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Phil Service
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Previous investigations have treated identical ancestry as a qualitative phenomenon. It is shown here by simulation that genealogies also become quantitatively nearly identical. In a finite population with bi-parental reproduction, sufficiently distant ancestors must occur multiple times in the genealogy of every present-day individual. The number of different paths that connect an ancestor and descendant is a measure of quantitative ancestry. In a random-mating population of size N, all individuals will have nearly identical quantitative genealogies within about 3 log2N generations, or less, after an arbitrarily chosen base generation (for population sizes 64 – 512). This is a consequence of the fact that the qualitative genealogies of all present-day individuals are exactly the same for all generations prior to the cohort of most recent identical ancestors (MRIA). Representation of base-generation individuals in the quantitative genealogies of later generations is largely determined by the reproductive success of base-generation members and their immediate descendants. The proportion of population-wide ancestry attributable to a given base-generation individual becomes nearly fixed by slightly more than 2 log2N generations after the base generation. The genetic consequences of quantitative near identical ancestry appear to be limited, at least at the level of descendant individuals. Continue reading
Posted 5 days ago at Log2(N)
A random mating, sexually reproducing population of constant size was simulated with two different computer programs. Time to most recent common ancestor (MRCA) for the entire population was approximately log2N generations, where N is population size. The time to most recent identical ancestry (MRIA) was approximately 2 log2N generations before the present. These results agree well with a previously published analytical and simulation study. For random pairs of individuals, the time to a most recent common ancestor (MRCA2) was about 0.5 log2N generations. The interval between successive identical ancestry cohorts was slightly more than two generations, and was independent of population size. Continue reading
Posted Mar 30, 2017 at Log2(N)
Images of a ColorChecker Classic were taken in direct sunlight and shade. Consistency of color reproduction across lighting conditions was best when using ColorChecker Passport profiles made specifically for each situation. The general-purpose dual-illuminant Adobe Standard profile for the Sony A6500 camera gave less color consistency between sun and shade images. However, the effect of camera profile on consistency, as measured by CIEDE2000, was not large. Average CIEDE2000 between sun and shade images was approximately 2.0 for the Passport profiles; a value that is well above the threshold of just noticeable difference when colors are compared side-by-side. It is likely that no profiling procedure can produce absolute color consistency across different illuminants and shooting situations, given current camera sensor technology. Continue reading
Posted Jan 17, 2017 at f/optimum
A raw image of a ColorChecker Classic chart was processed with three camera profiles: an Adobe Standard profile, a profile made with the ColorChecker Passport application, and a “do-it-yourself” profile made by the author. Accuracy of color reproduction was evaluated with the CIEDE2000 color-difference metric. The ColorChecker Passport profile resulted in the least accurate colors, as measured by average CIEDE2000. The author’s “do-it-yourself” profile produced the most accurate colors. Passport profile colors are systematically biased in the direction of increased chroma (C*). The bias is almost certainly intentional, and possible reasons for it are discussed. Continue reading
Posted Jan 6, 2017 at f/optimum
Data from the red-, green-, and blue-sensitive receptors of a camera sensor must be converted to a standard color space in order for images to be displayed properly. The CIE 1931 XYZ space, which encompasses the entire gamut of human color vision, serves as an intermediate between camera raw RGB and practical color spaces such as sRGB and AdobeRGB. The process of converting camera raw data to XYZ coordinates is explained. Empirically, the conversion seems never to be perfect. Color difference metrics (ΔΕ) are reviewed, and several color pairs designed to be near the threshold of just noticeable difference are illustrated. Reproduction of a 24-patch Color Checker Classic by a Sony A6300 camera is evaluated. Even with the best transformation investigated, the colors reproduced by the A6300 are generally distinguishable from their Color Checker standards. Camera profiling is the process of finding a set of linear equations that transform camera raw data to XYZ coordinates. The data necessary for constructing a profile are extracted from an A6300 raw image, and the transformation equations are estimated by multiple linear regression. Continue reading
Posted Dec 3, 2016 at f/optimum
The following two files contain the CIE 1931 xyY and L*a*b* coordinates of the MacAdam limits with D65 illumination. Methods for their computation can be found here. Continue reading
Posted Jun 22, 2016 at f/optimum
Three “wide-gamut” RGB color spaces — AdobeRGB, DCI-P3, and Rec. 2020 — are compared to the gamut of real surface colors (Pointer’s gamut) and the optimal color solid (MacAdam limits). Only Rec. 2020 covers Pointer’s gamut completely,or very nearly so, at all luminance levels examined. Rec. 2020 also approaches or exceeds the MacAdam limits in all color regions at high luminance (L* ≥ 80). Better coverage of the MacAdam limits at lower luminance levels will most likely require multi-primary displays. Continue reading
Posted Jun 20, 2016 at f/optimum
The derivation of the CIE 1931 RGB color space from the Wright – Guild color matching data is described. Emphasis is placed on explaining the underlying logic of the process. The transformation of the RGB space to the CIE 1931 XYZ space is briefly described. An argument is made that the principal intent of the color matching experiments was to develop a rigorous, quantitative framework for describing all visible colors. For that purpose, negative chromaticity coefficients or imaginary primary colors are not problematic. Neither of the CIE 1931 color spaces can be displayed on a physical device; and it seems possible that little, if any, consideration was given in 1931 to practical applications of that sort. In general, digital cameras are sensitive to all visible wavelengths, and in principal can encode all humanly visible colors in raw image files. Color spaces with gamuts that exceed the AdobeRGB gamut — currently about the widest that can be reproduced on specialized displays — may be useful for processing raw images. A wide-gamut space such as ProPhotoRGB will, in theory, minimize compression of image color information during editing; thus maximizing head-room for color adjustment. Archived raw image files may also be useful in the future if very-wide gamut displays — possibly using 4, 5, or 6 primary colors — become available. Continue reading
Posted Apr 26, 2016 at f/optimum
A Sigma DP2 Merrill and dp2 Quattro were used to photograph a 24-patch ColorChecker Classsic. Image processing (X3F and TIFF) was limited to global adjustments to white balance and exposure. Color reproduction of each of the 18 non-gray-scale patches was evaluated by comparison to the published ColorChecker (2005) L*a*b* coordinates. For both cameras, portrait color mode produced the best match to the ColorChecker. Overall, the Merrill images were a closer match to the reference values, although the difference between cameras was generally not large. Continue reading
Posted Mar 22, 2016 at f/optimum
Sigma Merrill (DP1M and DP2M) and Sigma Quattro (dp1Q and dp2Q) cameras were used to make six raw image comparisons. For each pair, comparisons were made at Merrill native resolution (4,704 x 3,136) and at Quattro native resolution (5,424 x 3,616). Images were evaluated for rendition of fine detail. Quattro images were generally superior to Merrill images when the latter were up-sampled to Quattro dimensions. Images were much more closely matched when Quattro images were down-sampled to Merrill dimensions. In that case preference may be largely subjective, and perhaps significantly influenced by post-processing adjustments. Actual-pixels central-area crops are provided for each of the twelve comparisons. Continue reading
Posted Mar 3, 2016 at f/optimum
A transmission step wedge was photographed with a Sigma DP1 Merrill and dp1 Quattro. Signal mean value and signal-to-noise ratio (SNR) were estimated for each color channel from the raw (X3F) files using RawDigger. Quattro raw image signal strength is substantially more uniform across color channels. Quattro images also have higher SNR (i.e., less noise) than do Merrill raw images in the red and green channels. Quattro blue channel signal strength was substantially less, and SNR was slightly inferior to that of the Merrill, particularly in the well-exposed portions of the images. Averaged across color channels, the difference in SNR is equivalent to about 1/2 EV advantage in dynamic range for the Quattro. In practice, the Quattro may have a ≥ 1 EV superiority because, in contrast to the strongly blue-channel biased overexposure of the Merrill sensor, the Quattro blue channel is relatively resistant to overexposure; and all three color channels tend to become overexposed in concert. These differences in signal strength and SNR can be understood at least partly by reference to differences in sensor design. They suggest that the Quattro design was chosen to improve signal strength and SNR in the red and green channels, while sacrificing some signal quality in the blue channel. The net effect is substantial improvement in overall signal characteristics: in particular better balance among color channels. ISO series were made with a DP2 Merrill and a dp2 Quattro. Raw files were processed through Sigma Photo Pro, exported as TIFFs, and taken into Photoshop. As expected from the signal strength and SNR analyses, Quattro images had an approximately 1-stop advantage in high ISO image quality. That is, Quattro images exposed at ISO 1600 were similar to, or slightly better than, Merrill images at ISO 800. Continue reading
Posted Feb 27, 2016 at f/optimum
Larger sensors can achieve higher resolution than smaller ones only if they have more photosites. To date, the potential of full-frame sensors, in terms of image resolution, has been limited by the fact that FF sensors typically have relatively large photosites. A 36 MP FF sensor has 4.9 µm photosites, and a linear resolution of about 7,400 photosites on its long axis (3:2 aspect ratio). On the other hand, a 20 MP 1-inch sensor has 2.4 µm photosites, and a linear resolution of about 5,500 photosites on the long axis. The 35% increase in resolution provided by the FF sensor is much less than the actual difference in linear dimensions of the sensors — 172%. In order to exploit the full potential of larger sensors with respect to image resolution, it will be necessary to keep photosites small. That will involve some sacrifices in low-light performance, and entail costs in processing very large (> 100 MP) images. Also, in order to fully realize the benefits of photosite spacing of 2 – 3 µm, lenses must perform exceptionally well at f/2.8 – f/4: perhaps close to the theoretical limits set by diffraction. Assuming such FF lenses can be manufactured at reasonable cost, use of such relatively large apertures will compromise the ability to obtain appreciable depth of field while at the same time realizing increased resolution. Continue reading
Posted Jan 23, 2016 at f/optimum
Posted Dec 17, 2015 at f/optimum
Photosite spacing of 1.5 µm or less is common for smart phone cameras; and 1-inch sensors in cameras such as the the Sony RX100 III have photosite spacing of 2.4 µm. Diffraction-limited line-pair resolution is given for photosite spacing as little as 0.5 µm. Two micron photosite spacing implies APS-C and “full-frame” (FF) sensors with 94 and 216 MP, respectively. In order to approach the theoretical resolution limits of sensors with 2 – 3 µm photosites, it will be necessary to have lenses that perform exceptionally well at apertures f/2.8 – f/4. It is not clear if such lenses can be manufactured at reasonable cost for APS-C and FF sensors. If it is, it may be necessary to sacrifice large maximum apertures, such as f/1.4, in order to make “slower” but sharper lenses. With current technology, 2 – 3 µm photosites will entail a trade-off between resolution and low noise, when compared to current FF and APS-C sensors. For most image uses, 100 MP or greater resolution implies capture oversampling. That is, images will be down-sampled for “final”use. It is suggested that such down-sampling may produce a sharper and less noisy final image than could otherwise be obtained by capturing images at lower initial resolution. Continue reading
Posted Jul 6, 2015 at f/optimum
High-resolution (64MP) and standard-resolution (16MP) images from the Olympus OM-D E-M5 II are compared. The high-resolution images have substantially more fine detail “information” than do the standard-resolution images. That is demonstrably true even when the high-resolution images are down-sampled to 16MP for 1:1 comparison with the standard images. The Olympus high-resolution images are also compared to images from a Sigma DP2 Merrill (14.75MP). Images from both cameras were resampled to dimensions appropriate for making 18-inch-long prints. When viewed on-screen at 100% magnification, the E-M5 II images were clearly superior to the DP2M images. However, when printed, the DP2M images exhibited greater apparent detail. Much of the difference in detail rendition in these comparisons is qualitative rather than quantitative. The E-M5 II high-resolution mode appears to offer considerable advantage if the images are intended for display at large sizes. That advantage must be balanced against the fact that it is appropriate only for tripod shooting of motionless subjects, and the fact that the improved detail is not apparent when images are sized for typical viewing on a computer display. Continue reading
Posted Apr 16, 2015 at f/optimum
The high definition shooting mode of the Olympus OM-D E-M5 Mark II is investigated by simulation of line-pair images. The half-photosite shift that is employed to produce 64MP images from a 16MP sensor can, in principle, double the linear resolution that is otherwise achieved by the camera without sensor shifting. However, image pixels obtained at the shifted sensor position are not independent of pixels obtained at the unshifted position: the two sets of pixels overlap completely, with a half-photosite offset. The result is that micro contrast is reduced in comparison with a true doubling of the linear photosite resolution. Given the 3.73 µm photosite pitch of the E-M5 II sensor, the resolution benefits of sensor shifting decrease rapidly at apertures smaller than about f/5.6 because any potential gain in resolution is overwhelmed by diffraction blur (coupled with non-independence of shifted and un-shifted image pixels). Continue reading
Posted Feb 27, 2015 at f/optimum
The maximum amount of detail that can be contained in an ink-jet print is set by the native resolution of the printer and the size of the print. In the case of the Epson Stylus Pro 3880, maximum detail requires that the image be printed at 720 pixels per inch (ppi). Printing at 720 ppi ensures that the visible detail in the print will be oversampled, and therefore printed with greater sharpness than if the image were captured with fewer pixels and printed to the same size at 360 ppi. In order to fully exploit the capabilities of the Epson 3880 in a modestly large (~ 50 cm) print, it is necessary to have an image on the order of 110 – 120MP. At present, the only way to make maximum-detail prints larger than 25-30 cm, is by stitching multiple frames to make larger images. Continue reading
Posted Feb 8, 2015 at f/optimum
Resolution limits (lp/mm) for a perfect lens as a function of aperture and sensor photosite size are determined by simulation. Results are shown for 20, 50, and 80% line-pair contrast ratios. Smaller photosites always increase resolution and contrast (within the range of sizes and aperture values simulated), although marginal gains are small if diffraction blur is appreciably larger than photosite size. If diffraction blur diameter is reduced to less than photosite size, resolution can improve substantially, particularly when line-pair contrast is high. With respect to “total” resolution (LW/PH), the usual rules of “camera equivalence” are obeyed. The result is that, with perfect lenses, larger sensors can achieve higher total resolution than smaller ones only if they have more photosites (or if depth of field is sacrificed). Resolution limits for a perfect lens are compared to published test results for the Nikkor 85mm f/1.4G on a Nikon D3x, and for the Zeiss Otus 55mm f/1.4 on a Nikon D800E. Test results for the two lens/camera combinations are inconsistent with each other and with simulations, even at smaller apertures where diffraction rather than lens performance is assumed to limit resolution. Reasons for the inconsistencies are discussed. Given that resolution limits depend upon both diffraction blur and photosite size, it is generally not useful to erect a dichotomy between “diffraction-limited” and “sensor-limited” resolution. Continue reading
Posted Jan 12, 2015 at f/optimum
The effect of diffraction blur on resolution of black and white line-pair images is modeled by aggregation of a large number of blur circles. The contrast ratio of the line-pairs declines to 50% when the blur circle diameter is about 79% of the width of a line-pair. Contrast falls to about 7% when the blur circle diameter reaches 115% of the line-pair width. These results imply aperture-specific diffraction-limited resolutions. For example, with 50% contrast, the theoretical limit of image resolution is approximately 209 line-pairs per millimeter (lp/mm) at f/4. No other sources of blur are considered. In order for cameras with “full-frame”, APS-C, and m4/3 sensors to achieve resolutions closer to the theoretical diffraction limits, it will be necessary to increase the number of sensor photosites several-fold. It is not clear how much, if any, improvement will be required in lenses. Continue reading
Posted Dec 19, 2014 at f/optimum
The Gorman-Holbert black and white conversion is discussed with reference to Gorman’s instructions for creating a Photoshop action. The conversion is based on the lightness channel of the L*a*b* color model. As such, it appears to be particularly useful for images with limited color palettes. The conversion also employs the Photoshop High Pass filter as a “finishing” step. The principal effect of the High Pass filter in this application appears to be to increase contrast, which enhances image detail, particularly in the mid tones. The High Pass filter is compared to the Clarity adjustment of Adobe Camera Raw/Lightroom, and to the Unsharp Mask filter that is also available in Photoshop. Two simple modifications to the Gorman-Holbert conversion are suggested. The first substitutes a Gradient Map adjustment layer to enable true split toning. The second implements the High Pass filter as a smart filter so that its radius setting can be easily adjusted. Continue reading
Posted Dec 4, 2014 at f/optimum
A camera sensor samples the image that is produced by the lens. The sampling frequency is determined by the photosite pitch of the sensor. In the hypothetical case where a lens is producing images of black and white line-pairs without blur, it is shown that the sampling frequency must be at least four photosites per line-pair in order to recover both the frequency and original contrast of the line-pairs. That is twice the Nyquist sampling rate. Higher sampling frequencies more faithfully reproduce the image formed by the lens. In the case of line-pairs of a given frequency, the edges of the lines become sharper with higher frequency sampling. Published resolution data for the Zeiss 55mm f/1.4 Otus lens on the Nikon D800E are discussed with regard to theoretical limits based on sampling frequency. A case is made that resolution is sensor-limited in that test. Continue reading
Posted Sep 27, 2014 at f/optimum
Matched pictures were taken with a Nikon D90 and a Nikon D7100. Both images were upsampled for printing at 20 x 13.3 inches (51 x 34 cm) and 360 pixels per inch (ppi) on an Epson Stylus Pro 3880. PhotoZoom Pro 6 (BenVista, Ltd.) was used for upsampling. The resulting prints from the two cameras were nearly indistinguishable with respect to amount and quality of fine detail. As a “control”, prints were also made with upsampling left in the background, under the control of the printer driver or Lightroom. The resulting prints were quite good, but noticeably lower-quality than those made from images upsampled with PhotoZoom Pro. The “native” resolution of the D90 image at 20 x 13.3 inches is only 214 ppi. These results argue that the frequently cited 300-pixel-per-inch rule for high quality ink jet printing is overly conservative when using current software and hardware. Continue reading
Posted Sep 20, 2014 at f/optimum
Summary: A previous experiment indicated that the effective size of the diffraction blur circle is substantially smaller than customarily assumed. Smaller size can be accounted for in optimal aperture calculations by incorporating a diffraction blur coefficient, α. The present paper describes a simple procedure to estimate α. Although simple in... Continue reading
Posted May 20, 2014 at f/optimum
Summary: The fundamental requirement of optimal aperture calculations is that diffraction and defocus blurs be interchangeable — that they make equivalent contributions to total blur. Examples are shown in which images that contain exclusively diffraction blur are indistinguishable from images that contain mostly defocus blur. However, the conditions under which... Continue reading
Posted May 7, 2014 at f/optimum