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#1




[LONG] Theoretical estimates for filmequivalent digital sens
This or similar topics appear quite often, but most treatments avoid starting "from first principles". In particular, the issues of photon Poisson noise are often mixed up with electron Poisson noise, thus erring close to an order of magnitude. Additionally, most people assume RGB sensors; I expect that nonRGB can give "better" color noise parameters than (high photon loss) RGB. [While I can easily detect such errors in calculations of others, I'm in no way a specialist, my estimates may be flawed as well... Comments welcome.] Initial versions of this document were discussed with Roger N Clark; thanks for a lot of comments which lead to major rework of calculations; however, in order of magnitude the conclusions are the same as in the beginning of the exchange... [I do not claim his endorsement of what I write here  though I will be honored if he does ;] I start with conclusions, follow with assumptions (and references supporting them), then conclude by calculations, and consideration of possible lenses. CONCLUSIONS: ~~~~~~~~~~~ Theoretical minimal size of a color sensor of sensitivity 1600ISO, (which is equivalent to Velvia 50 36x24mm in resolution and noise) is 13mm x 8.7mm. Similar B&W sensor can be 12x8mm. Likewise, theoretical maximum sensitivity of 3/4'' 8MP color sensor is 1227 ISO. [All intermediate numbers are given with quite high precision; of course, due to approximations in assumptions, very few significant digits are trustworthy.] These numbers assume QE=1, and nonRGB sensor (to trade noncritical chrominance noise vs. critical luminance noise). For example, in a 2x2 matrix one can have 2 cells with "white" (visibletransparent) filter, 1 cell with yellow (passes R+G) filter, another with cyan (passes G+B) filter. ASSUMPTIONS: ~~~~~~~~~~~ a) Photopic curve can be well approximated by Gaussian curve V(lambda) = 1.019 * exp( 285.4*(lambda0.559)^2 ) see http://home.tiscali.se/pausch/comp/radfaq.html b) Solar irradiation spectrum on the sea level can be well approximated by const/lambda in the visible spectrum (at least for the purpose of integration of photopic curve). See http://www.jgsee.kmutt.ac.th/exell/Solar/Intensity.html http://www.clas.ufl.edu/users/emarti...irradiance.htm In the second one lower horizontal axis is obviously in nm, and the upper one complete junk. Sigh...) c) Sensitivity of the sensor is noisebound. Thus sensitivity of a cell of a sensor should be measured via certain noise level at image of 18% gray at normal exposure for this sensitivity. d) The values of noise given by Velvia 50 film and Canon 1D Mark II at 800ISO setting at image of 18% gray are "acceptable". These two are comparable, see http://clarkvision.com/imagedetail/d...gnal.to.noise/ Averaging 15 and 28 correspondingly, one gets 21.5 as the "acceptable" value of S/N in the image of 18% gray. e) Noise of the sensor is limited by the electron noise (Poisson noise due to discrete values of charge); other sources of noise are negligeable (with exposition well below 40sec). See http://www.astrosurf.com/buil/d70v10d/eval.htm f) The AE software in digital cameras is normalizing the signal so that the image of 100% reflective gray saturates the sensor. [from private communication of Roger Clark; used in "d"] g) Normal exposure for 100ISO film exposes 18% gray at 0.08 luxsec. See http://www.photo.net/bboard/qanda...?msg_id=004kMM h) The color "equivalent resolution" numbers in http://clarkvision.com/imagedetail/f...digital.1.html may be decrease by 25% to take into account recent (as of 2005) improvements in demosaicing algorithms. E.g., see http://www.dpreview.com/reviews/koni...200/page12.asp Taking largest numbers (Velvia 50 again, and Tech Pan), this gives 16MP B&W sensor, and 12MP color sensor. i) Eye is much less sensitive to the chrominance noise than to luminance noise. Thus it makes sense to trade chrominance noise if this improves luminance noise (up to some limits). In particular, sensors with highertransparency filter mask give much lower luminance noise; the increased chrominance noise (due to "large" elements in the toRGBtranslation matrix) does not "spoil" the picture too much. j) To estimate Poisson noise is very simple: to get S/N ratio K, one needs to receive K^2 particles (electrons, or, assuming QE=1, photons). METAASSUMPTION ~~~~~~~~~~~~~~ In any decent photographic system the most important component of performance/price ratio is the lenses. Since the price of the lens scales as 4th or 5th power of its linear size, decreasing the size of the sensor (while keeping S/N ratio) may lead to very significant improvements of performance/price. Details in the last section... [This ignores completely the issue of the price of accumulated "legacy" lenses, so is not fully applicable to professionals.] Since sensor is purely electronic, so (more or less) subject to Moore law, the theoretical numbers (which are currently an order of magnitude off) have a chance to be actually relevant in not so distant time. ;) PHOTON FLOW OF NORMAL EXPOSURE ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ First, we need to recalculate 0.08 luxsec exposure into the the photon flow. Assume const/lambda energy spectral density (assumption b), integration of photonic curve gives const*0.192344740294 filtered flow. With constant spectral density at 1 photon/(sec*mkm*m^2), const = h * c, so the eyecorrected energy flow is 3.82082403851941e20 W/m^2 = 2.60962281830876e17 lux. Thus 0.08 luxsec corresponds to (constant) spectral density 3065.57711860622 photon/(mkm*mkm^2). This is the total photon flow of the image of 18% gray normally exposed for 100ISO film. B&W SENSOR ~~~~~~~~~~ One can imagine (among others) 3 different theoretical types of B&W sensor: one giving "physiologically correct" response of the photopic curve, one accepting all photons in the "extended visible" range of the spectrum 380 to 780nm, and an intermediate one, one accepting all photons in the "normal visible" range of the spectrum 400 to 700nm. See http://en.wikipedia.org/wiki/Visible_light To cover the first case, one needs to multiply the value obtained in the previous section by the integral of the photopic curve, 0.106910937 mkm; for the other, one needs to multiply by the width of the window, 0.4 mkm, and 0.3 mkm. Resulting values are 327.7437, 1226.23, and 919.673 photon/mkm^2 as the flow of 18% gray normally exposed for 100ISO film. However, since photopic curve should not produce any particularly spectacular artistic effect, it makes sense to have the sensor of maximal possible sensitivity, and achieve the photopic response (if needed) by application of a suitable onthelens filter. So we ignore the first value, and use the other two. For example, the smaller value gives photon Poisson noise S/N ratio of 21.5 with a square cell of 0.70896 mkm. The larger value of the window, 0.4 mkm, results in a square cell of 0.613977 mkm. These are smallest possible sizes of the cell which can provide the required S/N ratio at exposure suitable for 100ISO film. To have 1600ISO sensor, these numbers should be quartupled; 16MP 3:2 ratio sensor based on the 0.4mkm spectral window results in 12x8mm sensor. OPTIMIZING THE COLOR MASK ~~~~~~~~~~~~~~~~~~~~~~~~~ For color sensor, theoretical estimates are complicated by the following issue: different collections of spectral curves for the filter mask can result in identical sensor signal after suitable postprocessing. (This ignores noise, and demosaicing artefacts.) Indeed, taking a linear combination of the R,G,B cells is equivalent to substituting the transparency curves for mask filters by the corresponding linear combination. (This assumes the linear combination curve fits between 0 and 1.) As we saw in B&W SENSOR section, a more transparent filter results in higher S/N at the cell; if the filter is close to transparent, cell's signal is close to luminance, thus higher transparency results in improvement of luminance noise. To estimate color reproduction, take spectral sensitivity curves of the different types of sensors cells. Ideally, 3 linear combinations of these curves should match the spectral sensitivity curves of cones in human eyes. Assuming 3 different types of sensor cells, this shows that spectral curves of cells should be linear combinations of spectral sensitivity curves of cones. In principle, any 3 independent linear combinations can be used for sensors curves; recalculation to RGB requires just application of a suitable matrix. However, large matrix coefficients will result in higher chrominance noise. (Recall that we assume that [due to high transparency] the luminance is quite close to signals of the sensors, thus matrix coefficents corresponding to luminance can't be large; thus all that large matrix coefficients can do is to give contribution to CHROMINANCE noise.) Without knowing exactly how eye reacts to chrominance and luminance noise it is impossible to optimize the sensor structure; however, one particular sensor structure is "logical" enough to be close to optimal: take 2 filters in a 2x2 filter matrix to be as transparent as possible while remaining a linear combination of cone curves. This particular spectral curve is natural to call the W=R+G+B curve. Take two other filters to be as far as possible from W (and from each other) while keeping high transparency; in particular, keep the most powerful (in terms of photon count) G channel, and remove one of R and B channels; this may result, for example, in the following filter matrix W Y W Y W Y W Y C W C W C W C W W Y W Y W Y W Y C W C W C W C W here C=G+B, Y=R+G. Since the postprocessing matrix R=WC, B=WG, G=C+YW does not have large matrix coefficients, the increase in chrominance noise is not significant. Above, W means the combination of the cone sensitivity curves with maximal integral among (physically possible) combinations with "maximal transparency" being 1. While we cannot conclude that this results in the optimal mask, recall the following elementary fact: to estimate the maximal *value* f(x) one can make quite large errors in the *argument* x, and still get good approximation for f(xMAX). Thus choosing the matrix above gives a pessimistic estimate, AND one should expect that it is not very far of the correct one. TRANSPARENCY OF THE COLOR MASK ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Actually, what is R, G, B in colorimetry are in turn linear combinations of responses of cones. Use cones sensitivity curves from http://www.rwc.uc.edu/koehler/biophys/6d.html Now use RR, GG, and BB to denote *these* curves, not "usual" R, G, B of colorimetry. Since I could not find these data in table form, the values below are not maximal possible, but just first opportunities which come to mind. Using 0.9RR+0.35GG, one gets a quite flat curve; one may assume that in range 0.420.65nm the sensitivity is above 0.9. with one at 700nm going down to 0.6, and 400nm going down to 0.8. So the filter "compatible" with cone sensitivity curves can easily achive 0.9 transparency in the range 400700nm, which would give photon count 827.705822 photon/mkm^2 in the W (R+G+B) type cell. Taking GG and 0.9RR+0.35BB curves as other types of sensors, one gets average transparency about 0.8 and 0.85. Taking average transparency of the filter over a 2x2 WCWY matrix cell 0.85, one gets photon count averaged over different kinds of colorsensitive cells as 781.722165 photon/mkm^2. As above, we assume that this average photon count is the count giving contribution into luminance noise. FINAL ESTIMATES ~~~~~~~~~~~~~~~ With above average photon count at a cell, to get S/N ratio 21.5 one needs a square cell of 0.768975 mkm. Recall that this is the the smallest possible cell which can provide the required S/N ratio at exposure suitable for 100ISO film. Quadrupling to get sensitivity 1600ISO, and taking 12MP equivalent of 36x24mm Velvia 50, one gets the 13 x 8.7 mm sensor. HOW GOOD CAN 36x34mm SENSOR GO? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In other direction, 36x24 mm color sensor at sensitivity 1600ISO can (theoretically) be equivalent (or better) than 10 x 6.6 cm Velvia 50 film; that is 1/2 frame of 4x5 in film. In yet other words, take 36x24mm sensor with resolution and noise better than 4x5 Velvia 50 film; it has theoretical maximum of sensibility at 800ISO. Likewise, to achieve resolution and noise of 8x10in Velvia 50 film, the maximal sensibility of 36x24mm sensor is 200ISO. THE QUESTION OF LENSES ~~~~~~~~~~~~~~~~~~~~~~ Of course, preceeding section completely ignores the issue of lenses; on the other hand, a cheap prosumer zoom lens with 28200mm equivalent paired with a digital sensor easily gives resolution of 3.3mkm per single line (with usable image diameter about 11mm, see http://www.dpreview.com/reviews/koni...200/page12.asp ); so we know it is practically possible to create a lens which saturates the theoretical resolution of 1600ISO sensor (but probably not 800ISO and 200ISO sensor!). It is natural to expect that a nonzoom lens could saturate resolution of 800ISO sensor. This gives theoretical resolution limit of a "practical" lens + 800ISO digital 36x24mm sensor: it is equivalent to best 4x5in 50ISO film (with nonzoom lens). With zoom lense, one can achieve quality of 2.5x4in 50ISO film; sensor is at 1600ISO, lense is 28200mm zoom. Some more estimates of how practical is "practical": the zoom mentioned above is bundled with $600 street price camera which weights about 580g. Assume the lens takes 1/2 of the price, and 1/4 of the weight. Rescaling from 11mm diagonal image size to the 36x24mm image size will increase price to $70K$280K (assuming that price is proportional to 4th5th power of the size [these numbers were applicable 20 years ago, I do not know what holds today]), and will increase the weight to 9kg. On the other hand, the 4:3 aspect ratio sensor of the same area as the mentioned above 13 x 8.6 mm sensor (1600ISO sensor equivalent in quality to Velvia 50 at 36x24mm) is 12.2 x 9.17mm, diagonal is 15.26mm. It is 0.9'' sensor (in the current  silly  notation). Rescaling the mentioned above lens to this size gives lens price $1100$1500, and weight about 750g; both quite "reasonable". Recall that this 28200 equivalent zoom lens will saturates resolution of an equivalent of Velvia 50 36x24mm film. 
#2




A very nice write up, I will admit I have not gone through all of it
yet in detail. One thing to consider is that CCD have a read out noise of around 10 electrons, whereas this noise level will not greatly effect the signal to noise when looking at 400 detected photons with an noise level of 20 electrons it will start to dominate in darker parts of the scene. For instance by the time you are down 5 stops from full white the readout noise will be larger then the photon noise, by a small amount. The idea of using nonRGB filters is sound and a number of CCD sensors have used filters more like C, Y and M. Why RGB is used on digital cameras I am not sure. Scott 
#3




[A complimentary Cc of this posting was sent to
Scott W ], who wrote in article .com: A very nice write up, I will admit I have not gone through all of it yet in detail. One thing to consider is that CCD have a read out noise of around 10 electrons, whereas this noise level will not greatly effect the signal to noise when looking at 400 detected photons with an noise level of 20 electrons it will start to dominate in darker parts of the scene. For instance by the time you are down 5 stops from full white the readout noise will be larger then the photon noise, by a small amount. This is a very valid remark. However, note that these were *theoretical* estimates; after translation into this language your remark becomes: Readout noise should be decreased too; otherwise shadows noise is going to be well above Poisson noise. Thanks, Ilya 
#4




Scott W wrote:
The idea of using nonRGB filters is sound and a number of CCD sensors have used filters more like C, Y and M. Why RGB is used on digital cameras I am not sure. My guess is that to do otherwise would increase the chroma noise too much. Chroma noise in digital cameras at high ISO is already intrusive, and anything that increases it may be unwelcome, even if sensitivity improved. Without direct experimental data it's hard to say. The other issue is how well nonRGB filters could be made to approximate the colour matching functions of typical display systems. Red and green are quite well matched by sensors of a typical camera, but the blue is quite a way off because its spectral sensitivity is too broad.[1] It would be a matter of measuring some physically realizable filters and seeing what colour matching functions resulted. Andrew. [1] The Reproduction of Colour, 6th Edition, Robert Hunt, p556. 
#5




Ilya Zakharevich wrote:
PHOTON FLOW OF NORMAL EXPOSURE ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ First, we need to recalculate 0.08 luxsec exposure into the the photon flow. Assume const/lambda energy spectral density (assumption b), integration of photonic curve gives const*0.192344740294 filtered flow. With constant spectral density at 1 photon/(sec*mkm*m^2), const = h * c, so the eyecorrected energy flow is 3.82082403851941e20 W/m^2 = 2.60962281830876e17 lux. What unit is 'mkm', wavenumber?  Hans 
#6




[A complimentary Cc of this posting was sent to
], who wrote in article : My guess is that to do otherwise would increase the chroma noise too much. Chroma noise in digital cameras at high ISO is already intrusive, and anything that increases it may be unwelcome, even if sensitivity improved. Without direct experimental data it's hard to say. When I look on the digital images of a gray surface (those of "compare two cameras" kind), it looks like my perception of noise is not related to chrominance noise at all. At least a camera with higher measured individualchannel R/G/B noise can produce much lower visible noise if its noise reduction algorithm favors luminance noise (as confirmed by luminance noise graph). Of course, it is in no way scientific conclusion, but I may have seen about ten such comparisons... The other issue is how well nonRGB filters could be made to approximate the colour matching functions of typical display systems. AFAIU, this has nothing to do with display (output) system, but only with input system (cones). As far as the filters match cones, you can postprocess colors into *any* display system (if the initial color is in the gamut of the display system). And if you do not match the cone sensitivity, colors which look the same will get different when stored. After this no amount of postprocessing will be able to fix this. Hope this helps, Ilya 
#7




[A complimentary Cc of this posting was sent to
HvdV ], who wrote in article : filtered flow. With constant spectral density at 1 photon/(sec*mkm*m^2), const = h * c, so the eyecorrected energy flow is 3.82082403851941e20 W/m^2 = 2.60962281830876e17 lux. What unit is 'mkm', wavenumber? Yes (?); Wavelength. IIRC, wavenumber is 1/wavelength (or some such; 2pi comes to mind...). [BTW, because of nonlinearity of wavelength vs wavenumber, spectral density which constant per wavelength becomes very nonconstant when measured per wavenumber.] Yours, Ilya 
#8





#9




Hi Ilya,
Yes (?); Wavelength. IIRC, wavenumber is 1/wavelength (or some such; 2pi comes to mind...). Yes, wavenumber is 2 * pi / lambda, units m^1 [BTW, because of nonlinearity of wavelength vs wavenumber, spectral density which constant per wavelength becomes very nonconstant when measured per wavenumber.] Yes, but the choice is arbitrary. Since wavenumber which is proportional to photon energy, an interesting quantity for many applications, spectroscopy people tend towards wavenumber whereas optical people like wavelength since resolving power scales with that. BTW, can you substantiate your interesting assumption:  In any decent photographic system the most important component of performance/price ratio is the lenses. Since the price of the lens scales as 4th or 5th power of its linear size, decreasing the size of the sensor (while keeping S/N ratio) may lead to very significant improvements of performance/price.  with some examples? The tradeoff of lens aperture and expense vs sensor size determines ultimately the size and shape of the digital camera. After the 'fashion factor' of course.  hans 
#10




[A complimentary Cc of this posting was sent to
], who wrote in article : Scott W wrote: The idea of using nonRGB filters is sound and a number of CCD sensors have used filters more like C, Y and M. Why RGB is used on digital cameras I am not sure. My guess is that to do otherwise would increase the chroma noise too much. Chroma noise in digital cameras at high ISO is already intrusive, and anything that increases it may be unwelcome, even if sensitivity improved. Without direct experimental data it's hard to say. Judge for yourself: visit http://ilyaz.org/photo/randomnoise Yours, Ilya 
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