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 55-250 mm f/4-5.6 lens are rarely as detailed as those I take with my SX30. Even those taken with a 100-400 mm f/4.5-5.6 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.
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 micro-radians. 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 line-pairs produce an angle of 30 micro-radians, the image is blurred but can still be distinguished. At 27 micro-radians (smaller test pattern), the image of the test pattern is completely extinguished as a washed-out 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 line-pairs per mm as a function of f-stop. This table can be summarized by the following formula:
LensRes = 1600 / f# (1a)
where LensRes is in line-pairs per mm (of the test-pattern image on the surface of the sensor) and f# is the f-stop.
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 micro-radians. (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 micro-radians, D is the aperture in mm, f# is the f-stop 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 test-pattern used in photography, which has alternate black and white bars of equal width.
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 washed-out 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 line-pair 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 line-pairs 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 micro-radians is:
SensorAngRes = 2,000,000 x (sensor width / # horizontal pixels) / f (2b)
Total Resolution of Lens and Sensor
The total resolution of the lens-sensor 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 line-pairs 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)
To convert from line-pairs per mm to micro-radians:
Angle = 1,000,000 / (f x Line-Pairs per mm) (4a)
To convert from micro-radians to line-pairs per mm:
Line-Pairs 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
55-250 mm f/4-5.6: 250 5.6 5184 22.3
70 or 75 - 300 mm f/4-5.6: 300 5.6 5184 22.3
100-400 mm f 4.5-5.6: 400 5.6 5184 22.3
*1 smallest f-stop 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
Line-Pairs per mm
Superzooms Lens Sensor Total
SX30: 276 350 217
SX40: 276 324 210
55-250 mm f/4-5.6: 286 116 108
70 or 75 - 300 mm f/4-5.6: 286 116 108
100-400 mm f 4.5-5.6: 286 116 108
Note: Larger number of line-pairs per mm means better resolution.
Table 3: Lens, Sensor and Total Angular Resolution
Using Equations 1b, 2b and 3b
Angular Resolution (micro-radians)
Superzooms Lens Sensor Total
SX30: 24.1 19.0 30.7
SX40: 24.1 20.5 31.6
55-250 mm f/4-5.6: 14.0 34.4 37.2
70 or 75 - 300 mm f/4-5.6: 11.7 28.7 31.0
100-400 mm f 4.5-5.6: 8.8 21.5 23.2
Note: Smaller number for angular resolution means better resolution.
Table 2 gives the resolutions in line-pairs 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 micro-radians, which is a more useful indication of which lens-sensor 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 micro-radians. (The ".1" is meaningless, as the actual value might be say 25 or 26 micro-radians.) The sensor resolution is 19 micro-radians, 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 micro-radians. Perhaps fortuitously, this agrees fairly well with the value of 30 micro-radians that I measured. Because Equation 1 describes ideal diffraction-limited 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 100-400 mm lens. It shocks me that, although this wonderful lens can resolve an angle as small as 9 micro-radians, the limitations of the 18 MP APSC sensor produce a total resolution of 23 micro-radians, which is not a huge improvement over the SX30/40 with its 4.3-150.5 mm lens.
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 low-light 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 present-day 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 Nearly-Perfect Superzoom Lenses
In some of the forum-postings cited below, people have commented that SLR lenses are less nearly-perfect 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.
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 Close-up 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:
Forum Postings Related to the Article
1) Birds and Angular Resolution - Canon Talk Forum (DPReview) Dec 2/11 (before article was written)
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)
3) Birder's Superzoom - Canon Talk Forum July 20/12
(FZ200 and some speculation about future large-aperture superzooms)
4) FZ150 to FZ200: Lenses - Ron Tolmie July 26/12 Panasonic Talk Forum
5) FZ200 Industrial Espionage? - Canon Talk Forum July 27/12
(More speculation about future large-aperture superzooms)
6) A small alternative to superzoom bridge cameras - Billx08 July 27/12 Fujifilm Talk Forum
7)SX50 Lens on Canon Rumors - Canon Talk Forum July 29/12 1:10 AM
8) Three Birding Cameras to Watch - Canon Talk Forum July 30/12
(Includes resolution tables considering "sweet spot" of SLRs)
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