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Imatest Documentation SQF Options CSF equation selects the equation for the Contrast Sensitivity function. There are two choices. 1. 2. Flat from 3 to 12 cpd (where cpd is cycles per degree) is the simplifying approximation originally used by Granger and Cupery in 1972, and by a number of authors since. (Granger actually used 10 to 40 cycles/mm at the retina, which translates to 3-12 cycles/degree when the eye's focal length of 17 mm is tossed into the equation.) This setting is recommended only for comparing results between the new and old explanation. Equation: (a + bf) exp(-bf) (the default setting) is discussed in detail in the section on Contrast Sensitivity function. This setting is recommended. SQF integral is the form of the integral used in the SQF calculation. The default setting, 2. Integral( CSF(f) MTF(f) d(ln f) ), is recommended. The other values are experimental and not recommended; they are discussed below. Viewing distance allows four choices. 1. 2. 3. 4. Fixed assumes a fixed viewing distance, which can be selected in the Base viewing distance... box. Square root of PH (15 cm min.) (the default setting) assumes that viewing distance is proportional to the square root of the picture height, (d = (base distance) (PH /10) 1/2 ), with a minimum of 15 cm. This is the recommended setting, appropriate for typical gallery viewing. We tend to look at large prints at greater distances than small prints. With this assumption, if you viewed a 4x6 inch print at 12 inches (a number often found in the literature), you would view a 16x24 inch print at 24 inches. Cube root of PH (15 cm min.) assumes that viewing distance is proportional to the cube root of the picture height, (d = (base distance) (PH /10) 1/3 ). With this assumption, if you viewed a 4x6 inch print at 12 inches (a number often found in the literature), you would view a 32x48 inch print at 24 inches. This seems to be a little close, but it could be appropriate in some situations. Fixed print height; Distance from 1 cm to max. This option is different from the others. Maximum viewing distance (cm) and Print height (cm) are entered instead of Base viewing distance and Maximum print height. SQF is plotted for viewing distances from 1 cm to the maximum for the fixed print height. Of course Viewing distance is a broad average: we often move in and out when we critically examine a print. But some assumption must be made for the SQF calculation to proceed. We believe that choice 2, Viewing distance proportional the square root of the picture height, best represents the typical impression of sharpness for a range of prints. If we come across research that suggests a different fuction, we'll add it to the choices. Base distance Defined according to the Viewing distance setting. It should be left at its default value of 30 cm (12 inches) unless there is good reason to change it. 34 centimeters has also been used for measuring perceived quality in 10 cm high (4x6 inch) prints. For Viewing distance = 1. Fixed, base distance is the viewing distance in cm. For Viewing distance = 2. Square root or 3. Cube root (of picture height), base distance is the viewing distance for 10 14 of 451
Imatest Documentation cm (4 inch) high prints. Maximum print height is the maximum to plot. (Note that the height setting assumes landscape orientation: wider than tall.) It has no effect on the calculations. The default is 40 cm (16 inches), which is about as large as prints from consumer digital cameras get. Picture heights of 60 cm (24 inches) and larger are of interest to users of professional-quality digital SLRs. 20 cm is a stretch for cameraphone images. Reset resets settings to their default values. (It doesn't affect Maximum print height, which has no effect on the calculations.) We recommend keeping all settings at their default values, unless there is good reason to change them. This will help ensure standardized measurements and minimize confusion. The SQF equation Following the convention elsewhere in the Imatest site, we put the math in green boxes, which can be skipped by non-technical readers. The gist of the box below is that Granger presented the full equation for calculating SQF in 1972, but he used a simplified approximation for his calculations. Although the exact equation is strongly recommended, Imatest can use the simplified approximation where needed for comparing new and old calculations. In reviewing older publications, you should determine which calculation was used. The SQF equation The exact equation for SQF implied by Granger is, where SQF = K ∫CSF( f ) MTF( f ) d(log f ) = K ∫(CSF( f ) MTF( f ) / f ) d f for all angular frequencies f CSF( f ) is the contrast sensitivity function of the human eye, discussed below; CSF( f ) is close to zero for f > 60 cycles/degree), MTF( f ) is the Modulation Transfer function (equivalent to Spatial Frequency Response) of the optical system: the primary output of Imatest SFR, log (often written ln) is the natural logarithm, i.e., loge. d(log f ) = d f / f . K = 100% / ∫ CSF( f ) d(log f ) = 100% / ∫(CSF( f ) / f ) d f is the normalization constant: SQF would equal 100% for MTF( f ) with a constant value of 1. The Nyquist frequency is the upper frequency limit for the integral; energy above Nyquist does not contribute to image quality. Since Granger had limited access to sophisticated computers (the average personal computer today has about as much power as the entire Pentagon had in 1972), he used an an approximation for his calculations assuming that CSF( f ) is roughly constant from 3 to 12 cycles/degree. SQF = K ∫ MTF( f ) d(log f ) = K ∫(MTF( f ) / f ) d f for 3 cpd < f < 12 cpd; 0 otherwise; K = 100% / ∫ d(log f ) = 100% / ∫ d f /f for 3 cpd < f < 12 cpd is the normalization constant. K = 100% / (log(12)-log(3)) = 72.1348% Although Imatest offers the option of using this approximation (as a check on older calculations), the exact equation is recommended. The integration limits used by Granger and Cupery were 10 and 40 cycles/mm in the retina of the eye, which translate to 3 and 6 cycles/degree when the eye's focal length (FL = 17 mm) is considered. f (cycles/degree) = f (cycles/mm) (π FL) / 180. To calculate SQF it is necessary to relate spatial frequency in cycles/degree, which is used for the eye's response CSF( f ), to spatial frequency in cycles/pixel, which is calculated by Imatest SFR. where f (cycles/degree) = f (cycles/pixel) (π nPH d ) / (180 PH ) nPH is the number of vertical pixels (along the Picture Height, assuming landscape orientation), d is the viewing distance PH is the picture height in units of distance (centimeters is used in Imatest displays). Note that nPH / PH is the actual value of pixels per distance. The "pixels per inch" value of image files rarely corresponds to real prints. 15 of 451
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