In which ways, and why, are smaller sensors more efficient than larger?

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Anders W
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In which ways, and why, are smaller sensors more efficient than larger?
9 months ago

In a prior thread over on the PST (photographic science and technology) forum, I showed here and here that smaller sensors tend to be more efficient than larger for a number of different performance measures. In a recently expired thread on the MFT forum (as well as some earlier ones), Joe (aka Great Bustard) argued a) that there is in fact little or no difference in quantum efficiency across sensors of different sizes and b) that the difference in read noise at higher ISOs is largely a matter of pixel count rather than sensor size per se (i.e., that with pixel count held constant, there is little or no relationship with sensor size). You find his own words on the subject here.

This post of his inspired me to try to test these two hypotheses more systematically on the sample of bodies/sensors that I had used to demonstrate the general relationship between sensor size and efficiency in the posts to which I link above. Regrettably, I didn't have the opportunity to post the results in a reply to Joe in the previous thread before it expired so I do so here instead.

In order to test Joe's two hypotheses (as well as some others while I was at it), I ran a series of multiple-regression analyses using each of the three efficiency measures I had developed for the earlier thread on the PST forum as dependent variables and measures of sensor size and pixel count as predictors.

The sample of bodies/sensors is the same as that presented in the posts to which I link above except that I added (as a special courtesy to Joe ) the Canon 6D for which I already happened to have entered all the relevant data. The data used correspond to those summarized in the second of the two posts I link to above (which differs from the first with regard to "normed max SNR" only). It should additionally be mentioned that in the present case I used an unlogged rather than logged version of the measure I call "normed high-ISO DR" for consistency with the other two efficiency measures (which are both unlogged).

As independent variables, I used the square root of the sensor area (with the area measured in square mm) and the pixel count (measured in MP). I tried alternative measures of both quantities (sensor area rather than the square root of the sensor area and the square root of the pixel count rather than the pixel count) and the two I eventually chose are those that fit the data best (as indexed by the proportion of variance explained). The measure of sensor size employed ranges from 5.28 for the Panasonic LF1 to 29.39 for the Canon 6D and from 10.21 for the Nikon S1 to 36.37 for the Sony A7R.

The regression coefficients (with associated standard errors in parentheses) and adjusted R-squares are as follows.

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Normed ISO-100 DR

Sensor size -2.61 (0.56)

Pixel count 0.74 (0.77)

Adjusted R-square 0.643

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Normed High-ISO DR

Sensor size -9.22 (2.55)

Pixel count -6.87 (3.50)

Adjusted R-square 0.758

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Normed Max SNR

Sensor size -2.02 (0.54)

Pixel count 0.00 (0.73)

Adjusted R-square 0.610

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The last of the three regressions is a fairly straightforward test of Joe's first hypothesis since the normed max SNR is largely a measure of QE with read noise hardly contributing at all. The results in this case prove Joe's hypothesis wrong. There is a clear negative relationship between sensor size and normed max SNR. The coefficient estimate is about -2, which means that the expected decline in normed max SNR for a one-unit increase in sensor size is about 2 units. If we choose to regard the sample as simple random, the ratio between the regression coefficient and its standard error (about 4) shows the relationship to be highly statistically significant by any conventional criterion (p = 0.002). There is no indication of a relationship with pixel count. The adjusted R-square shows that sensor size accounts for about 61 percent of the variance in normed max SNR.

To test Joe's second hypothesis is a bit more complicated since the two other efficiency measures are not a measure of read noise only but are also affected by QE and I did not bother, in this first attempt, to try to isolate the former from the latter. On the other hand, the normed DR measures are of considerable interest in their own right (DR is an indicator of how well the shadows, the most problematic part of an image from a signal-noise point of view, are rendered) and we can get some leverage on Joe's hypothesis by considering how the impact of the predictors varies between the two DR measures, one at low ISO and one at high, the difference between the two measures being a matter of variations in read noise across the ISO range only.

Starting with the normed ISO-100 DR, we find a clear negative relationship with sensor size in this case as well. The expected decline in normed ISO-100 DR for a one unit increase in sensor size is in this case about -2.6 units and the estimate statistically significant under even more stringent criteria than in the previous case (p = 0.0003). There is some indication that the pixel count has a positive rather than negative impact on normed ISO 100-DR but the estimate is not significantly different from zero according to conventional criteria. The adjusted R-square indicates that the predictors (and largely the sensor size) accounts for about 64 percent of the variance in the dependent variable.

Turning to normed high-ISO DR, we again find a clear negative relationship with sensor size. The coefficient is in this case about -9.2. Note however, that the larger size of the estimate in this case than in the prior two cannot be taken to mean that the effect is substantively larger. The size of the coefficients are affected by the units in which the dependent variables are expressed, and these units are not directly comparable across the three efficiency measures. The estimate is statistically significant at roughly the same level as in the case of max normed SNR (p = 0.002).

With regard to pixel count, there is in this case more of an indication that it makes a difference than in the other two. The estimate is negative, in line with Joe's hypothesis, and reaches a magnitude of -6.87. Note, however, that this estimate is not directly comparable to that for sensor size since the two predictors are not measured in directly comparable units. Since, however, the two predictors have similar observed range from top to bottom (about 25 units), their relative magnitude do give us an idea of their relative impact within this sample, in which case the results suggest that sensor size is of greater importance than pixel count. With regard to statistical significance, the estimate for pixel count does not quite pass the criterion most commonly used (p = 0.05) but it comes fairly close (p = 0.067). The adjusted R-square suggests that the two predictors together account for about 76 percent of the variance in the dependent variable, which is somewhat higher than in the other two cases, and at least partly due to pixel count rather than sensor size alone.

In conclusion, a few words are in order about the relationship between the two predictors: sensor size and pixel count. As we know, there is a positive relationship between these two. Larger sensors tend to have more pixels although there are certainly exceptions to that norm. From a sensor-design point of view, this relationship can be regarded as a causal one. Arguably, sensor designers first decide to make a sensor of a certain size and only then how many pixels to include. Consequently, pixel count cannot really be regarded as an exogenous factor on a par with sensor size. Rather, it is a factor that mediates the impact of sensor size. It follows that when pixel count appears to have an impact, as it possibly has with regard to normed high-ISO DR and perhaps normed ISO-100 DR as well, it shows a specific indirect effect of sensor size. The direct effect of sensor size, indexed by its own regression coefficient, shows the impact of sensor size that is due to mechanisms other than pixel count.

Comments and questions are welcome as usual.

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