# In which ways, and why, are smaller sensors more efficient than larger? Part 2

Anders W wrote:

In a recently expired thread (whose OP is summarized here), some of us discussed (among other things) the extent to which the greater efficiency of smaller sensors compared to larger generalizes across different measures of efficiency. In order to answer some questions that still remain in that regard, I proposed here that I supplement the three indicators I have previously used to index efficiency by a fourth one. I am happy to present the results of so doing in this new thread.

The idea behind the four measures is as follows: The efficiency of a sensor can vary depending on which light level we consider (shadows versus highlights) as well as the ISO range we are looking at (low versus high ISOs). The three measures I previously used considered shadow rendering separately for low and high ISOs but highlights at a single point along the ISO range only. Since the efficiency with regard to highlight rendering turns out to vary a bit across the ISO range as well, my new set of indicators consider all four combinations, shadows as well as highlights at low as well as high ISOs.

How the measures are defined is described in detail in the technical appendix at the end of the post. The two indictors that index the efficiency with regard to shadow rendering, i.e., those that focus on dynamic range (DR), are the same as before. Those that index the efficiency with regard to highlight rendering, i.e., those that focus on max SNR, are both new although the first of them can be seen as slightly modified version of the single measure focusing on max SNR used in earlier analyses. The sample of bodies/sensors is the same as that used in the previous thread, i.e., that described here supplemented by the Canon 6D (since I already happened to have entered the data for that body as well). The data source is also unchanged (DxOMark).

As in the previous thread, I capture the relationship between efficiency and sensor size by a set of regression analyses with each of the four efficiency indicators as the dependent variable and sensor size as the independent. The measure of sensor size used is the square root of the square root of the sensor area (as expressed in millimeters squared). In all four cases, this provides a better fit (as measured by the adjusted R-square) than using the sensor area or the square root of the sensor area. The results are as follows (the regression coefficient, its standard error, and the adjusted R-square).

Normed ISO-100 DR: -18.21, 2.63, 0.723

Normed High-ISO DR: -102.59, 13.70, 0.754

Normed ISO-100 max SNR: -1.84, 0.27, 0.710

Normed High-ISO max SNR: -2.42, 0.67, 0.404

The coefficients show that the expected negative relationship with sensor size obtains in all four cases. If we choose to regard the sample as simple random, all effects are statisticially significant at the .01-level or better. Consequently, the conclusion that smaller sensors are more efficient than larger generalizes to all four efficiency measures.

Comments and questions welcome, especially, of course, from some of those who participated in the thread to which this is a continuation.

......................................................................................................................................

Technical appendix

Normed ISO-100 DR: This is the unlogged "print-mode" (8 MP) DR at a DxOMark measured ISO of 100 divided by the sensor area. Before unlogging and performing the division described, the logged DR at ISO 100 was extra- or interpolated from the closest observations available using the following formula:

Logged DR at ISO 100 = BIDR + (LOG10(100) - LOG10(BI))*(BIDR - NIDR) / (LOG10(BI) - LOG10(NI))

where BIDR is the logged DR at the base measured ISO, NIDR the logged DR at the next measured ISO, BI is the base measured ISO, and NI is the next measured ISO.

Normed High-ISO DR: This is the unlogged "print-mode" (8 MP) DR obtained for equivalent photos (same amount of total light on the sensor, same DoF, same shutter speed) at a higher measured ISO given by

12,800 * SA/864

where SA is the sensor area and 864 is the sensor area of real FF, i.e., 24 x 36 mm.

Before unlogging, the DR at that particular ISO was interpolated from the closest observations available using a formula similar to the one used for "normed ISO-100 DR".

Normed ISO-100 Max SNR: This is the unlogged max SNR at a DxOMark measured ISO of 100 multiplied by the square root of MP/8 (where MP is the number of sensor megapixels) and divided by the square root of the sensor area (as expressed in millimeters squared).

Before unlogging and performing the multiplication and division described, the logged max SNR (expressed in decibels) at ISO 100 was extra- or interpolated from the closest observations available using a formula similar to the one used for "normed ISO-100 DR".

Normed High-ISO Max SNR: This is the unlogged max SNR obtained for equivalent photos (same amount of total light on the sensor, same DoF, same shutter speed) after multiplication by the square root of MP/8 and at a higher measured ISO given by

12,800 * SA/864

where SA is the sensor area and 864 is the sensor area of real FF, i.e., 24 x 36 mm.

Before unlogging and performing the multiplication described, the logged max SNR (expressed in decibels) at that particular ISO was interpolated from the closest observations available using a formula similar to the one used for "normed ISO-100 DR".

It struck me that logging the sensor area rather than taking the square root of the square root of it (as I originally did) might be a more "natural" solution to the problem of finding the best-fitting functional form (given the restriction of a single parameter). Since logging additionally turned out to improve the fit one notch further, I present the results of this slightly revised analysis below. The independent variable is now the log2 of the sensor area. Everything else is unchanged. The regression coefficients now all have a very simple interpretation. They indicate the expected decline in sensor efficiency per doubling of the sensor area.

Normed ISO-100 DR: -11.98, 1.45, 0.788

Normed High-ISO DR: -65.37, 8.44, 0.766

Normed ISO-100 max SNR: -1.20, 0.16, 0.757

Normed High-ISO max SNR: -1.65, 0.39, 0.480

All relationships are now statistically significant at the 0.001 level or better (if we choose to regard the sample as simple random).

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