Introduction  The Depth of Field (Distances) and More (Geometrical Optics)  The Lens  The Sensor  Resolution Measures, Pixel Count, Pixel Size  SensorRelated and Further Resolution Data  And Now to the Lenses...  References  Appendix
On this page, I attempt to shed some light on topics like sharpness, resolution, resolving power, and more... But I do not want to write a scientific paper here, just make the issue more understandable. Therefore, I will not refer to more advanced concepts like MTF here...
All we want as photographers is sharp photos. So we strive for sharp lenses (razor sharp, tack sharp, ...), sensors with huge pixel numbers (the more the better...), cameras with a fast and precise focusing system, cameras with "antishake" mechanisms, and so on... But many people do not really know how these factors work together in making an image appear sharp and why they sometimes are not sharp, or not as sharp as expected. Already the term "sharpness" is, however, defined differently in everyday life and in engineering, and the confusion goes on: resolution (Auflösung), resolving power (Auflösungsvermögen), depth of field (DOF), circle of confusion  alas, the confusion is perfect!
First of all, I want to exclude certain topics from my discussion, because these are "outside" of the optical aspects of image sharpness, such as camera shake and motion blur (camera shake may be counteracted with antishake mechanisms in the optics or the camera body).
In my discussion, I would like to focus on three aspects as determining factors for "sharpness":
In the following, I provide brief introductions into each of these aspects and ask for their role in determining image sharpness (for more information, see the References).
Depth of field is defined by the distance of the object in focus from the lens, the distance of other objects from the object in focus, and a criterion, called the circle of confusion (CoC). The latter acts as a convention for what we perceive as "acceptably sharp" and is based on the resolving power of the human eye.
All the calculations performed for determining depth of field are based on pure geometrical optics and do not take care of the sensor and the lens. In the end, the calculations deliver distance ranges within which all objects appear "acceptably" sharp for given aperture values (fnumbers).
The only other factor that might be considered here is the effect of diffraction. It is actually caused by a number of effects, but eventually leads to a similar phenomenon as the circle of confusion, namely to a "circle of fuzziness", which is called Airy disk (or airy disk) after its discoverer and optically less well defined as the CoC (that is, it looks more fuzzy). In practice, diffraction introduces an upper limit for the fnumbers that can be used without noticeable image degradation. Thus, there is an fnumber that you should not exceed for a certain camera/sensor combination in order to avoid an increase in fuzziness. This may lead to a conflict: The needed DOF range may require a large fnumber, but diffraction calculations tell you that you should not use it.
A concept that is related to depth of field is the hyperfocal distance (HFD): If we focus at this distance, everything is "acceptably" sharp from half the HFD to infinity. Whether this is sharp enough for us, depends on a number of factors, the most important one is probably, how much we blow up a photo when we view or print it. This method is often compared with the method of setting distance simply to infinity. The latter leads to best sharpness at infinity, whereas the HFD leads only to "acceptable" sharpness at infinity, but has advantages in the foreground (look here for more information).
People buy cameras probably not only based on price, features, or size, but also because of the lens and its characteristics.
Often, they do so because of the zoom range of a lens. The motto seems to be: The more the better. This is at least, what comes to mind when you observe the evolution on digital compact and bridge cameras. Starting from a zoom range of 1:3 they reached a range of about 1:10 a couple of years ago, and now they "must" have zoom ranges of 1:30 (the smaller ones) or even 1:50 (if they are a bit bigger) if they want to be competitive. There are some "premium compacts" with larger sensors that still deliver only zoom ranges of about 1:3, but many hobby photographers do not even know why the zoom range is so "poor" for these cameras...
Another argument for a camera would be that is has a wideaperture lens, particularly if it offers a wide aperture across the whole zoom range. But this is more a feature for the "experts"  and makes a lens expensive...
Finally, people want a sharp lens, but, despite the zillions of tests in magazines and on the Web, it is hard to find out how sharp a camera's lens really is. Most socalled tests just make some statements about the lens without ever having made a technical test of it. For example, I recently read tests about the Panasonic Lumix TZ7/ZS50, and the lens got high marks, except for one site that really had made an intensive lab test of the lens, leading to a devastating result.
Users of DSLRs and system cameras who buy exchangeable lens are a different breed of camera users  at least, most of them. They know the lenses, read extensively technical tests, and expect optimal sharpness across the whole image, "special" rendering qualities, wide apertures, manual focusing, and so on. They want simply the best for their purposes. Regrettably, existing lenses are a compromise of many design decisions, and today many of them have their faults corrected in software (which leads to heated debates among the users).
While geometrical optics more or less neglects most aspects of "real" lenses (except for their focal length and aperture), the performance of "real" lenses is the result of a complex interaction of many factors which cannot be calculated. Therefore, the performance of lenses is analyzed in test labs to learn what their limitations are. Resolution is just one aspect that is investigated, but it is the one that is in focus on this page. The resolution values of a lens can be so low that they are the limiting factors for the image quality, not the sensor, not diffraction. These limitations have many labels, such as "corner softness", "decentering", "aberration", "coma" and more. Typically, a lens's performance is weaker at the corners and the borders than at the center. Therefore, the empirically measured resolution of lenses is given in the form of lots of numbers or even better, as graphs. I show examples of this further below.
One of the questions in this context is, whether the lens outperforms the sensor, which was usually the case in the past (at least for quality lenses), or vice versa. In the recent past (as of 2016), fullframe format sensors were introduced with such a high pixel count that older lenses were not "good enough" for these sensors. So it's time to take a look at the role that sensors play in the "resolution/resolving power game".
The sensor is the recording medium in digital cameras, much like file is in analog cameras. Like the lens, it can be the limiting factor for the image quality (or what we call "sharpness"). There are two basic sensor characteristics that play a role in these considerations:
*) According to Wikipedia, the term resolution is often used for a pixel count in digital imaging, even though British, American, Japanese, and international standards specify that it should not be so used, at least in the digital camera field.
Together, this data specifies the pixel size, which is typically square, but need not be so. For rectangular pixels, the pixel size for a dimension is given as the sensor size divided by the number of pixels in that dimension. The calculated size is, however, larger than the actual active size (or area) of a pixel, which captures light, because of borders, and other obstructions. The sensor's active pixel size determines the quality of the signal that a sensor outputs: The smaller sensor pixel, the less light it collects, and the more amplification is needed.
What we are actually after, is, however, what is called the resolving power of a sensor, that is, its ability to resolve two objects of a certain size. This is not only limited by the sensor geometry (pixel size), but also by filters:
To make things short, the effect of these filters is typically "summarized" in a factor, with which the sensor size is multiplied (or by which it is divided), thus, the "effective" pixels are larger than the physical pixels. The factor differs for black&white (or onecolor) sensors and color sensors. For details, see the following section and the calculations below.
Higher ISO values can be regarded as "more amplification," meaning that, due to smaller pixel sizes, smaller sensors performs worse at higher ISO values (or low light conditions) than larger sensors. All in all, this leads to more noise in the images, which also reduces the resolving power of a sensor. Usually, this is only discussed qualitatively on the basis if images that were taken at different ISO values.
The resolution of lenses is typically given in line pairs per mm (lp/mm) or lines per mm (l/mm). As expected, the number of line pairs per millimeters is half the number of lines per millimeter (for example, 100 lp/mm = 200 l/mm).
To be able to compare this measure with the pixel size of sensors, you can ask how wide a line is. This is the reciprocal of the l/mm measure. For example, 200 l/mm correspond to a line width of 5 μm (or 0.005 mm).
Many test sites and labs, such as dpreview.com specify resolution (or resolving power) in lines per height (LPH; horizontal LPH, vertical LPH) or lines per picture height (LPPH). The following adapted text was taken from the dpreview.com glossary for the term "resolution":
dpreview.com measured resolution "using the widely accepted PIMA/ISO 12233 camera resolution test chart" up to about 2013; for a short while, they used DxOMark test charts. Since then, they use their own studio setup and do not give away numbers any more. Regrettably, the dpreview.com glossary entry does not clearly distinguish between the terms resolution (which it uses differently depending on the context) and resolving power. See the dpreview.com glossary page "resolution" for example images and calculations.
If you divide the vertical LPH by the sensor height, you get lines/mm. Dividing this value by 2 leads to line pairs/mm.
The pixel count of a sensor is typically given in Megapixels (e.g. 24 Megapixels) and often also given in pixels per horizontal and vertical dimensions (e.g. 6000 x 4000 pixels), then sloppily called sensor resolution.
Some argue that the pixel size of a sensor (or the effective pixel size) is the most relevant characteristic to calculate a sensors resolution or resolving power from. The pixel size for a sensor can be calculated from the pixel count (in Megapixels) and the dimensions of the sensor (sensor size). See page About Focal Length, Aperture, and Depth of Field for Different Sensor Sizes for more information about how sensor size affects photographic parameters (the crop factor seems to be the most important measure here).
You can use the "advanced" calculator on the "Cambridge in Colour" Website for this purpose. You can also divide the sensor width by the number of pixels in the horizontal dimension, and the sensor height by the number of pixels in the vertical dimension. For quadratic pixels, both numbers should be the same.
To compare the results with resolution measures (lp/mm, l/mm) or with "circle of confusion" (CoC) measures (mm or μm), you have to take into account that usual sensors for color photography have a Bayer filter on top of them (and also an antialiasing filter). I found different "factors" that take this into account:
Applications:
According to dpreview (RX100 M1 lens test), they "want to show ... how well the camera is able to resolve the detail in [their] standard test chart compared to the theoretical maximum resolution of the sensor."
For the charts they used (and do no longer use) it is "simply the number of vertical pixels (the number of single lines per picture height = LPPH)." The theoretical limit is 1 line per pixel and usually referred to as the Nyquist frequency.
Even though the theoretical limit "may be effectively
unattainable with normal equipment in normal shooting situations, an understanding
of a sensor's theoretical limit provides a useful benchmark for best
possible performance."
(See the full citation below)
In the following, I try to put this information together and present a table that collects sensorrelated and resolutionrelated data for my/our own cameras, but it may be useful for similar (with respect to the sensor) cameras as well. For the reasons just given, I will also list the Nyquist frequency, that is, the theoretical resolution limit given by the sensor, in my table.
In the following, I try to collect, which resolution data can be gained from the sensor data for my cameras (the sensor table on my site served as the starting point for this), including the Nyquist frequency, that is, the theoretical resolution limit given by the sensor. I also add CoC data and diffraction data so that this data can be compared more easily with the sensor data.
Please note that this table may change over time...
Format >

FullFrame Format  APSC (DX)  1"  1/2.3" 
Dimensions (mm)  36 x 24 (35.8 x 23.9)  23.7 x 15.6  13.2 x 8.8 / 12.8 x 9.3  6.2 x 4.62 
Area (mm2)  864  370  116 / 119  29 
Diagonal (mm)  43.3  28.4  16  7.7 
Crop Factor  1.0  1.5  2.7  5.6 
Camera Example  Leica M (Typ 240)  Leica X Vario, Ricoh GR  Sony RX 100 M1  Ricoh CX4 
Megapixels  23.9  16  20  10 
Pixels  6000 x 4000 (5976 x 3992 (DNG) 5952 x 3968 (JPG) 
4944 x 3274 (DNG) 4928 x 3264 (JPG) 
5472 × 3648  3648 × 2736 
Pixel x size  6 μm (83.33 lp/mm166.67 l/mm)* 
4.8 μm (104.17 lp/mm208.33 l/mm)* 
2.4 μm*
(208.33 lp/mm416.67 l/mm)* 
1.7 μm (294.12 lp/mm588.24 l/mm)* 
Pixel y size  6 μm (83.33 lp/mm166.67 l/mm)* 
4.8 μm (104.17 lp/mm208.33 l/mm)* 
2.4 μm*
(208.33 lp/mm416.67 l/mm)* 
1.7 μm (294.12 lp/mm588.24 l/mm)* 
Color (f = 2)  Line Pairs (LP) / Lines (L) 

Per Sensor Width*  1491.67 LP / 2983.33 L  1234.37 LP / 2468.75 L  1375 LP / 2750 L  911.76 LP / 1823.53 L 
Per Sensor Height**  995.83 LP / 1991.67 L  812.5 LP / 1625 L  916.67 LP / 1833.33 L  679.42 LP / 1358.82 L 
Per Sensor Width*  1494 LP / 2888 L  1236 LP / 2472 L  1368 LP / 2736 L  912 LP / 1824 L 
Per Sensor Height*  998 LP / 1996 L  818.5 LP / 1637 L  912 LP / 1824 L  684 LP / 1368 L 
Nyquist Frequency**  3992 LPPH (83.33 lp/mm166.67 l/mm)* 
3274 LPPH (104.17 lp/mm208.33 l/mm)* 
3648 LPPH (208.33 lp/mm416.67 l/mm)* 
2736 LPPH (294.12 lp/mm588.24 l/mm)* 
CoC Based on Pixel Size 

Diam. b&w (f = 1.41)  8.49 μm (58.93 lp/mm)***  6.79 μm (73.66 lp/mm)  3.39 μm (147.31 lp/mm)  2.4 μm (207.97 lp/mm) 
Diam. color (f = 2)  12 μm (41.67 lp/mm)  9.6 μm (52.08 lp/mm)  4.8 μm (104 lp/mm)  3.4 μm (147.06 lp/mm) 
Diam. color (f = 2.5**)  15 μm (33.33 lp/mm)  12 μm (41.67 lp/mm)  6 μm (83.33 lp/mm)  4.25 μm (117.65 lp/mm) 
Diam. color (f = 3)  18 μm (27.78 lp/mm)  14.4 μm (34.72 lp/mm)  7.2 μm (69.44 lp/mm)  5.1 μm (98.04 lp/mm) 
For Comparison Purposes:
Standard CoC (Depth of Field, Hyperfocal Distance) 

Diameter  30 μm (16.67 lp/mm)  20 μm (25 lp/mm)  11 μm (45.45 lp/mm)  5.6 μm /6 μm (89.29/83.33 lp/mm) 
fNumber  For
Comparison Purposes: Diffraction Limited Spot Size /
Airy Disk (Independent of Sensor Size) 

f/4  5.33 μm
(93.81 lm/mm) 

f/5.6  7.53 μm (66.40 lp/mm) 

f/8  10.65 μm
(45.95 lp/mm) 

f/11  15.07 μm (33.18 lp/mm) 

f/16  21.31 μm
(23.46 lp/mm) 

f/22  30.14 μm (16.59 lp/mm) 

f/32  42.62 μm
(11.73 lp/mm) 
*) The Cambridge
in Colour diffraction calculator delivers 2.5 μm for 20 MP and
2.4 μm for 21 MP On the same page, another calculator delivers 2.4μm
for the Sony RX100 M1. Therefore, I decided for 2.4μm, which is also
the result of my own calculations.
**) The Cambridge
in Colour diffraction calculator uses a factor of 2.5; the formulae below use
a factor of 2 for sensors with Bayer filter
***) Line pairs were calculated using the formulae below.
*) Calculated as 1000 / pixel size / 2 *
sensor width for lines and as 1000 / pixel size / 2 / 2 * sensor width
for line pairs; has rounding errors
**)
Calculated as 1000 / pixel size / 2 * sensor height for lines and as 1000
/ pixel size / 2 / 2 * sensor height for line pairs; has rounding errors
*) Calculated as pixel count / 2 (per dimension)
for lines; calculated as pixel count / 2 / 2 (per dimension) for line pairs
**)
As vertical pixel count, i.e. corresponds to lines that are 1 pixel wide
= Nyquist frequency (see dpreview.com)
*) Corresponding to Nyquist frequency (lines
that are 1 pixel wide); one cycle are two lines
The resolving limit for a "normal" human observer was found to be about 8 line pairs per millimeter (1 arc minute), if a periodic blackandwhite pattern is seen from a distance of 250 mm. This was relaxed to 4 line pairs per millimeter (2 arc minutes) for an "acceptable" sharpness. Given a magnification factor of 8 between paper viewed from a distance of 250 mm and 35 mm film or a fullframe format sensor, we arrive at 32 line pairs/mm. Taking care of the Bayer filter using a factor of 2, we finally arrive at 16 line pairs/mm and are about there (the table says 16.67 line pairs/mm).
This corresponds to a CoC of 1/32 mm (exactly) or 1/30 mm (most often referred to). Mostly, however, 30 μm are used for the "fullframe" CoC these days. Therefore the discrepancy between 16 and 16.67 line pairs/mm...
CoC comparisons: The CoC based on the pixel size is always smaller for each of my cameras than the standard CoC that is used for DoF calculations and calculations of the hyperfocal distance. Thus, from a sensor resolution point of view, I might make my CoC criterion stricter and use about the CoC for a "camera class" smaller (max. factor 2.5). Or I could base it on a larger paper size, or whatever I want...
Diffraction: Airy Disk versus standard and sensorbased CoC: The diffraction limit for the standard CoC is given elsewhere but can also be read from the table above (it is, for example, f/22 for fullframe format, f/8 for 1" sensor format). If we would base the diffraction limit on the sensor, we would have to open the lens for two fstops (that is, for example, f/11 for fullframe format, f/4 for 1" sensor). According the Cambridge in Colour Website, this would correspond to viewing the photos on a computer screen at 100%. Thus, I should realize the effects of diffraction on the computer screen at 100% much earlier (that is, for smaller fnumbers) than I thought...
In this section, I try to accumulate sharpness data for my cameras or lenses in order relate the data to the data above. Regrettably, there is little "hard data" on the Internet...
Note: For more information on the lens, see the lens page for the Leica X Vario.
Jim Fisher (PC Magazine) used Imatest to check the sharpness of the X Vario's zoom lens. Below is a tabular overview of his results, with lp/mm data added by me*:
Focal Length 

Equivalent 
28  35  50  70  28  35  50  70  Equivalent 

Actual 
18  23  32  46  18  23  32  46  Actual  
Minimum fNumber  3.5  4.5  5.1  6.4  3.5 
4.5 
5.1 
6.4 
Maximum fNumber 

Lines  Center  1,774    1,978  2,043  56.86*    63.40* 
65.48* 
Line pairs/mm  Center 

Lines  Edges  1,441    just below 1,800 
1,900  46.19*    just below 57.69* 
60.90*  Line pairs/mm  Edges 

Stopped down to f/5.6 
Lines  Center  1,869        59.90*        Line pairs/mm  Center 

Lines  Edges  1,500        48.08* 
      Line pairs/mm  Edges 

Stopped down to f/8 
Lines      about the same 
about the same 
    ditto  ditto  
Nyquist Frequency  Lines  3,274
LPPH 
104.94 lp/mm, 209.87
l/mm* 104.17 lp/mm, 208.33 l/mm** 
Line pairs/mm,
Lines/mm 
*) Line pairs/mm were calculated from the Lines (per Height) data based
on a sensor height of 15.6 mm (APSC DX).
**) Line pairs/mm and lines/mm were calculated
from the pixel size.
Resolution lies between 46 and a little more than 65 line pairs/mm for the Leica X Vario. This corresponds to a CoC between 7.6 and about 11 μm (the "standard" CoC for this camera is 19 μm for a focal length of 18 mm, and 20 μm above that).
Note: For more information on the lens, see the lens page for the Sony RX 100 M1.
In its review of the Sony RX100 M1, dpreview.com reports a resolution above 2600 LPPH for this camera (3648 LPPH would correspond to the number of vertical pixels or the socalled Nyquist frequency, that is, the upper physical limit).
LPPH 
Line pairs/mm, Lines/mm 
Line pairs/mm, Lines/mm 

Calculated from Pixel Size (2.4 μm) 
Calculated from Sensor Height (8.8 mm) 

Lines (dpreview)  2600 
148.48 lp/mm, 296.97 l/mm  147.73 lp/mm, 295.45 l/mm 
Nyquist Frequency  3648 
208.33 lp/mm, 416.67 l/mm  207.27 lp/mm, 414.55 l/mm 
It is hard for me to relate this number 2600 LPPH (corresponding to 150 lp/mm or 300 l/mm) to other, more qualitative test results that reveal soft corners and diffraction effects... The lens is not as good as this number suggests. The DxOMark tests assigns only 6 "perceptual" Megapixels to the lens (with 20 MP)...
More detailed test results can be found here:
DxOMark, Sony RX100 M2 (same lens):
The German photography Website digitalkamera.de published a test of the Sony RX100 M1 (in German). Here is an excerpt of the result for the lens:
The lp/mm values have been scaled to 35 mm values, but I do not quite understand what this means.
They also published thorough technical test of the Sony RX100 M1 (in German), which can be downloaded for a fee from this page. Since this is pay content, I can not report on this test here.
A few comments from the dpreview test on their older test method, which used the PIMA/ISO 12233 camera resolution test chart until about 2013 (considered as outdated today; for a short while, they used DxOMark test charts (see the Ricoh GR below), since then, they use their own studio setup):
Note: For more information on the lens, see the lens page for the Ricoh GR.
For comparisons: The Nyquist frequency for the Ricoh GR is 3274 LPPH.
The Ricoh GR resolution test at dpreview.com is based on a "classic" DxOMark test. Here are some results:
The dpreview.com results suggest not to go beyond f/8 for best image quality.
DxOMark test of GR lens (Ricoh GR Lens mounted on Ricoh GR : Measurements): www.dxomark.com/Lenses/Ricoh/RicohGRLensmountedonRicohGRMeasurements__874
The DxOMark results also suggest not to go beyond f/8 for best image quality.
Camera Test: Ricoh GR APSC Compact (Philip J. Ryan, Popular Photography):
Factor (considering b&w and color sensors):
Lines/mm versus Line pairs/mm: Line pairs equal twice the amount of lines, for example, 100 line pairs equal 200 lines.
The number of lines/mm and line pairs/mm can be calculated as follows:
Example: 5.86 or 6 μm pixel (23.9 Megapixels, fullframe format); factor 2 for line pairs, factor 2 for color or 1.414 for b&w
(Inspired by OptoWiki: www.optowiki.info/faq/howtoconvertpixelsizeinlinepairspermillimeter/?noredirect=en, adapted)
The number of line pairs is twice the number of lines. For example, 100 line pairs equal 200 lines.
Width of a line from number of lines for one mm (= 1000 μm): 1000 / (number of lines) [μm].
Example:
Question: Does a lens with 100 line pairs per millimeter support 5 μm large pixels?
Answer: For monochrome applications, we find that 3.5 μm pixels are supported. For color applications, even 2.5 μm pixels are supported (because several pixels act together).
(Inspired by OptoWiki: www.optowiki.info/faq/howtoconvertlinespairspermillimetertopixelsize/?noredirect=en, adapted)
25.02.2016 