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Imatest Documentation Noisy images In images with long transitions (10-90% risetime r1090 over 2 pixels) and high noise, increasing the noise can increase the SQF, unless we take preventative steps. What we do is to recognize that most signal energy is at spatial frequencies below 1/r1090. At frequencies over 1/r1090 we do not allow MTF to increase: MTF( fn ) = min( MTF( fn-1 ), MTF( fn ) ). This has little or no effect on the MTF due to the edge, but prevents noise spikes from unduly increasing SQF. Contrast sensitivity function CSF The human eye's contrast sensitivity function (CSF) is limited by the eye's optical system and cone density at high spatial (or angular) frequencies and by signal processing in the retina (neuronal interactions; lateral inhibition) at low spatial frequencies. Various studies place the peak response at bright light levels (typical of print viewing conditions) between 6 and 8 cycles per degree. We have chosen a formula, described below, that peaks just below 8 cycles/degree. You may learn something about your own eye's CSF by viewing the chart below at various distances and observing where the pattern appears to vanish. Chart contrast is proportional to (y/h) Contrast sensitivity function 2 , for image height h. To my eyes (which underwent Lasik refractive surgery in 1998 to correct for 10 diopter nearsightedness) it appears that the peak in the curve on the right should be considerably broader. But this isn't quite the case because of the eye's nonlinear response to contrast. Although the chart below appears to decrease in contrast linearly from top to bottom, the middle of the chart has 1/4 the contrast as the top. Log frequency-Contrast chart, created by Test Charts. We have chosen a formula that is relatively simple, recent, and provides a good fit to data. The source is J. L. Mannos, D. J. Sakrison, ``The Effects of a Visual Fidelity Criterion on the Encoding of Images'', IEEE Transactions on Information Theory, pp. 525-535, Vol. 20, No 4, (1974), cited on this page of Kresimir Matkovic's 1998 PhD thesis. 16 of 451
Imatest Documentation CSF( f ) = 2.6 (0.0192 + 0.114 f ) exp(-0.114 f ) 1.1 The 2.6 multiplier drops out of SQF when the normalization constant K is applied. This equation can be simplified somewhat. CSF( f ) = (0.0192 + 0.114 f ) exp(-0.1254 f ) The preferred SQF equation, SQF = K ∫CSF( f ) MTF( f ) d(log f ) = K ∫(CSF( f ) MTF( f ) / f ) d f , blows up when the dc-term (0.0192) is present. Fortunately, as the above plot shows, removing it makes very little difference. The formula used to calculate SQF in the preferred equation is CSF( f ) = 0.114 f exp(-0.1254 f ). More geek stuff (additional equations, not used, but can be selected) Links Some additional equations are included in the SQF options for experimental study. where SQF = K ∫CSF( f ) MTF( f ) d f for all angular frequencies f SQF = K log( ∫CSF( f ) MTF( f ) d f ) for all angular frequencies f K = 100% / ∫ CSF( f ) d f or K = 100% / log(∫ CSF( f ) d f ) is the normalization constant. log (often written ln) is the natural logarithm, i.e., loge. These equations were developed to address concerns about d(log f ) = d f /f , which becomes very large at low spatial frequencies. Integrals containing this term blow up if CSF( f ) contains a dc (constant) term. Fortunately, as noted above, removing the dc term from CSF makes very little difference and works quite well, so we only keep these equations for experimental purposes. Granger used d(log f ) because he noticed that a constant percentage change in MTF corresponded to a just noticeable difference, i.e., the eye responds logarithmically. Equation 1 is linear. Equation 2 is logarithmic, but presents some scaling problems: What to do about log(1) = 0? Should we use log10 (comparable to density measurements), log2 (comparable to exposure value or f-stop measurements), or loge (in accordance with Boulder, Colorado community standards, where only organic, natural logarithms are employed)? We've chosen not to pursue these equations for now. Bob Atkins' excellent introduction to MTF and SQF is highly recommended. Bror Hultgren, with 26 years of experience as an imaging scientist at the Polaroid Corporation, has developed a number of tools that relate SQF to viewer preferences, using category scaling and subjective testing. Bror is available for consulting. Popular Photography has been using SQF for testing lenses for years. Their test results are well worth exploring. 17 of 451
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