )

where f is the dimensionless product of Spatial Frequency multiplied by Wavelength * F-Number.

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Photosite Aperture (100% Fill) convolved together with Optical Lowpass Filter assembly:

Absolute Value ( ( A * Sin (pi * B * f) / (pi * B * f) ) + ( C * Sin (pi * D * f) / (pi * D * f) ) )

where: A = 1/2 + Offset; B = 1 + 2 * Offset; C = 1/2 - Offset; D = 1 - 2 * Offset

and f is the dimensionless product of Spatial Frequency multiplied by Photosite Aperture (100% Fill).

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Notes:

Offset of 0.250000 yields first zero response at 1.000000 times the Spatial Sampling Frequency.

Offset of 0.333333 yields first zero response at 0.750000 times the Spatial Sampling Frequency.

Offset of 0.375000 yields first zero response at 0.666667 times the Spatial Sampling Frequency.

Offset of 0.400000 yields first zero response at 0.625000 times the Spatial Sampling Frequency.

Offset of 0.500000 yields first zero response at 0.500000 times the Spatial Sampling Frequency.

Detail Man wrote:

All BLUE colored plots in the diagrams below represent the MTF resulting from lens-system diffraction through a circular aperture (only). The product of the Wavelength multiplied by the F-Number (in spatial units of distance) is equal to the reciprocal of the unit spatial sampling frequency (X-axis).

Note that for the BLUE colored plots, the Shannon-Nyquist spatial frequency limit (relating only to the RED colored plots, as explained below) does not apply. The BLUE colored plots do not represent spatially sampled information - they represent the MTF response of the lens-system only, and are as a result valid up to the spatial frequency at which complete "extinction" of the composite system MTF response occurs.

However, since any lens-camera system is comprised of photosites (as well as any existing AA Filters), the BLUE colored plots are useful only for comparing the difference between the effects of the lens-system (when considered in isolation) and the composite system MTF response.

All RED colored plots in the diagrams below represent the composite MTF resulting from lens-system diffraction through a circular aperture combined (convolved together) with reference photosites of dimensions equal to the reciprocal of the unit spatial sampling frequency (appearing along the X-axis).

In all but the first diagram appearing below, AA Filters of varying "strengths" (varying locations of the first zero response spatial frequency as a fraction of the appropriate spatial sampling frequency) are also combined (convolved) together with the lens-system diffraction and photosites in order to calculate the composite MTF response.

Note that for the RED colored plots, the Shannon-Nyquist spatial frequency limit exists in the center of the X-axis (denoted by a small green colored square). In those cases, the area of primary interest exists in the left half of the diagrams, and the comparative analysis ...

... relating specifically to differences between the RED and the GREEN colored plots ...

... describing the diagrams below will relate to the MTF response existing at that Shannon-Nyquist spatial frequency limit.

All GREEN colored plots in the diagrams below represent the composite MTF resulting from lens-system diffraction through a circular aperture combined (convolved together) with photosites of dimensions that are equal to one-half the dimensions of the reference photosites represented in the RED plots. The purpose is to compare results when photosite size is halved.

Note that for the GREEN colored plots, the Shannon-Nyquist spatial frequency limit exists at the right-most end of the X-axis. In those cases, the area of interest extends across the entire diagram. This additional spatial frequency resolution afforded by the higher spatial sampling frequency represents a significant portion of the benefits realized from using the smaller photosites.

As stated in the addendum above, the BLUE colored plots representing diffraction (in addition to the GREEN colored plots representing photosite as well as any existing AA Filtering for the case of one-half sized photosites) are valid across the entire domain (X-axis of the) diagram.

The BLUE colored plots represent the MTF response of the lens-system diffraction which multiplies together with the MTF of the photosites (combined with any existing AA Filters), reducing the composite system MTF to a value of zero at a spatial frequency corresponding to the product of the Wavelength multiplied by the F-Number - (regardless) of the size of the photosites.

Who knew ?

Detail Man wrote:

This is a continuation of previous efforts here: http://www.dpreview.com/forums/thread/3447868

How lens diffraction effects, sensor photosite size, and optical low-pass filter (OLPF, AA Filter) effects influence the net composite MTF of a lens/camera system when the product of Wavelength and F-Number is increased (for the purpose of reducing the effects of certain types of lens aberrations, and/or realizing deeper Depth of Field) is a matter of interest to me.

The following analysis is not intended to be a rigorous accounting for all factors involved in determining the net composite Modulation Transfer Function (MTF) of a lens/camera system.

For the sake of simplicity, it assumes photosite aperture equal to photosite pitch (100% fill factor).

It does not account for the effects of micro-lenses, and it does not account for effects of the other optical filtering located within an image-sensor's "filter stack". It appears (from some things that I have read) that effects of lens aberrations - as well as diffusion in epitaxial layers (at least when photosite aperture does equal photosite pitch) - are not directly multiplicative in the spatial frequency domain, thus requiring ray-tracing analysis in order to determine their effects upon the composite system MTF.

While it is indeed true that a number of factors (lens aberrations, focus integrity, and camera movement, in addition to post-capture processing, print/display limitations, etc.) will all act to decrease the composite MTF, the relevant contributions of diffraction, photosite aperture, and optical low-pass filtering can still be compared in order to determine their composite effects.

Because de-mosaicing algorithms are complicated and differing in how many individual photosites are combined in the process of rendering their image-data into pixels in some RGB color-space, they are not considered here. The analytical model used relates to a monochromatic image-sensor.

.

All BLUE colored plots in the diagrams below represent the MTF resulting from lens-system diffraction through a circular aperture (only). The product of the Wavelength multiplied by the F-Number (in spatial units of distance) is equal to the reciprocal of the unit spatial sampling frequency (X-axis).

.

All RED colored plots in the diagrams below represent the composite MTF resulting from lens-system diffraction through a circular aperture combined (convolved together) with reference photosites of dimensions equal to the reciprocal of the unit spatial sampling frequency (appearing along the X-axis).

Note that for the RED colored plots, the Shannon-Nyquist spatial frequency limit exists in the center of the X-axis (denoted by a small green colored square). In those cases, the area of primary interest exists in the left half of the diagrams, and the comparative analysis describing the diagrams below will relate to the MTF response existing at that Shannon-Nyquist spatial frequency limit.

.

All GREEN colored plots in the diagrams below represent the composite MTF resulting from lens-system diffraction through a circular aperture combined (convolved together) with photosites of dimensions that are equal to one-half the dimensions of the reference photosites represented in the RED plots. The purpose is to compare results when photosite size is halved.

Note that for the GREEN colored plots, the Shannon-Nyquist spatial frequency limit exists at the right-most end of the X-axis. In those cases, the area of interest extends across the entire diagram. This additional spatial frequency resolution afforded by the higher spatial sampling frequency represents a significant portion of the benefits realized from using the smaller photosites.

An alternative way that the GREEN and the BLUE colored plots can be interpreted is as (rather than representing a one-half sized photosite with or without any indicated AA Filter included), instead representing the full sized reference photosite described (with or without any indicated AA Filter included), and in a situation where the product of the Wavelength multiplied by the F-Number of the lens-system instead equals twice the numerical value (as compared to its numerical value in the original case).

In the alternative case described, the modified resolution "extinction" spatial frequency of the BLUE colored lens-system diffraction MTF response (existing at the far right edge of the X-axis, at X=1.0) coincides with the Shannon-Nyquist spatial frequency limit for the full sized photosite aperture (instead of coinciding with two times the Shannon-Nyquist spatial frequency limit for the full sized photosite aperture).

As a result, regardless of the presence or absence of an AA Filter of any "strength", the MTF response of the lens-system itself limits the composite system MTF response (shown in the GREEN colored plot) to the Shannon-Nyquist spatial frequency limit. Therefore, in the case of this alternative way of reading the diagrams, the affects of any AA Filter cease to be relevant.

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. No AA Filter exists in the model in the case of this first diagram. The use of the smaller photosite results in a moderate (25% > 35%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 3/4 (75.00%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a more significant (13% > 30%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 2/3 (66.67%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a slightly larger (10% > 29%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 5/8 (62.50%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a slightly larger (8% > 28%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

In this diagram, the BLUE plot represents the lens-system diffraction, the RED plot represents the lens-system diffraction combined with the reference photosites, and the GREEN plot represents the lens-system diffraction combined with the smaller (one-half size) photosites. AA Filters with a first spatial frequency of zero response equal to 1/2 (50.00%) of the spatial sampling frequency are included in the model. The use of the smaller photosite results in a more significant (0% > 25%) improvement in the MTF response at the Shannon-Nyquist frequency:

.

Comments:

Degradation of composite system Modulation Transfer Function (MTF) due to lens-system diffraction through a circular aperture, though gradual in nature, is indeed tangible. When spatial frequency equals the product of the Wavelength multiplied by the F-Number, the composite system MTF is reduced to 0% (complete "extinction" of the MTF composite response).

Nothing will make the effects of lens-system diffraction upon the composite system MTF better without exacting some cost in terms of performance. Sharpening techniques may be able to improve matters somewhat by selectively boosting higher frequencies using subtractive high-pass or deconvolution filtering techniques - but at some cost of reduced signal/noise ratio. Attempting to increase the signal/noise ratio using noise-reduction techniques reduces the composite system MTF.

Reducing the physical dimensions of photosites (and any associated AA Filters) will (up to a point) increase the composite system MTF somewhat, and more importantly will extend the spatial frequency response to higher spatial frequencies. The lower limit of photosite size exists where the photosite size equals one-half of the product of the Wavelength multiplied by the F-Number. Further, meaningful resolution also depends upon the existence of a viable system signal/noise ratio.

Rather than thinking in terms of Airy disk diameters in relation to photosite dimensions, thinking in terms of the composite system MTF response is more revealing and indicative of actual results.

In a case where the F-Number equals the photosite size divided by a worst-case wavelength (such as 700nM), the effects of the photosite (combined with any AA Filter) on the composite system MTF response are going to be more significant than the effects of lens-system diffraction.

In a case where the F-Number equals two times the photosite size divided by a worst-case wavelength (such as 700nM), the effects of the lens-system diffraction are going to be as significant as the photosite on the composite system MTF response, and the effects of any AA Filter are not significant - as the lens-system diffraction causes "extinction" of the MTF response at the Shannon-Nyquist spatial frequency limit, anyway.

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The above being said, focus integrity, camera movement, post-capture processing, print/display limitations, and the limitations of the human vision of viewers (represented by a given COC diameter) are all factors that will further reduce the magnitude of the composite system MTF - all of which tend to make the effects of lens-system diffraction less predominant a factor.

In addition, lens-system aberrations will also reduce the magnitude of the composite system MTF, and increasing the F-Number to the point where the composite system MTF response is maximized makes sense (regardless of any other factors, such as calculated Airy disk diameters). At F-Numbers above that point (for any given Focal Length), the system is "diffraction limited".

DM ...

In a clean-up after a disrespectful post, 2 of my answers were deleted as well. So let me repeat, that I don't find these things boring at all and want to thank you for the effort you put in there. I did a similar exercise a while ago, so I know what it means to present these things as you did. I would have shown you my results, but they were in a german forum, written german. And even worse, their server crashed a few weeks ago and all the images, graphs and attachment from years are lost.

I did the exercise to understand, and possibly explain to others, how the AA-filter works and why it is a "low-pass filter" (which it isn't). But I had to realize, that almost nobody "out there" can read these curves. Therefore I now prefer to explain the combined effect of photosite size, diffraction, lens aberrations, motion/shake blur, defocus etc. as a combination of individual blurs, usually by just summing up their areas. That is a much more simplistic model, but it works to explain some basic principles. And a few people even start to understand, that the pixel pich is not any magic limit in such a way, that an image is sharp as long as alle the blurs are smaller. No, the pixel pitch stands on the same side of the equation and is an image blur contributor in itself.

Gruß, masi1157

Detail Man wrote:

Rather than thinking in terms of Airy disk diameters in relation to photosite dimensions, thinking in terms of the composite system MTF response is more revealing and indicative of actual results.

In a case where the F-Number equals the photosite size divided by a worst-case wavelength (such as 700nM), the effects of the photosite (combined with any AA Filter) on the composite system MTF response are going to be more significant than the effects of lens-system diffraction.

In a case where the F-Number equals two times the photosite size divided by a worst-case wavelength (such as 700nM), the effects of the lens-system diffraction are going to be as significant as the photosite on the composite system MTF response, and the effects of any AA Filter are not significant - as the lens-system diffraction causes "extinction" of the MTF response at the Shannon-Nyquist spatial frequency limit, anyway.

The above being said, focus integrity, camera movement, post-capture processing, print/display limitations, and the limitations of the human vision of viewers (represented by a given COC diameter) are all factors that will further reduce the magnitude of the composite system MTF - all of which tend to make the effects of lens-system diffraction less predominant a factor.

In addition, lens-system aberrations will also reduce the magnitude of the composite system MTF, and increasing the F-Number to the point where the composite system MTF response is maximized makes sense (regardless of any other factors, such as calculated Airy disk diameters). At F-Numbers above that point (for any given Focal Length), the system is "diffraction limited".

Here are two graphs that demonstrate how much Photosite Size (and an AA Filter with zero response at the Nyquist frequency) effects can dominate over Diffraction effects [between F-Number = (Photosite Size) / Wavelength and F-Number = (2)*(Photosite Size) / Wavelength].

The first graph shows F-Number = (Photosite Size) / Wavelength. The BLUE colored plot shows the diffraction effects only. The RED colored plot is the composite system MTF with Diffraction + AA Filter + Photosite all taken into account. The GREEN colored plot is the composite system MTF with Diffraction + (1/2 sized) AA Filter + (1/2 sized) Photosite all taken into account (improving the composite system MTF response fairly significantly as a result):

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The next graph shows F-Number = (2)*(Photosite Size) / Wavelength. The BLUE colored plot shows the diffraction effects only. The RED colored plot is the composite system MTF with Diffraction + Photosite all taken into account. The GREEN colored plot is the composite system MTF with Diffraction + (1/2 sized) Photosite taken into account.

Note that (now) diffraction effects dominate over any Photosite effects, and using a smaller (1/2 sized) Photosite does almost nothing whatsoever to improve the system MTF response:

One can calculate and use this "F-Number window" to make practical and meaningful estimates (without ever staring at Airy disk diameters, and wondering about just how big may be too big).

Note that the effects of diffraction through a circular aperture (the BLUE colored plots) act to reduce composite system MTF response (the spatial frequency contrast resolution) from zero spatial frequency onward. When increasing F-Number is continuing to reduce the negative effects upon the composite system MTF of certain lens-aberrations, increasing the F-Number is a "net positive" effect (on the the composite system MTF), however (regardless of the particular Airy disk diameter).

DM ...

... the relationship between diffraction softening and pixel density is largely misunderstood. For a given sensor size and lens, more pixels always result in more detail -- that's a fact. As we stop down and the DOF deepens, we reach a point where we begin to lose detail due to diffraction softening. As a consequence, photos made with more pixels will begin to lose their detail advantage earlier and quicker than images made with fewer pixels, but they will always retain more detail. Eventually, the additional detail afforded by the extra pixels becomes trivial (most certainly by f/32 on FF).

josephjamesphotography.com/equivalence/index.htm#diffraction

I think that it is important to point out that "more detail" from "more pixles" does not imply spatial frequency resolution extending beyond diffraction caused extinction of the composite system MTF response.

It can mean a larger area under the composite system MTF response curve up to the relevant Shannon-Nyquist spatial frequency sampling limit (considering the particular Photosite-pitch), however.

Here are two graphs that demonstrate how rapidly Diffraction effects begin to dominate over Photosite Size (where Photosite Aperture equals Photosite Pitch, and without any AA Filter), with respect to differences in area under the MTF curve as well as its highest spatial frequency.

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The first graph shows unit Photosite Size = (Wavelength) * (F-Number)

The BLUE colored plot shows the diffraction effects only.

The RED colored plot is the composite system MTF with Diffraction convolved together with Photosite Aperture taken into account. The Nyquist spatial frequency limit exists at F=0.5 (halfway across the X-axis).

The GREEN colored plot is the composite system MTF with Diffraction convolved together with a 1/2 sized Photosite. The Nyquist spatial frequency limit exists at F=1.0 (at the far-right of the X-axis).

In the first case, a larger area under the (relevant sections of the) composite system MTF response curves exists in the case of the smaller (1/2 size) Photosite. The fact that the smaller Photosite provides one octave higher spatial frequency resolution before reaching the Nyquist spatial frequency limit is a significant factor in extending spatial frequency resolution, as well as increasing the integral.

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The second graph shows Photosite Size = (Wavelength) * (F-Number) / 2

The BLUE colored plot shows the diffraction effects only.

The RED colored plot is the composite system MTF with Diffraction convolved together with Photosite Aperture (here equal to Photosite-Pitch) taken into account. The Nyquist spatial frequency limit now exists at F=1.0 (at the far-right of the X-axis).

The GREEN colored plot is the composite system MTF with Diffraction convolved together with a 1/2 sized Photosite. The Nyquist spatial frequency limit now exists at F=2.0 (past the far-right of X-axis).

In the second case, note that the fact that the smaller (1/2 size) Photosite has one octave higher spatial frequency response before reaching the Nyquist spatial frequency limit is a virtually insignificant factor in extending the spatial frequency resolution as well as in increasing the integral.

Note that (now) diffraction effects clearly dominate over any Photosite effects, and using a smaller (1/2 sized) Photosite does almost nothing whatsoever to improve the system MTF response:

Adding AA Filters results in some additional differences - but only arising out of AA Filter responses.

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... the effects of diffraction softening affect all systems equally at the same AOV and DOF, not the same f-ratio.

josephjamesphotography.com/equivalence/index.htm#diffraction

True enough in describing the effects of the Entrance Pupil Diameter resulting from scaling the F-Ratio by the ratio of the Crop Factors (as well as scaling the Focal Length by the ratio of the Crop Factors in order to maintain a constant AOV) in the case of differing Sensor Sizes requiring differing Enlargement Factors in order to produce a given Circle of Confusion (COC) diameter at a common Viewing Size. This relates specifically to the effects of defocusing due to COC diameter.

Diffraction effects themselves (as they contribute to composite system MTF response) are (at any given Wavelength) a function of F-Ratio, Photosite Size, and AA Filter beam-splitting offsets.

However, that does not mean that all systems record the same detail at the same AOV and DOF. Even though diffraction softening affects all systems equally at the same AOV and DOF, the system that began with more detail will retain more detail (although, as the DOF deepens, all systems asymptotically approach zero detail).

josephjamesphotography.com/equivalence/index.htm#diffraction

While it could be said that the ratio of the areas under the composite system MTF responses of the compared systems asymptotically approaches unity (1.0), the composite system MTF responses of both systems (at least in the case of the ideal "Chinese hat" function representing the Fourier transform of diffraction through a circular aperture) does more than asymptotically approach zero - they actually equal zero at a spatial frequency equal to or greater than the reciprocal of the product of the Wavelength multiplied by F-Ratio.

I recognize that other optical effects also exist in lens-system and image-sensor systems. Would the individual (or combined) nature of any such additional effects somehow manage to result in a situation where the composite system MTF response would not equal zero at or above a spatial frequency equal to the reciprocal of the product of the Wavelength multiplied by F-Ratio ? If so, would such additional effects be meaningful in terms of a generalized analytical model - or would they represent optical effects that are in some way specific only to particular designs ?

DM ...

Detail Man wrote:

... the relationship between diffraction softening and pixel density is largely misunderstood. For a given sensor size and lens, more pixels always result in more detail -- that's a fact. As we stop down and the DOF deepens, we reach a point where we begin to lose detail due to diffraction softening. As a consequence, photos made with more pixels will begin to lose their detail advantage earlier and quicker than images made with fewer pixels, but they will always retain more detail. Eventually, the additional detail afforded by the extra pixels becomes trivial (most certainly by f/32 on FF).

josephjamesphotography.com/equivalence/index.htm#diffraction

I think that it is important to point out that "more detail" from "more pixles" does not imply spatial frequency resolution extending beyond diffraction caused extinction of the composite system MTF response.

.

This ^^ point in particular cold use some elaboration.

I may suggest that you try to summarize what you're presenting in layman's terms, and perhaps give examples if you have the equipment (and time) available to do so. (See Joseph James' paragraph you quote above as example, or Falk Lumo's summaries to his work.)

As masi1157 hinted at, you may have more responders to this thread if it were presented in a more accessible way, with your information summarized, and then made available as-is for reference.

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Here are a few of my favorite things...

---> http://www.flickr.com/photos/95095968@N00/sets/72157626171532197/

moving_comfort wrote:

Detail Man wrote:

... the relationship between diffraction softening and pixel density is largely misunderstood. For a given sensor size and lens, more pixels always result in more detail -- that's a fact. As we stop down and the DOF deepens, we reach a point where we begin to lose detail due to diffraction softening. As a consequence, photos made with more pixels will begin to lose their detail advantage earlier and quicker than images made with fewer pixels, but they will always retain more detail. Eventually, the additional detail afforded by the extra pixels becomes trivial (most certainly by f/32 on FF).

josephjamesphotography.com/equivalence/index.htm#diffraction

I think that it is important to point out that "more detail" from "more pixles" does not imply spatial frequency resolution extending beyond diffraction caused extinction of the composite system MTF response.

.

This ^^ point in particular cold use some elaboration.

"Detail" (like "sharpness") is a somewhat vague term. "Resolution" relates to the maximum spatial frequency of a MTF response - whereas "contrast" and "acuity" appear to relate more to the area under the MTF response at spatial frequencies lower than the absolute highest spatial frequencies. This thread provided links to some interesting resources for exploring that further:

http://www.dpreview.com/forums/thread/3135840

I may suggest that you try to summarize what you're presenting in layman's terms, and perhaps give examples if you have the equipment (and time) available to do so. (See Joseph James' paragraph you quote above as example, or Falk Lumo's summaries to his work.)

Ironically, both of which I (personally) find to be (sometimes) vague in their intended brevity.

This stuff is hard to adequately encompass in brief summaries. Read the content if you choose. If you have specific questions, post them here, and I will try my best to provide you with specific answers.

As masi1157 hinted at, you may have more responders to this thread if it were presented in a more accessible way, with your information summarized, and then made available as-is for reference.

Maybe. My primary goal is to communicate to and with readers willing and able to comprehend. I do my level best to write as clearly and as simply as possible, given the subject-matter that is involved.

Detail Man wrote:

moving_comfort wrote:

Detail Man wrote:

josephjamesphotography.com/equivalence/index.htm#diffraction

.

This ^^ point in particular cold use some elaboration.

"Detail" (like "sharpness") is a somewhat vague term. "Resolution" relates to the maximum spatial frequency of a MTF response - whereas "contrast" and "acuity" appear to relate more to the area under the MTF response at spatial frequencies lower than the absolute highest spatial frequencies. This thread provided links to some interesting resources for exploring that further:

http://www.dpreview.com/forums/thread/3135840

I may suggest that you try to summarize what you're presenting in layman's terms, and perhaps give examples if you have the equipment (and time) available to do so. (See Joseph James' paragraph you quote above as example, or Falk Lumo's summaries to his work.)

Ironically, both of which I (personally) find to be (sometimes) vague in their intended brevity.

This stuff is hard to adequately encompass in brief summaries. Read the content if you choose. If you have specific questions, post them here, and I will try my best to provide you with specific answers.

As masi1157 hinted at, you may have more responders to this thread if it were presented in a more accessible way, with your information summarized, and then made available as-is for reference.

Maybe. My primary goal is to communicate to and with readers willing and able to comprehend. I do my level best to write as clearly and as simply as possible, given the subject-matter that is involved.

.

It seems that a sophisticated reader such as yourself might be able to explain "Equivalence" to me:

Subject: A half-hearted Coup

moving_comfort wrote:

Detail Man wrote: .... long, long snip....

I recently proposed referring to such rough corollaries as "metametric" as opposed to "equivalent":

http://www.dpreview.com/forums/post/51322339

... but the participants involved in that particular discussion seemed too embroiled to pay attention.

Sounds like you really would like to replace "King James" with "King Detail Man," but no-one else is getting on board the boat to your new kingdom.

I think the concept of 'equivalency' as described by Bustard is succinct, easy to understand and 100% accurate - for what it is and tries to describe.

A technical note that may interest a few.

The part of the red curve to the right if the zero that is above the Nyquist frequency is phase reversed. If you were photographing a the converging tapered lines at these frequencies you would see the pattern go "Negative". It is called aliasing.

When you stop down and make that go away the picture may seem to lose detail but you really have not lost "information" This is what anti aliasing filter do

A simple way to relate diffraction to pixel size. As a rule of thumb the diffraction spot size about equal to the f/number. About 1.2 time the f/number for green and and about twice it for red. This at f/2 you spot is about 2 microns in diameter for blue light. For a 1/2 16million pixel sensor pitch or sample size is about 1.6 microns diffraction spot is thus a bit larger than a pixel. Nyquist sampling wants the spot to be about two pixels across. Thus, at f/2 if you had a perfect lens the 1/2" camera would be at its diffraction limit.

Most lenses under ideal conditions resolve to about 3 of 4 pixel pitches.

In this way of treating it the sizes add as the square root of the sum of the squares like each is a side a right triangle and the result is the hypotenuse. So if we had lens that was at 4 pixels and were at f/3 our combined spot size is about 5 pixels across. That is why I do;t like to shoot at f/8 even with my ultra zoom. Instead I use a polarizer and ISO 100.

As you get to larger pixels the diffraction stays the so lens performance becomes the more important factor.

>Nothing will make the effects of lens-system diffraction upon the composite system MTF better >without exacting some cost in terms of performance. Sharpening techniques may be able to >improve matters somewhat by selectively boosting higher frequencies using subtractive high-pass >or deconvolution filtering techniques - but at some cost of reduced signal/noise ratio. Attempting >to increase the signal/noise ratio using noise-reduction techniques reduces the composite system >MTF.

Is MTF a good way to generally describe noise reduction? If you use a highly structured test-chart, I would think that sharpening + noise reduction could, indeed, improve "MTF" and "SNR" - as measured.

Problem is, natural scenes are not as highly structured/predictable, and man-made sharpening + noise-reduction will tend to enhance things perceptually characterized as "detail/sharpness" and "noisiness" at the cost of e.g. "artifacts" in a non-linear, scene-dependent manner.

-h

hjulenissen wrote:

>Nothing will make the effects of lens-system diffraction upon the composite system MTF better >without exacting some cost in terms of performance. Sharpening techniques may be able to >improve matters somewhat by selectively boosting higher frequencies using subtractive high-pass >or deconvolution filtering techniques - but at some cost of reduced signal/noise ratio. Attempting >to increase the signal/noise ratio using noise-reduction techniques reduces the composite system >MTF.

Is MTF a good way to generally describe noise reduction? If you use a highly structured test-chart, I would think that sharpening + noise reduction could, indeed, improve "MTF" and "SNR" - as measured.

Laplace/Fourier/Z transforms, etc. only apply to linear systems that are describable by homogenous differential/difference equations having constant coefficients. While that would apply to a linear sharpening process such as Unsharp Masking, it does not apply to NR processes (being non-linear).

(As I understand it), the multiplication of MTF responses cannot be used to describe (at least some) lens-system aberrations, or the optical response of epitaxial diffusion-layers of photosites in cases where the photosite-aperture does not equal the photosite-pitch (< 100% fill-factor).

Problem is, natural scenes are not as highly structured/predictable, and man-made sharpening + noise-reduction will tend to enhance things perceptually characterized as "detail/sharpness" and "noisiness" at the cost of e.g. "artifacts" in a non-linear, scene-dependent manner.

And, of course, human visual (as with aural) perception itself is anything but linear in nature ...

Detail Man wrote:

Laplace/Fourier/Z transforms, etc. only apply to linear systems that are describable by homogenous differential/difference equations having constant coefficients. While that would apply to a linear sharpening process such as Unsharp Masking, it does not apply to NR processes (being non-linear).

Even USM is non-linear (description from wikipedia):

"The same differencing principle is used in the unsharp-masking tool in many digital-imaging software packages, such as Adobe Photoshop and GIMP.[2] The software applies a Gaussian blur to a copy of the original image and then compares it to the original. If the difference is greater than a user-specified threshold setting, the images are (in effect) subtracted. The threshold control constrains sharpening to image elements that differ from each other above a certain size threshold, so that sharpening of small image details, such as photographic grain, can be suppressed."

In the case of such simple non-linearities, it might be possible to do frequency-domain analysis for the two regions separately ("sharpening mode" and "passthrough mode") and gain the desired insight from that.

-h

hjulenissen wrote:

Detail Man wrote:

Laplace/Fourier/Z transforms, etc. only apply to linear systems that are describable by homogenous differential/difference equations having constant coefficients. While that would apply to a linear sharpening process such as Unsharp Masking, it does not apply to NR processes (being non-linear).

Even USM is non-linear (description from wikipedia):

"The same differencing principle is used in the unsharp-masking tool in many digital-imaging software packages, such as Adobe Photoshop and GIMP.[2] The software applies a Gaussian blur to a copy of the original image and then compares it to the original. If the difference is greater than a user-specified threshold setting, the images are (in effect) subtracted. The threshold control constrains sharpening to image elements that differ from each other above a certain size threshold, so that sharpening of small image details, such as photographic grain, can be suppressed."

True enough when one uses a lower threshold/masking criteria (which is indeed a wise idea).

In the case of such simple non-linearities, it might be possible to do frequency-domain analysis for the two regions separately ("sharpening mode" and "passthrough mode") and gain the desired insight from that.

On a per-photosite level, yes. However, it seems that such an approach would break down on a spatial level when considering the output of multiple photosites along a horizontal/vertical path.

Human visual perception itself adds a final non-linear twist that is unavoidable. Our perception of "sharpness" seems to be somewhat intangible itself - and may well relate much more to the area under the MTF response curve at lower and mid spatial frequencies than how far out in spatial frequency the MTF response extends (when the area under the curve of that extension is relatively small). You might find some of the references in this thread to be interesting:

An in-depth and quite interesting 2008 paper entitled, "Does resolution really increase image quality?", authored in part by Frederic Guichard, Chief Scientist at DxO Labs, and that DxOLabs provides only upon specific request is downloadable here:

- Fujifilm X-T223.6%
- Nikon D50025.4%
- Nikon AF-S 105mm F1.4E8.2%
- Olympus M.Zuiko 12-100mm F47.5%
- Panasonic Lumix DMC-G857.2%
- Sigma 85mm F1.4 Art6.7%
- Sigma 50-100mm F1.8 Art5.1%
- Sony a63006.4%
- Sony Cyber-shot RX10 III3.7%
- Sony Cyber-shot RX100 V6.3%