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Imatest Documentation frequencies below its correct value. For this reason, until we solve the scaling issues we use the simpler, less accurate, but less error-prone approximation, C = W log2((S/N) 2 +1) where bandwidth W is traditionally defined as the channel's -3 dB (half-power) frequency, which corresponds to MTF50,S is standard (white - black) signal voltage, andN is RMS noise voltage. The square term converts voltage into power. S/N (the voltage signal-to-noise ratio) is displayed by Imatest. (S/N can refer to voltage or power in the literature; you have to read carefully to keep it straight.) Strictly speaking, this approximation only holds for white noise and a fairly simple (usually second-order) rolloff. It holds poorly when P( f ) has large response peaks, as it does in oversharpened digital cameras. The standardized sharpening algorithm comes to the rescue here. Imatest uses MTF50C (the 50% MTF frequency with standardized sharpening) to approximate W. This assures that P( f ) rolls off in a relatively consistent manner in different cameras: it is an excellent relative indicator of the effective bandwidth W. RMS (root mean square) noise voltage N is the standard deviation (sigma) of the linearized signal in either smooth image area, away from the edge. It is relatively easy to measure using the slanted edge pattern because the dynamic range of digital cameras is sufficient to keep the levels for the white and black regions well away from the limiting values (pixel levels 0 and 255). Typical average (mean) pixel values are roughly 18-24 for the dark region and 180-220 for the light region, depending on exposure. Imatest uses the average of the noise in the two regions to calculate Shannon capacity. It displays noise as N/S: normalized to (divided by) the difference between mean linearized signal level of the white and black regions, S. Noise power N doesn't tell the whole story of image quality. Noise spectral density plays an important role. The eye is more sensitive to low frequency noise, corresponding to large grain clumps, than to high frequency noise. To determine the precise effect of grain, you need to include its spectral density, the degree of enlargement, the viewing distance, and the MTF response of the human eye. High frequency noise that is invisible in small enlargements may be quite visible in big enlargements. Noise metrics such as Kodak's print grain index, which is perceptual and relative, takes this into account. Fortunately the noise spectrum of digital cameras varies a lot less than film. It tends to have a gradual rolloff (unless significant noise reduction is applied), and remains fairly strong at the Nyquist frequency. It's not a major factor in comparing cameras— the RMS noise level is far more important. Very geeky: The limiting case for Shannon capacity. Suppose you have an 8-bit pixel. This corresponds to 256 levels (0-255). If you consider the distance of 1 between levels to be the "noise", then the S/N part of the Shannon equation is log2(1+256 2 ) ≈ 16. The maximum possible bandwidth W— the Nyquist frequency— is 0.5 cycles per pixel. (All signal energy above Nyquist is garbage— disinformation, so to speak.) So C = W log2(1+ (S/N) 2 ) = 8 bits per pixel, which is where we started. Sometimes it's comforting to travel in circles. History Of course I didn't think of it first. Links R. Shaw, "The Application of Fourier Techniques and Information Theory to the Assessment of Photogrphic Image Quality," Photographic Science and Engineering, Vol. 6, No. 5, Sept.-Oct. 1962, pp.281-286. Reprinted in "Selected Readings in Image Evaluation," edited by Rodney Shaw, SPSE (now SPIE), 1976. The University of Texas Laboratory for Image & Video Engineering is doing some interesting work on image and video quality assessment. They approach the problem using information theory, natural scene statistics, wavelets, etc. Challenging material! Education: Digital X-ray: Image Quality Parameters for Digital Detector from Wipro GE Healthcare. Interesting material related to an application where low noise (high SNR) is more important than high MTF. The exact definition of DQE is hard to find on the web. 64 of 451
Imatest Documentation Blur units, MTF, and DXO Analyzer's BxU Blur operations, MTF, and DxO ® Analyzer's BxU ® * In early 2004, DxO Labs of Boulogne, France introduced a program for analyzing digital image quality called DxO analyzer ® . It has aroused a great deal of interest, but its price is far beyond the means of most photographers. (If you have to ask...) Several publications, including Luminous-landscape.com, pictchallenge.com (web), and France's outstanding Chasseur d'Images (print) use it for testing cameras and lenses. DxO analyzer measures sharpness using the Blur Experience Unit (BxU ® ), which was defined by Luc Marin of DxO in a PMA 2005 PowerPoint presentation. "Our convention : BxU = -1 for “Adobe PhotoShop 7.0” sharpen function. Therefore “Sharpen more” BxU = -2.7" It may seem counterintuitive to define blur in terms of sharpening (yes, it's confusing), but there's a reason: familiarity. Photographers rarely apply blurs, but sharpening is an important part of digital image processing. For best results sharpening needs to be applied to nearly every digital image (in the camera, RAW converter, and/or image editor). In a conversation in late 2006, Dr. Frederic Guichard of DxO indicated that the BxU is proportional to the variance (σ 2 ) of the line spread function LSF— the derivative of the average pulse— a convenient measurement that does not require an MTF calculation. Variance is displayed by Imatest SFR when Line spread function is selected for the Edge plot (the upper plot of the MTF figure). In this page we derive an approximate relationship between BxU, MTF, and variance. We present it here, before we get into the details, which most readers will prefer to skip. The details One BxU (DxO's Blur Experience Unit) is equivalent to 1 (or perhaps 1.25) applications of the Photoshop Blur filter. Approximately (using 1 Bxu = 1 Blur), MTF @ 0.25 cycles/pixel (half-Nyquist) = 0.862 * 0.78 BxU ). MTF50 (cycles/pixel) = 1 / sqrt(6.067 BxU + 3.6) These numbers were derived from the tables below using the observations that Blur is the approximate inverse of Sharpen and that the number of Blur applications is proportional to 1/MTF50 2 . Alternatively, BxU is approximately equal to 4.6 * LSF variance (σ 2 ) in pixels In the paper, "Uniqueness of blur measure" by Jerome Buzzi and Frederic Guichard, IEEE International Conference on Image Processing, 2004 (which contains some truly formidable math), blur is is equated with the variance (σ 2 ) of gaussian blur, exp(-x 2 / (2σ 2 )). (This one dimensional equation is somewhat simplified. DxO uses two-dimensions: Blur = (σxx 2 + σyy 2 ) / 2 ). Since σ has units of distance, this suggests that BxU is proportional to the square of the 10-90% rise distance. And since the Fourier transform of a gaussian is itself a gaussian, this implies that BxU is proportional to 1/MTF50 2 . We examine this hypothesis below. The effects of the Photoshop 6 Blur More and Blur filters were analyzed by applying each operation five times to an ideal edge of moderate contrast (pixel levels 202 and 50), shown on the right unmagnified and magnified 4x. The image on the lower right is the edge after two Blur Mores. Imatest SFR produced the following MTF results. Blur more Blur N MTF50 (C/P) m1 m2 R10-90 R10- Pixels 90 2 MTF @ .25 c/p σ 2 MTF50 Pixels (C/P) m1 m3 MTF @ .25 c/p σ 2 Pixels 0 0.526 3.61 0 0.953 0.91 0.862 0.121 0.526 3.61 0 0.862 0.121 1 0.229 19.1 1.07 2.13 4.54 0.431 0.637 0.332 9.07 0.90 0.665 0.343 Ideal edge 65 of 451
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