General: Resolution, Sharpness, and More

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...

Introduction

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 "anti-shake" 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 counter-acted with anti-shake mechanisms in the optics or the camera body).

In my discussion, I would like to focus on three aspects as determining factors for "sharpness":

Depth of field (geometrical optics)
The lens
The sensor

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).

The Depth of Field (Distances) and More (Geometrical Optics)

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 (f-numbers).

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 f-numbers that can be used without noticeable image degradation. Thus, there is an f-number 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 f-number, 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).

The Lens

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 wide-aperture 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 so-called 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), full-frame 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

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:

The physical size (width, height, area, diagonal)
The number of pixels (or the pixel count) given in Megapixels for the complete count.
The (pixel_number_x) x (pixel_number_y) is typically called the resolution* of a sensor (for example 6,000 x 4,000 for a 24 MP sensor), although this use of the term can be misleading.

*) 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:

Since sensors capture only one color (ar better, all...), a so-called Bayer filter is placed on the sensor with a regular distribution of red, green, and blue filters. From this, a color image is calculated, resulting in a loss of sharpness (or resolving power). This is typically accommodated by a factor of two (and 1.41 for monochrome sensors). Only few sensors have a different arrangement of filters (Fuji) or none at all (Foveon), and I will not comment on these here.
Since the regular Bayer filter can cause moiré, typically an anti-aliasing filter (AA filter) is also placed on the sensor, resulting in a further loss of resolving power. In the more recent past, many cameras, such as the Ricoh GR and the Leica M (Type 240) do without an AA filter.

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 one-color) 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.

Resolution Measures, Pixel Count, Pixel Size

Resolution Measures

Line Pairs/mm, Lines/mm

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).

Line Per Height (Horizontal, Vertical)

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":

Horizontal LPH refers to the number of (vertical) lines measured along the horizontal (x) axis or width of the image.
Likewise, the Vertical LPH refers to the number of (horizontal) lines measured along the vertical (y) axis or height of the image.
Because the resolution is "normalized" to the picture height, the results of cameras with different aspect ratios can be compared easily.
Since we normalize on picture height, the absolute number of (horizontal) lines the camera is able to resolve along the vertical axis (image height) (LPPH, lines per picture height) is equal to the vertical LPH. The absolute number of (vertical) lines the camera is able to resolve along the horizontal axis (image width) is equal to the horizontal LPH multiplied by the aspect ratio.
Thus: LPPH = vertical LPH (absolute number of horizontal lines the camera is able to resolve along the vertical axis)

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.

Sensor Measures: Pixel Count, Sensor Resolution, Pixel Size

Pixel Count, Sensor Resolution

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.

Pixel Size

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 anti-aliasing filter). I found different "factors" that take this into account:

Color: 2 (OptoWiki), 2.5 (advanced calculator on the "Cambridge in Colour" Website), 3 (source unavailable)
Black & white: square root of 2 (1.414) (OptoWiki)

Applications:

Comparison with the airy disk to check whether diffraction limits resolution at given f-numbers
Comparison with resolution data for lenses to check whether the lens outresolves the sensor or vice versa

Theoretical Maximum Resolution of the Sensor (Nyquist Frequency)

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)

Putting Things together...

In the following, I try to put this information together and present a table that collects sensor-related and resolution-related 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.

Sensor-Related and Further Resolution Data

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 >	Full-Frame Format	APS-C (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/mm\|166.67 l/mm)*	4.8 μm (104.17 lp/mm\|208.33 l/mm)*	2.4 μm* (208.33 lp/mm\|416.67 l/mm)*	1.7 μm (294.12 lp/mm\|588.24 l/mm)*
Pixel y size	6 μm (83.33 lp/mm\|166.67 l/mm)*	4.8 μm (104.17 lp/mm\|208.33 l/mm)*	2.4 μm* (208.33 lp/mm\|416.67 l/mm)*	1.7 μm (294.12 lp/mm\|588.24 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/mm\|166.67 l/mm)*	3274 LPPH (104.17 lp/mm\|208.33 l/mm)*	3648 LPPH (208.33 lp/mm\|416.67 l/mm)*	2736 LPPH (294.12 lp/mm\|588.24 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)

f-Number	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

A Note on the Circle of Confusion (CoC)

The resolving limit for a "normal" human observer was found to be about 8 line pairs per millimeter (1 arc minute), if a periodic black-and-white 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 full-frame 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 "full-frame" CoC these days. Therefore the discrepancy between 16 and 16.67 line pairs/mm...

A few Conclusions from the Table

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 sensor-based 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 full-frame 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 f-stops (that is, for example, f/11 for full-frame 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 f-numbers) than I thought...

And Now to the Lenses...

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...

Sharpness Data for Leica X Vario

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 f-Number		3.5	4.5	5.1	6.4	3.5	4.5	5.1	6.4	Maximum f-Number
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
Stopped down to f/5.6	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 (APS-C 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).

Sony RX100 M1

Note: For more information on the lens, see the lens page for the Sony RX 100 M1.

dpreview

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 so-called 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:

dpreview, Sony RX100 M2 review (same lens, allows comparison with RX100 M1):
www.dpreview.com/reviews/sony-cybershot-dsc-rx100-m2/8

DxOMark

DxOMark, Sony RX100 M2 (same lens):

Sony Cyber-shot DSC-RX100 II lens review: does it live up to the Zeiss name? (DxOMark): www.dxomark.com/Reviews/Sony-Cyber-shot-DSC-RX100-II-lens-review-does-it-live-up-to-the-Zeiss-name/Sony-Cyber-shot-DSC-RX100-II-lens-performance
Sony Cyber-shot DSC-RX100 II Lens mounted on Sony Cyber-shot DSC-RX100 II : Measurements (DxOMark): www.dxomark.com/Lenses/Sony/Sony-Cyber-shot-DSC-RX100-II-Lens-mounted-on-Sony-Cyber-shot-DSC-RX100-II---Measurements__896

Amy Davies, TechRadar

Sony RX100 Review (Amy Davies, TechRadar): www.techradar.com/reviews/cameras-and-camcorders/cameras/compact-cameras/sony-rx100-1083683/review/3
Lens: The Sony RX100 is capable of resolving up to around 24 (line widths per picture height x100) in its highest quality JPEG files.
>> That is, 2400 lines/picture height

digitalkamera.de (Free and Pay Content)

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:

Die Auflösung erreicht bei allen Blendenstufen und Brennweiten im Bildzentrum sehr gute über 40 Linienpaare pro Millimeter (lp/mm), das Maximum liegt bei 48 lp/mm. Der Auflösungsverlust zum Bildrand hält sich in Grenzen und liegt zwischen 20 und 30 Prozent. Die Auflösung liegt hier stets bei oder über 30 lp/mm, im Maximum bei 38 lp/mm – und das wohlgemerkt bei Offenblende im Weitwinkel. Extrem unscharfe Ecken sind bei der RX100 also nicht zu verzeichnen.
Translated: The resolution achieved at all f-stops and focal lengths in the image center very good about 40 line pairs per millimeter (lp/mm), the maximum is 48 lp/mm. The loss of resolution for the corners is very limited and is between 20 and 30 percent. The resolution here is always at or above 30 lp/mm, the maximum at 38 lp/ mm - and this at maximum aperture at wide angle. Extremely blurred edges were therefore not found for the RX100

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.

Comments from dpreview

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):

What we want to show here is how well the camera is able to resolve the detail in our standard test chart compared to the theoretical maximum resolution of the sensor, which for the charts we shoot is easy to work out - it's simply the number of vertical pixels (the chart shows the number of single lines per picture height, the theoretical limit is 1 line per pixel). Beyond this limit (which when talking about line pairs is usually referred to as the Nyquist frequency) the sensor cannot faithfully record image detail and aliasing occurs.
This limit is rarely attained, because the majority of sensors are fitted with anti-aliasing filters. Anti-aliasing filters are designed to reduce unpleasant moiré effects, but in doing so, they also reduce resolution (the relative strength and quality of these filters varies from camera to camera). In theory though, a sensor without an AA filter, when coupled with a 'perfect' lens, will deliver resolution equal to its Nyquist limit. Therefore, even though it 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.

Ricoh GR

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.

dpreview

The Ricoh GR resolution test at dpreview.com is based on a "classic" DxOMark test. Here are some results:

f/2.8: Center: 2500 lines; 1/3: above 2000 lines; 2/3: slightly below 1500 lines; corner: even more below 1500 lines
f/4 (best results): center: 2700 lines; edges: slightly above 1500 lines
f/8: center: slightly above 2000 lines; corner: 1500 lines
f/16: center: 1400 lines, corner: 1100 lines

The dpreview.com results suggest not to go beyond f/8 for best image quality.

DxOMark

DxOMark test of GR lens (Ricoh GR Lens mounted on Ricoh GR : Measurements): www.dxomark.com/Lenses/Ricoh/Ricoh-GR-Lens-mounted-on-Ricoh-GR---Measurements__874

The DxOMark results also suggest not to go beyond f/8 for best image quality.

Popular Photography

Camera Test: Ricoh GR APS-C Compact (Philip J. Ryan, Popular Photography):

In our resolution tests, the GR showed a respectable result with 2425 lines per picture height at ISO 100, for an Extremely High rating. Much of this was retained at ISO 1600, where the camera captured 2375 lines. ISO 6400 brought a drop to 2290 lines - still within our Extremely High ratings band. At the camera's top sensitivity of ISO 25,600, it turned in 2150 lines, which makes for a Very High rating on our scale.

References

General

Photographic Optics (Bob Atkins)

Circle of Confusion

Circle of confusion (Wikipedia)
Zeiss formula (Wikipedia)

Hyperfocal Distance

Hyperfocal distance (Wikipedia)
Hyperfocal distance (Wikipedia, within depth of field)
Hyperfocal Distance Table Calculator (Panohelp.com, Jerry Jongerius)
Note that this calculator uses "standard" aperture values; therefore there are some difference between this calculator and my calculations (except for aperture values that are powers of 2)

Depth of Field (Near Limit, Far Limit)

Depth of field (Wikipedia)
Digital Depth of Field (Bob Atkins)
Depth of Field – An Insider's Look (PDF) (Camera Lens News #1. Carl Zeiss AG. 1997)
Schärfentiefe und Bookeh (H. H. Nasse, Zeiss AG, 2010, in German)

Resolution, Sharpness

Auflösung (Wikipedia): de.wikipedia.org/wiki/Auflösung_(Fotografie)
Optical resolution (Wikipedia): en.wikipedia.org/wiki/Optical_resolution
Henner: www.henner.info/vergl.htm
OptoWiki: www.optowiki.info/faq/how-to-convert-pixel-size-in-line-pairs-per-millimeter/?noredirect=en
OptoWiki: www.optowiki.info/faq/how-to-convert-lines-pairs-per-millimeter-to-pixel-size/?noredirect=en
VisionDoctor: www.vision-doctor.de/optik-berechnungen/berechnen-bildkreisdurchmesser-linienpaare-pro-mm.html
Mineralienatlas: Das Auflösungsvermögen digitaler Kameras: www.mineralienatlas.de/lexikon/index.php/Auflösungsvermögen%20digitaler%20Kameras?lang=de&language=german
Resolution (dpreview.com glossary): www.dpreview.com/glossary/digital-imaging/resolution
Understanding image sharpness part 1: Introduction to resolution and MTF curves (Norman Koren): www.normankoren.com/Tutorials/MTF.html
Understanding image sharpness part 1A: Resolution and MTF curves in film and lenses (by Norman Koren): www.normankoren.com/Tutorials/MTF1A.html
Sharpness - What is it and how is it measured? (Imatest): www.imatest.com/docs/sharpness/

Appendix

How to convert pixel size in line pairs per millimeter

Factor (considering b&w and color sensors):

Monochrome (black&white): The pixel size is multiplied by a factor 1.414 (= square root of two = the diagonal of the pixel).
Color: One can generally use a factor 2 (takes care of the feect of the Bayer filter).

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:

Number of lines/mm: 1000 / (pixel size in μm * factor)
Number of line pairs/mm: 1000 / (pixel size in μm * factor * 2) = 500 / (pixel size in μm * factor)

Example: 5.86 or 6 μm pixel (23.9 Megapixels, full-frame format); factor 2 for line pairs, factor 2 for color or 1.414 for b&w

Color: 1000 / (5.86 * 2 * 2) = 250 / 5.86 = 42.67 lp/mm; 1000 / (6 * 2 * 2) = 250 / 6 = 41.67 lp/mm
Black&white: 1000 / (5.86 * 1.414 * 2) = 60.3 lp/mm; 1000 / (6 * 1.414 * 2) = 500 / (6 * 1.414) = 58.93 lp/mm

(Inspired by OptoWiki: www.optowiki.info/faq/how-to-convert-pixel-size-in-line-pairs-per-millimeter/?noredirect=en, adapted)

How to convert lines pairs per millimeter to pixel size

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:

100 line pairs/mm = 200 lines/mm; one line has the thickness of 1000 μm / 200 lines = 5 μm/line.

Question: Does a lens with 100 line pairs per millimeter support 5 μm large pixels?

Monochrome: We can use the pixel diagonal and thus, divide the pixel size by square root of two (1.414):
5 μm / 1.414 = 3.5 μm
Color: A typical value is half the pixel size (that is, divide the pixel size by a factor of 2):
5 μm / 2 = 2.5 μm.

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/how-to-convert-lines-pairs-per-millimeter-to-pixel-size/?noredirect=en, adapted)