Image IQ

[Jack Hogan]

Hi Jack,

Besides me having a hard time seeing that as a common camera option, wouldn't sampling at every other pitch position reduce the resolution by half? That may not change the shape of the MTF but it is at a completely different scale, so the limiting resolution (or Nyquist) and MTF response for given feature sizes in cy/mm would change.

I'm not sure what's being overlooked.

Hey Bart,

What is being overlooked is that the slanted edge method produces a radial slice of the 2D MTF of the imaging system as a whole by first effectively projecting the 2D edge onto the 1D edge normal (ESF, its derivative being the LSF).  The LSF is therefore a 1D super-resolution projection of the 2D PSF.

The fact that it is super-sampled means that, within limits dictated by acceptable noise in the resulting curves, pitch more or less drops out of the equation, as Jim mentioned.  Incidentally, sampling aperture doesn't.  So when we show MTF in c/p or calculate Nyquist, we do so based on a priori knowledge of the layout of the sensor (e.g. a grid of known pitch).

Jack
 

[Jack Hogan]

Maybe because MTF mapper (using 10 bins per pixel) as a tool already takes care of it, dunno. The ISO states that one should only bin (they use 4 bins, 1/4th of a pixel resolution) "full phase rotations" if I recall their phrasing correctly. Meaning that if e.g. the slope of the slant is ArcTan(1/10) degrees, one should use a multiple of 10 edge pixels for the SFR evaluation. So a cropped Region of Interest (ROI) should have multiples of 10 pixels in dimension in the length. The data is also filtered with (if I recall correctly) a Hamming filter to avoid ripple effects.

Hi Bart,

Frans is the better person to ask, but MTF Mapper actually bins supersampled ESF data into bins 1/8th of a pixel wide.  The key is making sure that, for the given edge length and angle, there are enough samples to fill the bins.  MTF Mapper used to use a hamming window but has since evolved to better methods, including importance sampling and other neat improvements (see his blog for details).  With its recent releases I see cleaner curves than the ISO standard, sfrmat3 and probably (because I do not use it) Imatest.

Jack

[Jim Kasson]

Besides me having a hard time seeing that as a common camera option, wouldn't sampling at every other pitch position reduce the resolution by half? That may not change the shape of the MTF but it is at a completely different scale, so the limiting resolution (or Nyquist) and MTF response for given feature sizes in cy/mm would change.

I'm not sure what's being overlooked.

I think Jack is doing a better job of making my point than I was, but I do want to chime in with one explanation.

I initially introduced dropping samples as a thought experiment; a way to get to the comparison of the 4 um 100% FF and the 8 um 25% FF comparison with a single 4 um 100% FF sensor and a little post processing.

I now fear that that thought experiment has confused the issue. So let's go back to the 4um 100%/8um 25% comparison.

If the resolution of the binning of the slanted edge pixels is fixed at, say, 1 um (quarter pixel in the 4 um case and eighth pixel in the 8 um case), and the population of the 8 um sensor image is enough to average out the noise sufficiently, shouldn't the statistics of the binned edge be the same? I realize that the 4 um case will have less noise, since there are 4 times the number of samples.

Jim

[Jack Hogan]

let's go back to the 4um 100%/8um 25% comparison.

If the resolution of the binning of the slanted edge pixels is fixed at, say, 1 um (quarter pixel in the 4 um case and eighth pixel in the 8 um case), and the population of the 8 um sensor image is enough to average out the noise sufficiently, shouldn't the statistics of the binned edge be the same? I realize that the 4 um case will have less noise, since there are 4 times the number of samples.

Hi Jim,

I think the easiest way to think of this kind of question is to rely on the model for guidance.  Ignoring phase (reasonable in your ideal example) the system MTF can be modeled as the product of the MTFs due to diffraction (a known function of N and lambda) and due to pixel aperture (a function of pixel width and shape) convolved with the sampling grid (typically a comb, function of sampling interval):

MTFsys = MTFdiff x MTFpxAp ** COMBsamp

MTFdiff we know and it is what it is for the given setup;
MTFpxAp varies with the size and shape of the pixel, in your example size would be 4um in one case and 8um in the other; and
COMBsamp is not pixel pitch in the slanted edge method but the pitch of sampling-bin spacing (1/4th and 1/8th of a pixel in your example).  Either way it is immaterial in the interval of interest to us (o-1 c/p) because it pushes images of the system MTF out to 4 and 8 c/p respectively.

Pixel pitch is worked back into MTF when we specify the frequency axis.  Here is a clumsy attempt at explaining how this is done.

Jack

[Jim Kasson]

Hi Jim,

I think the easiest way to think of this kind of question is to rely on the model for guidance.  Ignoring phase (reasonable in your ideal example) the system MTF can be modeled as the product of the MTFs due to diffraction (a known function of N and lambda) and due to pixel aperture (a function of pixel width and shape) convolved with the sampling grid (typically a comb, function of sampling interval):

MTFsys = MTFdiff x MTFpxAp ** COMBsamp

MTFdiff we know and it is what it is for the given setup;
MTFpxAp varies with the size and shape of the pixel, in your example size would be 4um in one case and 8um in the other; and
COMBsamp is not pixel pitch in the slanted edge method but the pitch of sampling-bin spacing (1/4th and 1/8th of a pixel in your example).  Either way it is immaterial in the interval of interest to us (o-1 c/p) because it pushes images of the system MTF out to 4 and 8 c/p respectively.

Pixel pitch is worked back into MTF when we specify the frequency axis.  Here is a clumsy attempt at explaining how this is done.

Makes sense to me. Let's see if it does to Bart.

Jim

[Bart_van_der_Wolf]

Hi Bart,

Frans is the better person to ask, but MTF Mapper actually bins supersampled ESF data into bins 1/8th of a pixel wide.

Thanks for the correction, I knew it was more than Imatest (and the ISO standard) uses, so apparently 8 bins are used.

The key is making sure that, for the given edge length and angle, there are enough samples to fill the bins.

Yes, although 8 bins will need twice as many samples to populate to a similar statistical stability as 4 bins do. And longer edges means increased sensitivity for lens distortion. So I assume that Frans has considered that.

MTF Mapper used to use a hamming window but has since evolved to better methods, including importance sampling and other neat improvements (see his blog for details).  With its recent releases I see cleaner curves than the ISO standard, sfrmat3 and probably (because I do not use it) Imatest.

From what I've read, MTF Mapper's algorithms produce quite good results. I just don't know all the implementation details, so I was wondering if it could be even a hair better.

Cheers,
Bart

========================================
P.S.  This is what I could find after a short perusal in my files about what the ISO describes (short fragments from a draft version of ISO 16067-2) about the SFR function:

The selected region is converted from digital code values to an edge spread image of normalised photopic intensities via the OECF and colour weighting coefficients a, b, & c.

<<formula>>

Each row of the edge spread image is an estimate of the camera edge spread function (ESF). Each of these ESFs is differentiated to form its discrete line spread function (LSF). The position of the centroid (C) of each of R LSFs is determined along the continuous variable x.

<<formula>>

The slope of the best fit relating the x positions of the centroids to the r index of each row is computed.

<<formula>>

This slope m, is used to compute a shift S(r) to be applied to each row to bring each ESF to coincidence around a common origin at x=0. It effectively takes out the tilt out of the edge.

<<formula>>

The slope is also used to truncate the number of rows of data to the largest number R' that will have an integer number of full phase rotations. For example, if the fit to the centroid moves 0.1 pixels per row, then the largestmultiple of ten rows that is less than R will be used. A check is made that there is at least one full phase rotation.

The next step is the super-sampling and averaging. This step performs a composite requantized edge spread function (ESF) over the discrete variable j, where j is four times more finely sampled than p but is not a continuous variable like x. The super sampling factor is 4, so N=4X bins are created, each with width 0.25 pixels.

... followed by more formulas and conversions between cycles/pixel and cycles/mm.

[Jack Hogan]

Yes.