
Detail of SX30/40 vs Compact SLR
I was puzzled about the lack of detail in SLR moon photos posted on the Internet. SLR pictures taken with 55250 mm f/45.6 lens are rarely as detailed as those I take with my SX30. Even those taken with a 100400 mm f/4.55.6 lens look only slightly better. The reason I was puzzled is that the SLR lenses have larger focal lengths than the SX30 & SX40 and also larger apertures, so the resolution of the SLR lenses should be better too. The calculations below confirm this.
In addition to comparing the lens resolution of the superzooms with the SLR lenses, I decided to compare the sensor resolution as well. I also found a formula from Kodak for combining the two resolutions into a total resolution for the camera (lens and sensor together). Although these calculations do not capture all of the contributions to camera resolution, they describe the two most important contributions with enough accuracy to understand a lot of what is happening in these cameras. I found the results for compact SLRs quite surprising.
The following sections discuss the formulas for "Lens Resolution", "Sensor Resolution" and "Total Resolution". If you are not interested in formulas, skip to the "Results" and "Conclusions" sections.
Lens Resolution
In a previous thread on Canon Talk Forum, I discussed the angular resolution of the SX30IS at full optical zoom. My outdoor tests in bright sunlight, using the USAF 1951 test pattern, produced an angular resolution of 30 microradians. I got the same result with indoor tests on a bird feather, where the alternate dark and light lines of the hairs radiating from the central shaft are similar to the USAF 1951 test pattern except for colour. When the target is at a distance such that the linepairs produce an angle of 30 microradians, the image is blurred but can still be distinguished. At 27 microradians (smaller test pattern), the image of the test pattern is completely extinguished as a washedout grey blob.
The site http://www.dantestella.com/zeiss/resolution.html (Resolution of camera lenses where are the limits – and why?, Camera Lens News No. 2, fall 1997) provides a table (from Zeiss) of resolutions for perfect lenses in linepairs per mm as a function of fstop. This table can be summarized by the following formula:
LensRes = 1600 / f# (1a)
where LensRes is in linepairs per mm (of the testpattern image on the surface of the sensor) and f# is the fstop.
The angular resolution is the smallest angle that can be resolved. A small angle can be calculated with good accuracy, without the use of trigonometry, as the object size divided by its distance from the camera. This gives the angle in radians. If you multiply by 1 million, this gives the angle in microradians. (1 Radian = 57.3 degrees). The angular resolution of a lens is given by:
LensAngRes = 625 / D
= 625 x f# / f (1b)
where LensAngRes is in microradians, D is the aperture in mm, f# is the fstop and f is the focal length in mm.
Equation 1b is exactly equivalent to 1a (being derived from it using Equation 4a, below). Equation 1a has the same form as the Rayleigh criterion except that the Rayleigh criterion has a wavelength dependence and is for two point sources of light. Equations 1a and 1b, however, are for white light and apply to the type of testpattern used in photography, which has alternate black and white bars of equal width.
Sensor Resolution
Consider the image of a test pattern on the sensor. The test pattern consists of alternate black and white bars of equal width. On the sensor, the image of each bar can be obtained even if the image of each bar is as small as a pixel in width, assuming that the lens produces a sharp enough image. Normally, the camera is not exactly oriented with the target lines, with the result that some regions of the sensor have pixels that capture half a white bar and half a black bar so that the image in that region is a washedout grey. Other regions of the sensor have the pixels aligned with the black and white bars, so that the image in those regions looks quite good. So, in this situation, the linepair resolution (black bar to black bar or white bar to white bar) corresponds to a distance of 2 pixel widths and this is considered the limit of sensor resolution.
The site http://en.wikipedia.org/wiki/Image_resolution shows images of the the letter "R", composed of pixels. On a 10x10 pixel grid, where the pixel width matches the width of the hole in the "R", the letter looks blurred but it can be identified. On a 5x 5 pixel grid, where the pixel width is twice the width of the hole in the "R", the letter cannot be identified.
The resolution of the sensor, in linepairs per mm, is given by:
SensorRes = 0.5 x Horizontal Pixel # / Sensor Width (2a)
where Sensor Width is in mm.
The angular resolution of a sensor in microradians is:
SensorAngRes = 2,000,000 x (sensor width / # horizontal pixels) / f (2b)
Total Resolution of Lens and Sensor
The total resolution of the lenssensor combination is worse than both of the two resolutions individually. The site http://www.diax.nl/pages/Lens_res_uk.html
gives the following formula for the total resolution:
1 / TotalRes^2 = 1 / LensRes^2 + 1 / SensorRes^2 (3a)
where all of the resolutions are in linepairs per mm. The symbol "^2" means that the quantity is squared. The formula is attributed to Kodak and referenced to Higgins, G.C.Appl. Opt. 3, v.1, 9, Jan 1964.
The equivalent formula for total angular resolution is:
TotalAngRes = squareroot( LensAngRes^2 + SensorAngRes^2) (3b)
Conversions
To convert from linepairs per mm to microradians:
Angle = 1,000,000 / (f x LinePairs per mm) (4a)
To convert from microradians to linepairs per mm:
LinePairs per mm = 1,000,000 / (f x Angle) (4b)
Table 1: Lens and Sensor Data Used in the Formulas
Max. Widest *1 Max Horiz Sensor
Superzooms f (mm) f# Pixel # Width (mm)
Canon SX30: 150.5 5.8 4320 6.17
Canon SX40: 150.5 5.8 4000 6.17
Compact SLRs *2
55250 mm f/45.6: 250 5.6 5184 22.3
70 or 75  300 mm f/45.6: 300 5.6 5184 22.3
100400 mm f 4.55.6: 400 5.6 5184 22.3
Notes:
*1 smallest fstop number (widest aperture) at the maximum optical zoom.
*2 Compact SLR sensor data is for the 18 MP APSC of the Canon T2i, T3i or T4i.
Table 2: Lens, Sensor and Total Resolution
Using Equations 1a, 2a and 3a
LinePairs per mm
Superzooms Lens Sensor Total
SX30: 276 350 217
SX40: 276 324 210
Compact SLRs
55250 mm f/45.6: 286 116 108
70 or 75  300 mm f/45.6: 286 116 108
100400 mm f 4.55.6: 286 116 108
Note: Larger number of linepairs per mm means better resolution.
Table 3: Lens, Sensor and Total Angular Resolution
Using Equations 1b, 2b and 3b
Angular Resolution (microradians)
Superzooms Lens Sensor Total
SX30: 24.1 19.0 30.7
SX40: 24.1 20.5 31.6
Compact SLRs
55250 mm f/45.6: 14.0 34.4 37.2
70 or 75  300 mm f/45.6: 11.7 28.7 31.0
100400 mm f 4.55.6: 8.8 21.5 23.2
Note: Smaller number for angular resolution means better resolution.
Results
Table 2 gives the resolutions in linepairs per mm on the surface of the sensor and the numbers are the same for all of the Compact SLR lenses considered.
Table 3 gives the resolutions in microradians, which is a more useful indication of which lenssensor combination can capture the detail of distant objects.
Consider the Angular Resolutions of the SX30IS in Table 3. The SX30 lens can resolve details as small as 24.1 microradians. (The ".1" is meaningless, as the actual value might be say 25 or 26 microradians.) The sensor resolution is 19 microradians, which is quite good because of the high pixel density of the sensor. Because this number is so low, the sensor can capture most of the resolution achieved by the lens, so that the total resolution is 30.7 microradians. Perhaps fortuitously, this agrees fairly well with the value of 30 microradians that I measured. Because Equation 1 describes ideal diffractionlimited lenses, one would expect the measured resolution to be worse than calculated. The reason that this is not the case may be inaccuracies in any of Equations 1, 2 and 3.
For the Compact SLRs, all of the lenses considered have better angular resolution than the SX30 and SX40. The total resolution of lens and sensor, however, is better for only the 100400 mm lens. It shocks me that, although this wonderful lens can resolve an angle as small as 9 microradians, the limitations of the 18 MP APSC sensor produce a total resolution of 23 microradians, which is not a huge improvement over the SX30/40 with its 4.3150.5 mm lens.
Conclusions
1) The angular resolution of the SX30 and SX40 at full optical zoom is determined primarily by the lens. Because of high pixel density, the sensor resolution is good, so that most of the lens resolution is retained. I think that this is the basic explanation of why the SX30 and SX40 are so good.
2) The angular resolution of a compact SLR, however, is determined by the limitations of the 18 MP APSC sensor, which is not able to capture the full resolving power of the lenses at full zoom. To take full advantage of the lenses considered, the APSC sensor would need a higher pixel density. The pixel density (pixels per square millimetre) of the SX30 is about 9 times that of the APSC. The SX40 has about 8 times the pixel density of the APSC. Regarding the resolution a compact SLR with a telephoto lens, the relatively low pixel density of the APSC sensor seems like a deficiency, but perhaps there are compensating advantages in terms of lowlight capability, low noise and bokeh etc.
3) The calculations seem to confirm my impression from viewing pictures on the internet that, if you want to significantly improve on the resolution of the SX30 or SX40 with a presentday compact SLR, you need a 400 mm focal length or more, perhaps by using a teleconverter.
4) The SX30IS and SX40HS are great cameras.

Update of Article August 5, 2012
50% MTF vs 9% MTF
Most MTF charts for SLR lenses stop at 50% MTF (Modulation Transfer Function) as if nothing beyond that matters. The formulas in my article are based on approximately 10% or 9% MTF ( equivalent to the Rayleigh Criterion), where line pairs are distinguishable but blurred and the pattern would be completely washed out if the target were 12% more distant or smaller like the difference between elements on the USAF 1951 test pattern.
Sweet Spot of SLR Lenses and NearlyPerfect Superzoom Lenses
In some of the forumpostings cited below, people have commented that SLR lenses are less nearlyperfect than superzoom lenses because it is so difficult to engineer a lens for large sensors. Because of this, SLR lenses need to be stopped down by one or 2 stops to their "sweet spot" in order to obtain their best resolution. So, my calculations of lens resolution for Compact SLRs in the article are too optimistic. Reference 8, below, contains a table of angular resolutions considering the sweet spot for SLRs. The SX30/40 lens, on the other hand seems to be close to perfect. Firstly, I have verified by testing that the SX30 has its best resolution wide open (Ref 2). Secondly, the calculated angular resolution of the lens is 24.1 microradians, which is already close to the measured total (lens + sensor) of 30.
Total Resolution
Some sources indicate that the exponent 2 in Equation 3 for total resolution is incorrect and that it should be 1, e.g.: http://www.normankoren.com/Tutorials/MTF.html
In the first line of Table 1, if you were to add the Lens and Sensor angular resolutions together (exponent 1 and omit sqrt in 3b), it comes out to a total of 43.1 microradians, which is much more than the measured value of 30 microradians. Exponent 2 yields a calculated total angular resolution of 30.7, which is much closer.
Changes of Formulas for Closeup Photography
Equations 1b,2b, 4a and 4b contain f, which is the nominal lens focal length, which is appropriate for the long telephoto shots with infinite focus considered in this article. The equations become more generally applicable to cases where the focus is not infinite if f is replaced by the effective focal length F (see: How to Properly Use the "Thin Lens Formula" to Model a 35 mm Camera `Thick' Lens, by Jerry Jongerius, Revision 7A, September 2, 2012, http://www.panohelp.com/thinlensformula.html)
[August 27, 2012 Update:
Some examples of resolution calculations with the more general equations are provided in:
Macro vs TeleMacro with SX30/40", August 27, 2012:
http://www.dpreview.com/articles/8819494033/macrovstelemacrowithsx3040
]
Forum Postings Related to the Article
1) Birds and Angular Resolution  Canon Talk Forum (DPReview) Dec 2/11 (before article was written)
http://forums.dpreview.com/forums/readflat.asp?forum=1010&thread=39978400
2) Angular Resolution of Superzooms  preview Canon Talk Forum June 21/12
(Calculations for various brands; SLR lenses less nearly perfect than superzooms; SX40 stopped down vs SX30 wide open; test patterns & SX30 MTF curve)
http://forums.dpreview.com/forums/read.asp?forum=1010&message=41843993
3) Birder's Superzoom  Canon Talk Forum July 20/12
(FZ200 and some speculation about future largeaperture superzooms)
http://forums.dpreview.com/forums/read.asp?forum=1010&message=42071682
4) FZ150 to FZ200: Lenses  Ron Tolmie July 26/12 Panasonic Talk Forum
http://forums.dpreview.com/forums/readflat.asp?forum=1033&message=42124020
5) FZ200 Industrial Espionage?  Canon Talk Forum July 27/12
(More speculation about future largeaperture superzooms)
http://forums.dpreview.com/forums/readflat.asp?forum=1010&message=42129846
6) A small alternative to superzoom bridge cameras  Billx08 July 27/12 Fujifilm Talk Forum
http://forums.dpreview.com/forums/readflat.asp?forum=1012&thread=42124394
7)SX50 Lens on Canon Rumors  Canon Talk Forum July 29/12 1:10 AM
http://forums.dpreview.com/forums/read.asp?forum=1010&message=42139905
8) Three Birding Cameras to Watch  Canon Talk Forum July 30/12
(Includes resolution tables considering "sweet spot" of SLRs)
http://forums.dpreview.com/forums/readflat.asp?forum=1010&message=42150475
The views and opinions expressed in this article are those of the author and do not necessarily reflect the views and opinions held by dpreview.com or any affiliated companies.
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