MTF Mapper 0.5.2 was compiled using MSVC Express 2013, which Microsoft calls "vc12". The Windows binaries have been linked statically against the runtime, but this does not appear to be sufficient to run MTF Mapper under wine without further tweaks.

For me, running "winetricks vcrun2013" in the console seemed to do the trick. I would say that this is a necessary step to get MTF Mapper to work under wine.

In case you are wondering, without the winetricks step I get the following error:

wine: Call from 0x7b83c506 to unimplemented function msvcr120.dll.?_Trace_ppl_function@Concurrency@@YAXABU_GUID@@EW4ConcRT_EventType@1@@Z, aborting

Let me know if there are any other issues related to wine, and I'll see what I can do.

# MTF Mapper

## Monday, 13 June 2016

## Wednesday, 13 April 2016

### MTF Mapper vs Imatest vs Quick MTF

I recently noticed that Quick MTF now has an automated region-of-interest (ROI) detection function. This allows me (in theory) to perform the same type of automated testing that I applied to MTF Mapper and Imatest. Now would be a good time to read the Imatest comparison post to familiarise yourself with my testing procedure.

Anyhow, the automatic ROI functionality in Quick MTF is

I also discovered that QuickMTF's "trial" limit of 40 tests can be bypassed with relatively little fuss (Oleg, if you are reading this, I promise not to share the secret).

Lastly, note that I performed these tests using the "ISO 12233" mode of Quick MTF. The default settings produces much smoother plots, but these are severely biased, i.e., they report MTF50 values that are much too low. To illustrate: the default settings produce a 95th percentile relative error of 13% when measured using images with an expected MTF50 of 0.25 c/p; switching to ISO 12233 mode reduces the error to only 5%. As expected, the standard deviation of MTF50 error is lower in the default mode, but I maintain that bias and variance should

Figure 1 illustrates the relative MTF50 error boxplot, calculated as 100*(measured_mtf50 - expected_mtf50)/expected_mtf50. Firstly, Quick MTF should be commended for its unbiased performance between expected MTF50 values of 0.1 and 0.4 cycles/pixel; the median error is exactly zero. Unfortunately, a strong bias appears after 0.4 c/p, which is consistent with some (light) smoothing of the ESF. The boxes, and especially the whiskers, are a bit wide, which is more readily seen in Figure 2.

Things go a bit pear shaped when we look at the standard deviation of the relative MTF50 error. If we consider the "usable" range of 0.08 to 0.5 c/p, then Quick MTF contains the standard deviation below 3.5%, which is not bad, but Imatest and MTF Mapper perform a bit better here. A more useful (and my preferred) measure is the 95th percentile of relative MTF50 error magnitude, as illustrated in Figure 3.

The values in Figure 3 have a natural interpretation: the magnitude of the error will remain below the indicated value in about 95% of the edges measured with each tool. This measure combines the effects of bias (Figure 1) and variance (Figure 2) in one convenient value. Consider again the "usable" range of 0.08 to 0.5 c/p: Quick MTF only manages to keep the error below about 9% across the range. It does quite a bit better in the centre of the range, almost matching Imatest at 0.2 c/p.

Based on these versions, it would appear that MTF Mapper still produces competitive results. And you cannot beat MTF Mapper's price.

####

Anyhow, the automatic ROI functionality in Quick MTF is

*almost*able to work with the simulated Imatest charts I produced with mtf_generate_rectangle. I had to manually adjust about half of the ROIs to ensure that Quick MTF was using as much of each edge as possible, i.e., similar ROIs to what Imatest and MTF Mapper used. Since the edge locations remain the same across all the test images, I used the "open with the same ROI" option to keep the experiment as fair as possible.I also discovered that QuickMTF's "trial" limit of 40 tests can be bypassed with relatively little fuss (Oleg, if you are reading this, I promise not to share the secret).

Lastly, note that I performed these tests using the "ISO 12233" mode of Quick MTF. The default settings produces much smoother plots, but these are severely biased, i.e., they report MTF50 values that are much too low. To illustrate: the default settings produce a 95th percentile relative error of 13% when measured using images with an expected MTF50 of 0.25 c/p; switching to ISO 12233 mode reduces the error to only 5%. As expected, the standard deviation of MTF50 error is lower in the default mode, but I maintain that bias and variance should

*both*be managed well.#### The results

Figure 1: Quick MTF MTF50 relative error boxplot |

Figure 2: Standard deviation of relative MTF50 error |

Figure 3: 95th percentile of relative MTF50 error magnitude |

The values in Figure 3 have a natural interpretation: the magnitude of the error will remain below the indicated value in about 95% of the edges measured with each tool. This measure combines the effects of bias (Figure 1) and variance (Figure 2) in one convenient value. Consider again the "usable" range of 0.08 to 0.5 c/p: Quick MTF only manages to keep the error below about 9% across the range. It does quite a bit better in the centre of the range, almost matching Imatest at 0.2 c/p.

#### Conclusion

The Imatest results were not based on the latest version; I do not have an Imatest license, and my trial has expired, so it will take a fair bit of effort to refresh the Imatest results. The Quick MTF 2.09 results are current, though.Based on these versions, it would appear that MTF Mapper still produces competitive results. And you cannot beat MTF Mapper's price.

## Tuesday, 3 November 2015

### PffffFFTttt...

There is no doubt that FFTW is one of the fastest FFT implementations available. It can be a pain to include in a Microsoft Visual Studio project, though. Maybe I am "using it wrong"...

One solution to this problem is to include my own FFT implementation in MTF Mapper, thereby avoiding the FFTW dependency entirely. Although it is generally frowned upon to use a homebrew FFT implementation in lieu of an existing, proven library, I decided it was time to ditch FFTW.

One of the main advantages of using a homebrew FFT implementation is that it avoids the GPL license of FFTW. Not that I have any fundamental objection to the GPL, but the main sources of MTF Mapper are available under a BSD license, which is a less strict license than the GPL. In particular, the BSD license makes allowance for commercial use of the code. Before anyone asks, no, MTF Mapper is not going closed source or anything like that. All things being equal, the BSD license is just less restrictive, and avoiding FFTW brings MTF Mapper closer to being a pure BSD (or compatible) license project.

I tried to implement it in fairly straightforward c++, but taking care to use pointer walks in stead of array indexing, and using look up tables for both the bit-reversal process and the sine/cosine functions. These changes produced an algorithm that was at least as fast as my similarly optimized complex FFT implementation augmented with a two-for-the-price-of-one step for real-valued inputs.

One thing I did notice is that the FFT in its "natural" form does not lend itself to an efficient streaming implementation. For example, the first pass of the radix-2 algorithm looks like this:

Using the homebrew FFT of MTF Mapper 0.4.23 the total number of instruction fetches increase by about 1.34%, but this does not imply a 1.34% increase in runtime. The callgrind trace indicates that 0.31% of v0.4.23's instructions are spent in the new FFT routine.

In relative terms, this implies that the new routine is roughly 3.5 times slower, but this does not account for the additional overheads incurred by FFTW's memory allocation routines (the FFTW routine is not in-place, hence requires a new buffer to be allocated before every FFT to keep the process thread-safe).

Measuring the actual wall-clock time gives us a result of 22.27 ± 0.14 seconds for 20 runs of MTF Mapper v0.4.21 on my test image, versus 21.631 ± 0.16 seconds for 20 runs of v0.4.23 (each experiment repeated 4 times for computing standard deviations). These timings were obtained on a Sandy-bridge laptop with 8/4 threads. The somewhat surprising reversal of the standings (the homebrew FFT now outperforms the FFTW implementation) just goes to show that the interaction between hyperthreading, caching, and SSE/AVX unit contention can produce some surprising results.

Bottom line: the homebrew FFT is fast enough (at least on the two hardware/compiler combinations I tested).

I set up a simple test using FFTW version 3.3.4 built on gentoo using gcc-4.9.3, running on a Sandy-bridge laptop cpu (i7-2720QM) running at a base clock of 2.2 GHz. This was a single-threaded test, so we should see a maximum clock speed of 3.3GHz, if we are lucky.

For a 1024-sample real-valued FFT, 2 million iterations took 14.683 seconds using the homebrew code, and only 5.798 seconds using FFTW. That is a ratio of ~2.53.

For a 512-sample (same as what MTF Mapper uses) real-valued FFT, 2 million iterations took 6.635 seconds using the homebrew code, and only 2.743 seconds using FFTW. That is a ratio of ~2.42.

According to general impressions gathered from the Internet, you are doing a good-enough job if you are less than 4x slower than FFTW. I ran metaFFT's benchmarks, which gave a ratio of 2.4x and 2.1x relative to FFTW for size 1024 and 512, respectively (these were probably complex transforms, so not a straight comparison).

The MTF Mapper homebrew FFT at least appears to be in the right ballpark, at least fast enough not to cause embarrassment....

One solution to this problem is to include my own FFT implementation in MTF Mapper, thereby avoiding the FFTW dependency entirely. Although it is generally frowned upon to use a homebrew FFT implementation in lieu of an existing, proven library, I decided it was time to ditch FFTW.

One of the main advantages of using a homebrew FFT implementation is that it avoids the GPL license of FFTW. Not that I have any fundamental objection to the GPL, but the main sources of MTF Mapper are available under a BSD license, which is a less strict license than the GPL. In particular, the BSD license makes allowance for commercial use of the code. Before anyone asks, no, MTF Mapper is not going closed source or anything like that. All things being equal, the BSD license is just less restrictive, and avoiding FFTW brings MTF Mapper closer to being a pure BSD (or compatible) license project.

### FFT Implementation

After playing around with a few alternative options, including considering the my first c++ FFT implementation way back from first year at university, I settled on Sorenson's radix-2 real-valued FFT (Sorenson, H.B, et al, Real-Valued Fast Fourier Transform Algorithms, IEEE Transactions on Accoustics, Speech, and Signal Processing, 35(6), 1987). This algorithm appears to be a decent balance between complexity and theoretical efficiency, but I had to work fairly hard at the code to produce a reasonably efficient implementation.I tried to implement it in fairly straightforward c++, but taking care to use pointer walks in stead of array indexing, and using look up tables for both the bit-reversal process and the sine/cosine functions. These changes produced an algorithm that was at least as fast as my similarly optimized complex FFT implementation augmented with a two-for-the-price-of-one step for real-valued inputs.

One thing I did notice is that the FFT in its "natural" form does not lend itself to an efficient streaming implementation. For example, the first pass of the radix-2 algorithm looks like this:

for (; xp <= xp_sentinel; xp += 2) {Note that the value of x[i] (here *xp) is overwritten in the 3rd line of the code, while the original value of x[i] (copied into xt) is still required in the 4th line of the code. This write-after-read dependency causes problems for out-of-order execution. Maybe the compiler is smart enough to unroll the loop and intersperse the reads and writes to achieve maximal utilization of all the processing units on the CPU, but the stride of the loop and the packing of the values is not ideal for SSE2/AVX instructions either. I suppose that this can be addressed with better code, but before I spend time on that I first have to determine how significant raw performance of the FFT is in the context of MTF Mapper.

double xt = *xp;

*(xp) = xt + *(xp+1);

*(xp+1) = xt - *(xp+1);

}

### Real world performance in MTF Mapper

So how much time does MTF Mapper spend calculating FFTs? Well, one FFT for every edge. A high-density grid-style test chart has roughly 1452 edges. According to a "callgrind" trace produced using valgrind, MTF Mapper v0.4.21 spends 0.09% of its instruction count inside FFTW's real-valued FFT algorithm.Using the homebrew FFT of MTF Mapper 0.4.23 the total number of instruction fetches increase by about 1.34%, but this does not imply a 1.34% increase in runtime. The callgrind trace indicates that 0.31% of v0.4.23's instructions are spent in the new FFT routine.

In relative terms, this implies that the new routine is roughly 3.5 times slower, but this does not account for the additional overheads incurred by FFTW's memory allocation routines (the FFTW routine is not in-place, hence requires a new buffer to be allocated before every FFT to keep the process thread-safe).

Measuring the actual wall-clock time gives us a result of 22.27 ± 0.14 seconds for 20 runs of MTF Mapper v0.4.21 on my test image, versus 21.631 ± 0.16 seconds for 20 runs of v0.4.23 (each experiment repeated 4 times for computing standard deviations). These timings were obtained on a Sandy-bridge laptop with 8/4 threads. The somewhat surprising reversal of the standings (the homebrew FFT now outperforms the FFTW implementation) just goes to show that the interaction between hyperthreading, caching, and SSE/AVX unit contention can produce some surprising results.

Bottom line: the homebrew FFT is fast enough (at least on the two hardware/compiler combinations I tested).

### Are we done yet?

Well, surely you want to know how fast the homebrew FFT is in relation to FFTW in a fair fight, right?I set up a simple test using FFTW version 3.3.4 built on gentoo using gcc-4.9.3, running on a Sandy-bridge laptop cpu (i7-2720QM) running at a base clock of 2.2 GHz. This was a single-threaded test, so we should see a maximum clock speed of 3.3GHz, if we are lucky.

For a 1024-sample real-valued FFT, 2 million iterations took 14.683 seconds using the homebrew code, and only 5.798 seconds using FFTW. That is a ratio of ~2.53.

For a 512-sample (same as what MTF Mapper uses) real-valued FFT, 2 million iterations took 6.635 seconds using the homebrew code, and only 2.743 seconds using FFTW. That is a ratio of ~2.42.

According to general impressions gathered from the Internet, you are doing a good-enough job if you are less than 4x slower than FFTW. I ran metaFFT's benchmarks, which gave a ratio of 2.4x and 2.1x relative to FFTW for size 1024 and 512, respectively (these were probably complex transforms, so not a straight comparison).

The MTF Mapper homebrew FFT at least appears to be in the right ballpark, at least fast enough not to cause embarrassment....

## Sunday, 5 July 2015

### A critical look

Most of the posts on this blog are tutorial / educational in style. I have come across a paper published by an Imatest employee that requires some commentary of a more critical nature. With some experience in the academic peer review process, I hope I can maintain the appropriate degree of objectivity in my commentary.

At any rate, if you have no interest in this kind of commentary / post, please feel free to skip it.

A copy can be obtained directly from Imatest here.

By that I mean that Kohm's approach (and MTF Mapper's approach) is to compute an estimate of the edge normal, followed by projection of the pixel centre coordinates (paired with their intensity values) onto this normal. This produces a dense set of samples across the edge in a very intuitive way; the main drawback of this approach being the potential increase in the processing cost because it lends itself better to a floating point implementation.

The ISO12233:2000 approach rather attempts to project the edge "down" (assuming a vertical edge) onto the bottom-most row of pixels in the region of interest (ROI). Using the slope of the edge (estimated earlier), each pixel's intensity (sample) can be shifted left or right by the appropriate phase offset before being projected onto the bottom row. If the bottom row is modelled as bins with 0.25-pixel spacing, this process allows us to construct our 4x-oversampled, binned ESF estimate with the minimum amount of computational effort (although that might depend on whether a particular platform has strong floating-point capabilities).

The method proposed in the Imatest paper is definitely of the ISO12233:2000 variety. How can we tell? Well, the Imatest paper proposes that the ESF must be corrected by appropriate scaling of the x values using a scaling factor of cos(theta), where theta is the edge orientation angle. What this accomplishes is to "squash" the range of x values (i.e. pixel column) to be spaced at an interval that is consistent with the pixel's distance as measured along the normal to the edge. For a 5 degree angle, this correction factor is only 0.9962, meaning that distances will be squashed by a very small amount indeed. So little, in fact, that the ISO12233:2000 standard ignores this correction factor, because a pixel at a horizontal distance of 16 pixels will be mapped to a normal distance of 15.94. Keeping in mind that the ESF bins are 0.25 pixels wide, this error must have seemed small.

I recognize that the Imatest paper proposes a valid solution to this "stretching" of the ESF that would occur in its absence, and that this stretching would become quite large at larger angles (about a 1.5 pixel shift at 25 degrees for our pixel at a horizontal distance of 16 pixels).

My critique of this approach is that it would typically involve the use of floating point calculations, the potential avoidance of which appears to have been one of the main advantages of the ISO12233:2000 method. If you are going to use floating point values, then Kohm's method is more intuitive.

At any rate, if you have no interest in this kind of commentary / post, please feel free to skip it.

### The paper

The paper in question is : Jackson K. M. Roland, " A study of slanted-edge MTF stability and repeatability ", Proc. SPIE 9396, Image Quality and System Performance XII, 93960L (January 8, 2015); doi:10.1117/12.2077755; http://dx.doi.org/10.1117/12.2077755.A copy can be obtained directly from Imatest here.

### Interesting point of view

One of the contributions of the paper is a discussion of the impact of edge orientation on MTF measurements. The paper appears to approach the problem from a direction that is more closely aligned with the ISO12233:2000 standard, rather than Kohm's method ("Modulation transfer function measurement method and results for the Orbview-3 high resolution imaging satellite", Proceedings of ISPRS, 2004).By that I mean that Kohm's approach (and MTF Mapper's approach) is to compute an estimate of the edge normal, followed by projection of the pixel centre coordinates (paired with their intensity values) onto this normal. This produces a dense set of samples across the edge in a very intuitive way; the main drawback of this approach being the potential increase in the processing cost because it lends itself better to a floating point implementation.

The ISO12233:2000 approach rather attempts to project the edge "down" (assuming a vertical edge) onto the bottom-most row of pixels in the region of interest (ROI). Using the slope of the edge (estimated earlier), each pixel's intensity (sample) can be shifted left or right by the appropriate phase offset before being projected onto the bottom row. If the bottom row is modelled as bins with 0.25-pixel spacing, this process allows us to construct our 4x-oversampled, binned ESF estimate with the minimum amount of computational effort (although that might depend on whether a particular platform has strong floating-point capabilities).

The method proposed in the Imatest paper is definitely of the ISO12233:2000 variety. How can we tell? Well, the Imatest paper proposes that the ESF must be corrected by appropriate scaling of the x values using a scaling factor of cos(theta), where theta is the edge orientation angle. What this accomplishes is to "squash" the range of x values (i.e. pixel column) to be spaced at an interval that is consistent with the pixel's distance as measured along the normal to the edge. For a 5 degree angle, this correction factor is only 0.9962, meaning that distances will be squashed by a very small amount indeed. So little, in fact, that the ISO12233:2000 standard ignores this correction factor, because a pixel at a horizontal distance of 16 pixels will be mapped to a normal distance of 15.94. Keeping in mind that the ESF bins are 0.25 pixels wide, this error must have seemed small.

I recognize that the Imatest paper proposes a valid solution to this "stretching" of the ESF that would occur in its absence, and that this stretching would become quite large at larger angles (about a 1.5 pixel shift at 25 degrees for our pixel at a horizontal distance of 16 pixels).

My critique of this approach is that it would typically involve the use of floating point calculations, the potential avoidance of which appears to have been one of the main advantages of the ISO12233:2000 method. If you are going to use floating point values, then Kohm's method is more intuitive.

### Major technical issues

- The Point Spread Functions (PSFs) used to perform the "real world" and simulated experiments were rather different, particularly in one very important aspect. The Canon 6D camera has a PSF that is anisotropic, which follows directly from its square (or even L-shaped) photosites. The composite PSF for the 6D would be an Airy pattern (diffraction) convolved with a square photosite aperture (physical sensor) convolved with a 4-dot beam splitter (the OLPF). Of course I do not have inside information on the exact photosite aperture (maybe chipworks has an image) nor the OLPF (although a 4-dot Lithium Niobate splitter seems reasonable). The point remains that this type of PSF will yield noticeably higher MTF50 values when the slanted edge approaches 45 degrees. Between the 5 and 15 degree orientations employed in the Imatest paper, we would expect a difference of about 1%. This is below the error margin of Imatest, but with a large enough set of observations this systematic effect should be visible.

In contrast, the Gaussian PSF employed to produce the simulated images is (or at least is supposed to be) isotropic, and should show no edge-orientation dependent bias. Bottom line: the "real world" images had an anisotropic PSF, and the simulated images had an isotropic PSF. This means that the one cannot be used in the place of the other to evaluate the effects of edge orientation on measured MTF. Well, at least not without separating the PSF anisotropy from the residual orientation-depended artifacts of the slanted edge method. - On page 7 the Imatest paper states that "The sampling of the small Gaussian is such that the normally rotationally-invariant Gaussian function has directional factors as you approach 45 degree increments." This is further "illustrated" in Figure 13.

At this point I take issue with the reviewers who allowed the Imatest paper to be published in this state. If you suddenly find that your Gaussian PSF becomes anisotropic, you have to take a hard look at your implementation. The only reason that the Gaussian (with a small standard deviation) is starting to develop "directional factors" is because you are undersampling the Gaussian beyond repair.

The usual solution to this problem is to increase the resolution of your synthetic image. By generating your synthetic image at, say, 10x the scale, all your Gaussian PSFs will be reasonably wide in terms of samples in the oversampled image. For MTF measurement using the slanted edge method, you do not even have to downsize your oversampled image before applying the slanted edge method. All you have to do is to change the scale of your resolution axis in your MTF plot. That way you do not even have to worry about the MTF of the downsampling kernel.

There are several methods that produce even higher quality simulated images. At this point I will plug my own work: see this post or this paper. These approaches rely on importance sampling (for diffraction PSFs) or direct numerical integration of the Gaussian in two dimensions; both these approaches avoid any issues with downsampling and do not sample on a regular grid. These methods are implemented in mtf_generate_rectangle.exe, which is part of the MTF Mapper package.

### Minor technical issues

- On page 1 the Imatest paper states that the ISO 12233:2014 standard lowered the edge contrast "because with high contrast the measurement becomes unstable". This statement is quite vague, and appears to contradict the results presented in Figure 8, which shows no degradation of performance at high contrast, even in the presence of noise.

I would offer some alternative explanations: the ISO12233 standard is often applied to images compressed with DCT-based quantization methods, such as JPEG. A high-contrast edge typically shows up with a large-magnitude DCT coefficient at higher frequencies; exactly the frequencies that are more strongly quantized, hence the well-kown appearance of "mosquito noise" in JPEG images. A lower contrast edge will reduce the relative energy at higher frequencies, thus the stronger quantization of high frequencies will have a proportionately smaller effect. I am quite temtpted to go and test this theory right away.

Another explanation, one that is covered in some depth on Imatest's own website, is of course the potential intensity clipping that may result from incorrect exposure. Keeping the edge contrast in a more manageable range reduces the chance of clipping. Another more subtle reason is that a lower contrast chart allows more headroom for sharpening without clipping. By this I mean that sharpening (of the unsharp masking type) usually results in some "ringing" which manifests as overshoot (on the bright side of the edge) and undershoot (on the dark side of the edge). If chart contrast was so high that the overshoot of overzealous sharpening would be clipped, then it would be harder to measure (and observe) the extent of oversharpening. - The noise model is employed a little basic. Strictly speaking the standard deviation of the additive Gaussian white noise should be signal dependent; this is a more accurate model of photon shot noise, and is trivial to implement. I have not done a systematic study of the effects of noise simulation models on the slanted edge method, but in 2015 one really should simulate photon shot noise as the dominant component of additive noise.
- Page 6 of the Imatest paper states that "There is a problem with this 5 degree angle that has not yet been addressed in any standard or paper." All I can say to this is that Kohm's paper has presented an alternative solution to this problem that really should be recognized in the Imatest paper.

### Summary

Other than the unforgivable error in the generation of the simulated images, a fair effort, but more time spent on the literature, especially papers like Kohm's, would have changed the tone of the paper considerably, which in turn would have made it more credible.### Taking on Imatest

After having worked on MTF Mapper for almost five years now, I have decided that it is time to go head-to-head with Imatest. I downloaded a trial version of Imatest 4.1.12 to face off against MTF Mapper 0.4.18.

For the purpose of this comparison I decided to generate synthetic images using mtf_generate_rectangle. This allows me to use a set of images rendered using an accurately known PSF, meaning that we know exactly what the actual MTF50 value should be for those images. I decided to render a test chart conforming to the SFRPlus format, since that allows me to extract a fair number of edges for each test case. The approximately-sfrplus-chart looks like this:

Figure 1: SFRPlus style chart with an MTF50 value of 0.35 cycles/pixel |

### Validating the test images

(This section can be skipped if you trust my methodology)Although I have posted quite a few posts here on this blog regarding the algorithms used by mtf_generate_rectangle to render synthetic images, I will now show from first principles that the synthetic images truely have the claimed point spread functions (PSFs), and thus known MTFs.

I rendered the synthetic image using a command like this:

mtf_generate_rectangle.exe --b16 --pattern-noise 0.0085 --read-noise 2.5 --adc-gain 0.641 --adc-depth 12 -c 0.33 --target-poly sfrchart.txt -m 0.35 -p gaussian-sampled --airy-samples 100

This particular command renders the SFRPlus chart using a Gaussian PSF with an MTF50 value of 0.35. Reasonably realistic sensor noise is simulated, including photon shot noise, which implies that the noise standard deviation scales as the square root of the signal level; in plain English: we have more noise in bright parts of the image.

I ran a version of mtf_mapper that dumped the raw samples extracted from the image (normally used to construct the binned ESF); I specified the edge angle as 5 degrees to remove all possible sources of error. NB: the "raw_esf_values.txt" file produced by MTF Mapper contains the binned ESF, and is not suitable for this particular experiment because of the smoothing inherent in the binning.

Given that I specified an MTF50 value of 0.35 cycles per pixel, we know that the standard deviation of the true PSF should be 0.5354018 pixels [ sqrt( log(0.5)/(-2*pi*pi*0.35*0.35) ]. From this we can calculate the expected analytical ESF, which is simply erf(x/sigma)*(upper-lower) + lower, where erf() is the standard "error function", defined as the integral of the unit Gaussian. The values upper and lower merely represent the mean white and black levels, which were defined as lower = 65536*0.33/2 and upper = 65536 - lower. With these values, I can now plot the expected analytical ESF along with the raw ESF samples dumped by MTF Mapper.

Figure 2: Raw ESF samples along with analytical ESF |

Figure 3: ESF difference |

We see two things in Figure 3: The mean difference appears to be close to zero, and the noise magnitude appears to increase with increasing signal levels (to the right). The increase in noise was expected, since that follows from the photon shot noise model used to simulate sensor noise. We can normalize the noise by dividing the ESF difference (noise) by the square root of the analytical ESF, which gives us this plot:

Figure 4: Normalised ESF difference |

Figure 5: Normalized ESF difference density |

Now that we have confirmed that the distribution of the residuals are Gaussian, we can fit a line through them. This line comes out with a slope of -0.005765, which means that our normalized residuals are fairly flat. Lastly, we can perform some LOESS smoothing on the normalized residuals:

Figure 6: LOESS fit on normalized ESF difference |

The mean signal-to-noise ratio in the bright regions of the images comes out at around 15dB; because we compute the LSF (or PSF if you prefer)) from the derivative of the ESF, the bright parts of the image are representative of the worst-case noise. Alternatively, we can say that the noise is quite similar to that produced by a Nikon D7000 at ISO400, for an SRFplus test chart at a 5:1 contrast ratio.

I have shown that there is no systematic difference between the ESF extracted from a synthetic image and the expected analytical ESF. The simulated noise also behaves in the way that we would expect from properties of the simulated sensor. Based on these observations, we can safely assume that the synthetic images have the desired PSF, i.e., the simulated MTF50 values are spot-on. (In previous posts I examined the properties of the simulated ESF values in the absence of noise, but here I chose to demonstrate the PSF properties directly on the actual images used in the Imatest vs MTF Mapper comparison).

### The results

The results presented here were obtained by running Imatest 4.1.12 and MTF Mapper 0.4.18 on these images (about 100MB). SFRPlus (from Imatest, of course) was configured to enable the LSF correction that was recently introduced. Other than that, all settings were left to defaults, including leaving the apodization option enabled. I turned off the "quick mtf" option, although I did not check to see whether this affected the results. After a run of SFRPlus, the "save data" option was used to store the results, after which the "MTF50" column values were extracted, discarding the top and bottom row edges as explained before.MTF Mapper was run using the "-t 0.5 -r" settings; the "-t 0.5" option is required to allow MTF Mapper to work with the rather low 5:1 contrast ratio. The values output to "raw_mtf_values.txt" were used as the representative MTF50 values extracted by MTF Mapper.

Simulated images were produced over the MTF50 range 0.1 cycles/pixel to 0.7 cycles/pixel in increments of 0.05 cycles/pixel, with one extra data point at 0.08 cycles/pixel to represent the low end (which is quite blurry). For each MTF50 level a total of three images were simulated, each with a different seed to produce unique sensor noise. This gives us 19*3*4 = 228 samples at each MTF50 level.

As in previous posts, the results will be evaluated in two ways: bias and variance. The first plots to consider illustrate both bias and variance simultaneously, although it is somewhat harder to compare the variance of the methods on these plots.

Figure 7: Imatest relative error boxplot |

Figure 8: MTF Mapper relative error boxplot |

Figure 9: Median relative MTF50 error comparison |

So that illustrates bias. To measure variance we can plot the standard deviation at each MTF50 level:

Figure 10: Standard deviation of relative MTF50 error |

A final performance metric to consider is the 95th percentile of relative MTF50 error. By computing this value on the absolute value of the relative error, it combines both variance and bias into a single measurement that tells us how close our measurements will be to the true MTF50 value, in 95% of measurements. Here is the plot:

Figure 11: 95th percentile of MTF50 error |

### Conclusion

It took quite a bit of effort on my part to improve MTF Mapper to the point where it produces more accurate results than Imatest. There are some other aspects I have not touched on here, such as how accuracy varies with edge orientation. For now, I will say that MTF Mapper produces accurate results at known critical angles, whereas Imatest appears to fail at an angle of 26.565 degrees. Given that Imatest never claimed to work well at angles other than 5 degrees, I will let that one slide.I have also not included any comparisons to other freely available slanted edge implementations (sfrmat, Quick MTF, the slanted edge ImageJ plugin, mitreSFR). I can tell you from informal testing that most of them appear to perform significantly worse than Imatest, mostly because none of those implementations appear to include the finite-difference-derivative correction. Maybe I will back this opinion up with some more detailed results in future.

So where does that leave your typical Imatest user? Well, the difference in accuracy between Imatest and MTF Mapper is relatively small. What I mean by that is that these results do not imply that Imatest users have to switch over to using MTF Mapper, rather, these results show that MTF Mapper users can trust their measurements to be at least as good as those obtained by Imatest. And, of course, MTF Mapper is free, and the source code is available.

There are some fairly nifty features that I noticed in SFRPlus during this experiment. It appears that SFRPlus will perform lens correction automatically, meaning that radial distortion curvature can be corrected for on the fly. MTF Mapper currently limits the length of the edge it will include in the analysis as a means of avoiding the effects of strong radial distortion. But now that I am aware of this feature, I think it would be relatively straightforward to include lens distortion correction in MTF Mapper. So little time, so many neat ideas to play with ...

## Wednesday, 24 June 2015

### Truncation of the ESF

A really quick post to highlight one specific aspect: what happens to the MTF produced by the slanted edge method if the ESF is truncated.

To recap: The slanted edge method projects image intensity values onto the normal of the edge to produce the Edge Spread Function (ESF). Any practical implementation has to place an upper limit on the maximum distance that pixels can be from the edge (as measured along the edge normal). MTF Mapper, for example, only considers pixels up to a distance of 16 pixels from the edge.

Looking back at the Airy pattern that results from the diffraction of light through a circular aperture we can see that the jinc

We also know that the effective width of the Airy pattern increases with increasing f-number. Herein lies the problem: a slanted edge implementation that truncates the ESF will necessarily discard part of the Airy pattern. The discarded part is of course the samples furthest from the edge, and we know that those samples tend to contribute more to the lower frequencies in the MTF.

Simulating a slanted edge image using the Airy + photosite aperture model, with an aperture of f/8, light at 550 nm, a 100% fill-factor square photosite aperture, and 4.886 micron photosite pitch (something approximating the D810), we can investigate the impact of the truncation distance on the MTF as measured by the slanted edge method. Here goes:

The green dotted line represents the expected MTF curve (from our simple model). I have zoomed in on the low-frequency region, but we can see that both the truncated MTF measurements (red and black curves) tend to follow the green curve more closely after about 0.10 cycles per pixel. We also note that both the red and black curves contain a few points that are clearly above the green curve between 0 and 0.05 cycles per pixel. It is physically impossible for the measured MTF to exceed the diffraction MTF (blue curve), so we can state with confidence that this is a measurement error.

If we compare the red and the black curves we can see that a wider truncation window (red curve) reduces the overshoot at low frequencies. If we had the opportunity to use an even wider truncation window, we would be able to reduce the overshoot to even lower levels.

Lastly, if we introduce apodization into the mix we are compounding the problem even further by attenuating the edges of the PSF. This leads to even greater overshoot (at low frequencies) in our measured MTF curve.

Bottom line: The slanted edge method is constrained by practical limitations, most notably the desire to have a finite truncation window, and the desire to reduce the impact of image noise using apodization of the PSF. These constraints lead to overshoot in the lowest frequencies of the measured MTF. It may be possible to apply an empirical correction to minimize the overshoot, but only at the cost of making strong assumptions regarding the shape of the MTF, which is best avoided.

To recap: The slanted edge method projects image intensity values onto the normal of the edge to produce the Edge Spread Function (ESF). Any practical implementation has to place an upper limit on the maximum distance that pixels can be from the edge (as measured along the edge normal). MTF Mapper, for example, only considers pixels up to a distance of 16 pixels from the edge.

Looking back at the Airy pattern that results from the diffraction of light through a circular aperture we can see that the jinc

^{2}function has infinite support, in other words, it tapers off to zero but never quite reaches zero if we consider a finite domain.We also know that the effective width of the Airy pattern increases with increasing f-number. Herein lies the problem: a slanted edge implementation that truncates the ESF will necessarily discard part of the Airy pattern. The discarded part is of course the samples furthest from the edge, and we know that those samples tend to contribute more to the lower frequencies in the MTF.

Simulating a slanted edge image using the Airy + photosite aperture model, with an aperture of f/8, light at 550 nm, a 100% fill-factor square photosite aperture, and 4.886 micron photosite pitch (something approximating the D810), we can investigate the impact of the truncation distance on the MTF as measured by the slanted edge method. Here goes:

If we compare the red and the black curves we can see that a wider truncation window (red curve) reduces the overshoot at low frequencies. If we had the opportunity to use an even wider truncation window, we would be able to reduce the overshoot to even lower levels.

Lastly, if we introduce apodization into the mix we are compounding the problem even further by attenuating the edges of the PSF. This leads to even greater overshoot (at low frequencies) in our measured MTF curve.

Bottom line: The slanted edge method is constrained by practical limitations, most notably the desire to have a finite truncation window, and the desire to reduce the impact of image noise using apodization of the PSF. These constraints lead to overshoot in the lowest frequencies of the measured MTF. It may be possible to apply an empirical correction to minimize the overshoot, but only at the cost of making strong assumptions regarding the shape of the MTF, which is best avoided.

## Tuesday, 23 June 2015

### Anisotropy

In my post on "critical angles" I mentioned that there was one other factor to consider when looking at the influence of edge orientation on slanted edge analysis. I will refer to that phenomenon as the influence of

The simplest example of an anisotropic PSF is to consider just a square photosite aperture, without any lens aperture diffraction.

In figure 1 we can see the interaction between our slanted edge (shown in blue here) and the photosite aperture (orange). If the value

When the edge orientation angle theta is 0 degrees, then we obtain a linear function, which is what one would expect. If the edge is at a 45 degree angle (as shown in the right panel of Figure 1), then we obtain the other extreme. Angles between 0 and 45 degrees produce a curve that is somewhere in between these extremes.

What can we learn from these curves? Well, we can see that an edge orientation of 45 degrees will overlap with the photosite square from -√0.5 to √0.5, whereas the 0 degrees edge orientation only results in overlap between -0.5 and 0.5. From this we can infer that the square appears wider when approached by an edge with a 45 degree orientation. We also know that a square photosite acts as a low-pass filter, in the sense that the image captured by our sensor is the convolution of this low-pass filter and the analytical model of our scene. This might lead one to believe that the 45 degree case would result in a stronger low-pass filter, because it is clearly "wider" than the 0 degree case.

We can plot the derivative of the curves from Figure 2:

The 0 degree case is easy to visualize with the help of the left panel of Figure 1: clearly, the width of the photosite square (measured along the step edge) is constant. The 45 degree case is also readily visualized by noting that the we cross the widest part of the photosite square when t=0 (right panel of Figure 1); this nicely corresponds to the peak instantaneous width of √2 in Figure 3.

We can interpret the curve in Figure 3 as a weighting function, i.e., the relative contribution to the convolution of the edge and the photosite aperture at distance

This is a rather unexpected turn of events, since it implies that even though the 45 degree case starts overlapping with the edge sooner (the regions -√0.5 < t < -0.5 and 0.5 < t < √0.5), it represents only a small fraction of the total interaction with the edge. Instead of the 45 degree case being a stronger low-pass filter than the 0 degree case, we expect the opposite because the 45 degree case has roughly 20% (50.6/42) more of its weight located close to t=0.

We appear to have two mildly conflicting views:

a) the 45 degree case is "wider" at its widest point, thus it should be a stronger low-pass filter than the 0 degree case, and

b) more of the weight of the 45 degree case is close to the centre, hence it should present a

I am betting on outcome b), mostly because I already know what the empirical results will tell us ....

We can see that MTF50 overestimation steadily increase to about 5% as we approach 45 degrees.

Just to check, let us examine an isotropic PSF: a pure Gaussian without any photosite aperture simulation. This should yield a purely Gaussian MTF. Same simulation, but with the radially symmetric Gaussian PSF:

Other than a bit of a glitch at 2 degrees producing a few outliers, we see a fairly flat median MTF50 error with the Gaussian PSF. No systematically increasing MTF50 error with increasing angle appears.

Simulations were repeated using a simulated aperture at f/2.8, light at 550 nm, a photosite pitch of 4.73 micron and no AA (OLPF) filter. These settings give an expected MTF50 value of ~ 0.504 cycles per pixel, which is slightly lower than the expected MTF50 value of ~ 0.6 cycles per pixel seen in the previous section. Accordingly, the MTF50 errors may be slightly reduced (or at least the expected variance should be reduced).

The trend is clearly visible, but appears to be only about 60% of the magnitude of the case without diffraction (about 2.5% at 44 degrees, vs about 4% without diffraction). Smaller apertures (larger f-numbers) will reduce the anisotropy as the Airy component of the PSF will start to dominate the photosite aperture PSF.

Unfortunately, we have to at least be aware of this for certain types of testing and measurement. Because the error (overestimation) is systematic, it will show up in any measurement that sweeps through a range of angles, just like the MTF Mapper grid test chart, pictured here:

This chart can be used to produce Sagittal/Meridional MTF50 plots across your lens/sensor/camera. The chart aims to keep one edge perpendicular to the virtual line connecting that edge to the centre of the chart, which inevitably causes some of the squares to approach a 45 degree edge orientation.

I simulated this chart using mtf_generate_rectangle, using an aperture of f/4, an Airy+box PSF, green light and a photosite pitch of 4.73 micron. Passing this synthetic image through MTF mapper to produce a surface plot (-s option) produces this result:

The systematic distortion of MTF50 values is clearly visible, even though the range of values is quite small. The maximum value on the scale is 0.47, which is only about 2% higher than the expected MTF50 value of 0.46073 (at 0 degrees, of course). But the cross pattern is clearly visible. At least I have confirmed the cause.

Pushing for even greater realism I repeated the simulation using the "rounded-square" photosite aperture that MTF Mapper provides. Here is the surface plot:

We can see that the MTF50 values are slightly higher (I think the effective fill factor is slightly lower for my hand-crafted rounded corner photosite aperture), but ignore that bit for the moment. Instead, notice that the range is even smaller than the square aperture case (Figure 7), but the cross pattern is still visible.

Lastly, if we use a circular photosite aperture, we get this:

Other than the fact that the resulting image is appallingly ugly, we can see that the cross structure has disappeared, as expected.

I tried to construct an elegant analytical explanation for the interaction between the edge orientation and a square photosite aperture. This turned out to be harder than I expected, so I only have some interesting plots to offer for now. What did emerge from the theory is that we should not focus on the apparent width of the photosite aperture, but rather on the distribution of its weight relative to the centre. The somewhat startling conclusion is that we should observe higher MTF50 measurements when the orientation approaches 45 degrees.

This was supported by the actual experiments using simulated imagery.

So what can we do about this systematic distortion? Well, the only sound solution would be to stick to edges with a relative orientation of about 5 degrees. This is not a universal solution, though, because it makes it impossible to measure in the true Sagittal/Meridional directions. Imatest solved the problem by sticking to 5 degree angles and referring to "horizontal" and "vertical" MTF. This works well enough if you wish to measure peak astigmatism, but it does not allow you to measure MTF in the optically more appropriate sagittal/meridional directions.

I might add a 5-degree test chart to MTF Mapper in future, just to cover all bases.

*anisotropic*point spread functions. In this context, I use the term anisotropic to refer to point spread functions that are not radially symmetric.The simplest example of an anisotropic PSF is to consider just a square photosite aperture, without any lens aperture diffraction.

Figure 1: Edge orientation relative to photosite aperture |

*t*represents the distance from the centre of our photosite aperture to the right edge of our slanted edge (rectangle or step edge), then we can consider the overlapping area between the two as a function of t. The interesting range of values for*t*would be between -√0.5 and √0.5, if we assume the photosite is a square with sides of length 1. Plotting this overlapping area as a function of*t*gives us Figure 2:Figure 2: Fraction of square photosite covered by slanted edge as a function of edge distance to photosite centre |

What can we learn from these curves? Well, we can see that an edge orientation of 45 degrees will overlap with the photosite square from -√0.5 to √0.5, whereas the 0 degrees edge orientation only results in overlap between -0.5 and 0.5. From this we can infer that the square appears wider when approached by an edge with a 45 degree orientation. We also know that a square photosite acts as a low-pass filter, in the sense that the image captured by our sensor is the convolution of this low-pass filter and the analytical model of our scene. This might lead one to believe that the 45 degree case would result in a stronger low-pass filter, because it is clearly "wider" than the 0 degree case.

We can plot the derivative of the curves from Figure 2:

Figure 3: Instantaneous width of PSF |

The 0 degree case is easy to visualize with the help of the left panel of Figure 1: clearly, the width of the photosite square (measured along the step edge) is constant. The 45 degree case is also readily visualized by noting that the we cross the widest part of the photosite square when t=0 (right panel of Figure 1); this nicely corresponds to the peak instantaneous width of √2 in Figure 3.

We can interpret the curve in Figure 3 as a weighting function, i.e., the relative contribution to the convolution of the edge and the photosite aperture at distance

*t*from the centre of the photosite aperture. Looking at the problem this way reveals a new angle: the 45 degree case presents a fair amount of its total weight located close to t=0. Roughly 50.6% of its weight is located in the part where it is wider than the 0 degree case, corresponding to the central of Figure 3 where the red curve is above the gray curve. In contrast, only about 8.6% of the weight of the 45 degree curve is located in the two tail ends (t < -0.5 and t > 0.5). If we compare this to the 0 degree case, we obtain 42% in the centre (area under gray curve where the red curve is above the gray curve), and of course 0% in the tails.This is a rather unexpected turn of events, since it implies that even though the 45 degree case starts overlapping with the edge sooner (the regions -√0.5 < t < -0.5 and 0.5 < t < √0.5), it represents only a small fraction of the total interaction with the edge. Instead of the 45 degree case being a stronger low-pass filter than the 0 degree case, we expect the opposite because the 45 degree case has roughly 20% (50.6/42) more of its weight located close to t=0.

We appear to have two mildly conflicting views:

a) the 45 degree case is "wider" at its widest point, thus it should be a stronger low-pass filter than the 0 degree case, and

b) more of the weight of the 45 degree case is close to the centre, hence it should present a

*weaker*low-pass filter than the 0 degree case.I am betting on outcome b), mostly because I already know what the empirical results will tell us ....

### Empirical results for square photosites (no diffraction)

The prediction favoured by outcome b) in the previous section tells us that we should expect MTF50 values to increase as we progress from a relative edge orientation of 0 degrees through to 45 degrees. Simulations were performed in the absence of noise, using 30 repetitions over sub-pixel shifts. Keep in mind that the MTF50 value of a square photosite aperture is about 0.6033 cycles per pixel, which is quite high.Figure 4: Square (box) PSF relative MFT50 error as function of edge orientation |

Just to check, let us examine an isotropic PSF: a pure Gaussian without any photosite aperture simulation. This should yield a purely Gaussian MTF. Same simulation, but with the radially symmetric Gaussian PSF:

Figure 5: Gaussian PSF relative MTF50 error as a function of edge orientation |

### Somewhat real world: squares plus diffraction

We have seen that a box PSF (without diffraction) produces strong anisotropy, and that a Gaussian PSF (without photosite aperture) produces no noticeable anisotropy. Using a PSF consisting of an Airy pattern convolved with a square photosite aperture should put us somewhere in the middle of the anisotropy scale.Simulations were repeated using a simulated aperture at f/2.8, light at 550 nm, a photosite pitch of 4.73 micron and no AA (OLPF) filter. These settings give an expected MTF50 value of ~ 0.504 cycles per pixel, which is slightly lower than the expected MTF50 value of ~ 0.6 cycles per pixel seen in the previous section. Accordingly, the MTF50 errors may be slightly reduced (or at least the expected variance should be reduced).

Figure 6: Airy+box PSF relative MTF50 error as a function of edge orientation |

The trend is clearly visible, but appears to be only about 60% of the magnitude of the case without diffraction (about 2.5% at 44 degrees, vs about 4% without diffraction). Smaller apertures (larger f-numbers) will reduce the anisotropy as the Airy component of the PSF will start to dominate the photosite aperture PSF.

### Any practical implications?

The effect of PSF anisotropy on MTF measurements is real, but appears to be relatively small. At 2.5%, do we even have to worry about it?Unfortunately, we have to at least be aware of this for certain types of testing and measurement. Because the error (overestimation) is systematic, it will show up in any measurement that sweeps through a range of angles, just like the MTF Mapper grid test chart, pictured here:

MTF Mapper grid test chart |

I simulated this chart using mtf_generate_rectangle, using an aperture of f/4, an Airy+box PSF, green light and a photosite pitch of 4.73 micron. Passing this synthetic image through MTF mapper to produce a surface plot (-s option) produces this result:

Figure 7: MTF50 plot obtained from simulated rendering of the MTF Mapper grid test chart using the Airy-box MTF with a square photosite aperture |

The systematic distortion of MTF50 values is clearly visible, even though the range of values is quite small. The maximum value on the scale is 0.47, which is only about 2% higher than the expected MTF50 value of 0.46073 (at 0 degrees, of course). But the cross pattern is clearly visible. At least I have confirmed the cause.

Pushing for even greater realism I repeated the simulation using the "rounded-square" photosite aperture that MTF Mapper provides. Here is the surface plot:

Figure 8: MTF50 plot obtained from simulated rendering of the MTF Mapper grid test chart using the Airy-box MTF with a rounded-square photosite aperture |

Lastly, if we use a circular photosite aperture, we get this:

Figure 9: MTF50 plot obtained from simulated rendering of the MTF Mapper grid test chart using the Airy-box MTF with a circular photosite aperture |

### Conclusion

Anisotropy is a reality that we have to deal with if we apply the slanted edge method to edges that approach a relative orientation of 45 degrees with respect to the (presumed) square photosites. The isotropy of the Airy pattern helps to attenuate the overestimation of edges approaching 45 degrees, but the systematic effect is still clearly visible in simulated images.I tried to construct an elegant analytical explanation for the interaction between the edge orientation and a square photosite aperture. This turned out to be harder than I expected, so I only have some interesting plots to offer for now. What did emerge from the theory is that we should not focus on the apparent width of the photosite aperture, but rather on the distribution of its weight relative to the centre. The somewhat startling conclusion is that we should observe higher MTF50 measurements when the orientation approaches 45 degrees.

This was supported by the actual experiments using simulated imagery.

So what can we do about this systematic distortion? Well, the only sound solution would be to stick to edges with a relative orientation of about 5 degrees. This is not a universal solution, though, because it makes it impossible to measure in the true Sagittal/Meridional directions. Imatest solved the problem by sticking to 5 degree angles and referring to "horizontal" and "vertical" MTF. This works well enough if you wish to measure peak astigmatism, but it does not allow you to measure MTF in the optically more appropriate sagittal/meridional directions.

I might add a 5-degree test chart to MTF Mapper in future, just to cover all bases.

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