1.1.

Traditional Exposure Strategy

Traditional exposure strategy is designed to be independent of the sensor format used by the camera. For a typical photographic scene metered using average photometry, the same combination of $f$-number $N$ and ISO setting $S$ used on any format will provide the following:

Average photometry assumes the use of a hand-held light meter or a simple in-camera metering mode. As discussed in Ref. 5, a typical scene is indirectly assumed to have an average luminance that is $~18\%$ of the maximum. Provided the ISO setting is defined using the standard output sensitivity (SOS) method,^6,7 the average luminance for a typical scene metered using average photometry will be mapped to a standard lightness in the output JPEG file, specifically 50% lightness or middle gray. This is defined as a digital output level (DOL) of 118 for an 8-bit output JPEG image encoded using the sRGB color space.

The variables $N$, $S$, and $t$ are often collectively referred to as the “camera exposure.” Although the camera exposure is format independent, there are various aspects of the resulting image appearance that are not. When used on different formats with the lens focused at the same object-plane distance, the same camera exposure along with the same lens focal length $f$ will lead to an image with the following properties:

However, the following aspects of the image appearance will be different:

This is because the AFoV is dependent upon both $f$ and the dimensions of the imaging sensor, and the DoF is dependent upon the circle of confusion (CoC) diameter, which is itself format dependent.⁸

Two further image characteristics will also be fundamentally different, both of which relate to IQ:

This is because, first, the sensor area of the larger format will collect a greater amount of light compared to the smaller format for the same average photometric exposure $⟨H⟩$. Since photon shot noise obeys Poisson statistics and scales as the square root of the total amount of light collected, it follows that the larger format will produce a less noisy image since the overall signal-to-noise ratio (SNR) will be higher. Second, the image captured by the smaller format will suffer greater diffraction softening than the image captured by the larger format because the lens entrance pupil diameter will be smaller.

In summary, the use of the same camera exposure on different formats will not yield images that have the same appearance characteristics. One important consequence is that it is inappropriate to perform cross-format IQ comparisons using the same camera exposure on different formats.

1.2.

Equivalent Photos

Discussions of equivalence theory have appeared in books^3,5 and various online articles.^1,2,4 Equivalent photos are defined as photos taken using cameras based on different sensor formats that have the following characteristics:^3,4

It is crucial to point out that these are all image attributes that depend only upon sensor format and are independent from the underlying camera and lens technology.^3,4 In other words, equivalent photos are not expected to be identical. A detailed discussion of the six attributes of equivalent photos will be given in Sec. 2.

In order to define how to take equivalent photos, of fundamental importance is the equivalence ratio $R$, which is the ratio between the lengths of the sensor diagonals:⁴

Here, $d_1$ is the diagonal length of the larger sensor, and $d_2$ is the diagonal length of the smaller sensor, as illustrated in Fig. 1. For the special case that the larger format is 35-mm full-frame, the equivalence ratio is simply the traditional focal-length multiplier, informally referred to as “crop factor.” Note that precise equivalence cannot be achieved if the sensor formats have different aspect ratios.

Fig. 1

The equivalence ratio $R$ is defined as $R=d_1/d_2$. (a) Format 1 and (b) format 2.

Equivalent photos can be produced by different formats provided an “equivalent” combination of focal length $f$, $f$-number $N$, and ISO setting $S$ is used on each respective format. Provided the object plane upon which focus is set is positioned beyond macro object distances:

Again using subscripts to denote formats 1 and 2, where 1 is the larger format, the relevant equations are as follows:

It will be shown in Sec. 2.2 that $R$ must formally be replaced by the “working” equivalence ratio $R_{\mathrm{w}}$, when focus is set on an object plane positioned closer than infinity.⁵ The replacement is required only for equivalent focal lengths and equivalent $f$-numbers and is not required for equivalent ISO settings. However, $R$ can be used in general photographic situations because the practical significance of the replacement turns out to be negligible beyond macro object-plane distances.

As an example, let camera system 1 be based on the 35-mm full-frame format. In this case, Fig. 2 lists the corresponding value of $R$ for a selection of smaller formats. Using the above equations, two examples of combinations of $f$, $N$, $t$, and $S$ that result in an equivalent photo are listed in Tables 1 and 2. The focal lengths and ISO settings have been rounded to the nearest integer, and the $f$-numbers have been rounded to one decimal place. The shutter speed $t$ has been chosen arbitrarily in these examples since the required shutter speed depends upon the nature of the scene luminance distribution.

Fig. 2

The relative sizes of common sensor formats. Arbitrarily taking format 1 to be 35-mm full frame, the corresponding value of the equivalence ratio $R$ is shown for each smaller format 2. The 1/1.7 and 1/2.5 in. sensor formats are commonly used in compact and mobile phone cameras.

Table 1

Example 1: combinations of f, N, t, and S, which produce an equivalent photo.

Table 2

Example 2: combinations of f, N, t, and S, which produce an equivalent photo.

A real-world online demonstration that equivalent photos have the same DoF can be found in Ref. 1.

1.3.

Cross-Format IQ Comparisons

In Sec. 1.1, it was pointed out that the same camera exposure used on different formats leads to images with different levels of noise and diffraction softening, the advantage belonging to the larger format. By contrast, equivalent photos have the following properties:

As proven in Sec. 2, these properties arise from the fact that equivalent photos are produced by using the same lens entrance pupil diameter on each format instead of the same camera exposure.

The entrance pupil defines the flux entering the camera system and is typically the virtual image of the aperture stop seen through the front of the lens.⁹ Within Gaussian optics, the entrance pupil diameter $D$ is given by $D=f/N$, where $f$ is the front (anterior) effective focal length.⁵ Use of the same entrance pupil diameter automatically corresponds with the same level of diffraction softening, as discussed further in Sec. 4.6. Since equivalent photos are produced by using the same exposure duration (shutter speed) on each format, the total light received by each format will also be the same. It follows that equivalent photos will have total image noise of the same order of magnitude because generally the largest contribution to total image noise is photon shot noise, and this is proportional to the amount of light used to form the image.

Nevertheless, real world IQ differences (including total image noise) will inevitably occur in practice even when equivalent photos are taken. These will arise due to differences in the underlying camera and lens technology, such as:

In other words, since the total light received by each format is the same when equivalent photos are taken, it is factors such as those above that explain real-world cross-format IQ differences rather than format size. These factors will be discussed further in Sec. 4.

Although real-world IQ differences could favor any of the cameras being compared when equivalent photos are taken, the advantage of a larger format is that it offers extra photographic capability over a smaller format. This extra capability corresponds to situations in which the required equivalent camera exposure does not exist on the smaller format due to limitations in available $f$-number or ISO setting. Consequently, the smaller format is unable to provide the entrance pupil diameter and shutter speed required to match the larger format and produce an equivalent photo. For a given scene luminance distribution, the range of common entrance pupil diameters and shutter speeds available to two different formats can be termed the “equivalence overlap” between them.⁵ The equivalence overlap in turn defines the range of available equivalent camera exposures. The larger the disparity in size between two formats, the smaller the equivalence overlap.

When the extra photographic capability offered by the larger format is utilized, the resulting photo will be produced using a greater amount of light than that achievable using the smaller format, and this in turn leads to appearance characteristics, such as shallow DoF or long-exposure motion blur, which cannot be achieved using the smaller format. Significantly, this potentially offers an IQ advantage in terms of SNR and resolving power (RP).

Whether or not a photographer is able to make use of the extra photographic capability offered by a larger format is an important factor to consider when choosing an appropriate camera system. The situations where the extra photographic capability can be utilized generally fall into two categories:

As an example of the first type, consider a scenario in which the larger format is being used to take an action photo using a low $f$-number to isolate the subject from the background and provide a short exposure duration (fast shutter speed) in order to freeze the appearance of the moving subject. If an equivalent photo is attempted using the smaller format but an equivalent $f$-number does not exist, the smaller format will produce a photo with a deeper DoF and will be forced to underexpose in order to match the shutter speed used on the larger format. In this case, the extra exposure utilized by the larger format will, in principle, lead to a higher SNR. Furthermore, the image produced by the larger format will suffer less diffraction softening because the lens entrance pupil diameter will be larger.

As an example of the second type, consider the scenario where the larger format is being used to take a landscape photo with the camera set at the base ISO setting. If an equivalent photo is attempted using the smaller format but an equivalent ISO setting does not exist, the smaller format will not be able to produce a photo with a sufficiently long exposure duration (slow shutter speed) without overexposing the image. In other words, the smaller format will be unable to provide sufficient long-exposure motion blur. In this case, the extra exposure utilized by the larger format will again in principle lead to a higher SNR.

In summary, cross-format IQ comparisons should be carried out using equivalent camera settings over the equivalence overlap between the formats being compared. In this case, IQ for both formats is expected to be of the same order of magnitude. However, the extra photographic capability offered by the larger format should also be demonstrated beyond the equivalence overlap. In this regime, the larger format potentially offers higher IQ in terms of RP and SNR.

2.1.

Same Perspective

For cameras based on different formats focused on a specified object plane, the same perspective requires the same object-plane distance measured from the lens entrance pupil of each camera.⁹

For generality, consider a compound photographic lens with any valid pupil magnification $m_{\mathrm{p}}$. The pupil magnification is defined as follows:

Fig. 3

For a general pupil magnification, (a) the vector distance $s_1-s_{\mathrm{ep},1}$ for the larger format and (b) vector distance $s_2-s_{\mathrm{ep},2}$ for the smaller format must be equal in order to achieve the same perspective and framing. In the diagram, OP is the object plane, H and H’ are the first and second principal planes of the compound lens, EP is the entrance pupil, XP is the exit pupil, SP is the sensor plane, and $f^{\prime }$ is the rear (posterior) effective focal length. The principal planes and pupils are not required to be in the order shown. Figure reproduced from Ref. 5 courtesy of IOP Publishing Ltd.

Let $s_1$ denote the distance from the first principal plane H to the object plane for format 1, and let $s_{\mathrm{ep},1}$ denote the distance from H to the entrance pupil of format 1. Analogously, let $s_2$ denote the distance from H to the object plane for format 2, and let $s_{\mathrm{ep},2}$ denote the distance from H to the entrance pupil of format 2. Since the total distance from the entrance pupil to the object plane must be the same for both cameras in order that the perspective (and framing) be the same, the following condition must hold:⁵

This result will be utilized later in the proof.

2.2.

Same Framing: Equivalent Focal Lengths

Recall the standard expression for the AFoV:

For completeness, a derivation of this formula is given in Sec. 7 as an appendix. Note that the apex of the AFoV is situated at the lens entrance pupil.⁹ Here, $f$ is the front (anterior) effective focal length,⁵ and $d$ is the sensor length yielding the corresponding AFoV $\alpha$ measured in either the horizontal, vertical, or diagonal direction. The quantity $b$ is the so-called “bellows factor,” which depends upon both $|m|$ and $m_{\mathrm{p}}$:

A useful practical expression for $|m|$ can be derived (see Sec. 7) as follows:

At infinity focus, $s\to \infty$ and so $|m|\to 0$ and $b\to 1$. At closer focus distances (i.e., as the object plane upon which focus is set is brought forward from infinity) and assuming a traditional-focusing lens, the value of $b$ gradually increases from unity. For a fixed framing, the AFoV therefore becomes smaller. Consequently, the object appears to be larger than expected, particularly at close-focusing distances.

The proof of equivalence given in this section is based upon a traditional-focusing lens. Such lenses achieve focus by movement of the whole lens barrel and not by altering their focal length. On the other hand, lenses that utilize front-cell or internal focusing may alter their focal length depending on the object-plane distance, and this can result in very different AFoV behavior as a function of object-plane distance to that described above. Nevertheless, the proof of equivalence remains valid for such lenses provided the “new” value for the focal length $f$ is used in the bellows factor and AFoV formulae after focus has been set. In all cases, $s$ is the object-plane distance measured from the first principal plane after focus has been set. A more detailed discussion of focus breathing is given in Sec. 8 as an appendix.

In order to derive the formula relating the equivalent focal lengths required on different formats in order to achieve the same framing or AFoV, consider format 1 with a sensor diagonal $d$ and lens with front effective focal length $f_1$ focused at an object-plane distance $s_1$ measured from the first principal plane. Assuming a traditional-focusing lens, according to Eq. (6), the AFoV and bellows factor are as follows:

Now, consider format 2 with a smaller sensor diagonal $d/R$, where $R$ is the equivalence ratio introduced in Sec. 1.2. Assume the lens has front effective focal length $f_2$ and is focused on the same object plane positioned a distance $s_2$ from the first principal plane. Again, assuming a traditional-focusing lens, the AFoV and bellows factor in this case are as follows:

The requirement that the two systems have the same AFoV demands that ${\alpha }_1={\alpha }_2$ and therefore

Rearranging yields the following condition for equivalent focal lengths corresponding to the same AFoV on each format:

The “working” equivalence ratio⁵ denoted by $R_{\mathrm{w}}$ has been defined by

It is important to realize that $b_1\ne b_2$ when focus is set on an object plane positioned closer than infinity. This is because the total refractive power $\mathrm{\varphi }$ of a photographic lens is defined as the reciprocal of the effective focal length, $\mathrm{\varphi }=1/f_{\mathrm{E}}$, where $f_{\mathrm{E}}=f/n$ and $n\approx 1$ when the object-space medium is air. When equivalent photos are taken using cameras based on different formats, the equivalent front effective focal lengths $f_1$ and $f_2$ are not identical and therefore do not correspond to the same refractive power. Consequently, the systems have different bellows factors at the same perspective or object distance $s-s_{\mathrm{ep}}$. This is evident from Eq. (8) when considering the examples of equivalent focal lengths given in Sec. 1.2 earlier.

At infinity focus, $b_1\to 1$ and $b_2\to 1$, and so, $R_{\mathrm{w}}\to R$. Practical expressions for $R_{\mathrm{w}}$ are derived below.

2.3.

Same Display Dimensions

When viewing a photo, the level of detail resolved by an observer affects the perception of properties such as DoF. If a photo is viewed at a specified distance by an observer whose visual system has a known RP, the details resolved will depend upon the size of the photo, i.e., the enlargement factor from the optical image captured by the imaging sensor. Therefore, a fundamental requirement of equivalence theory is that equivalent photos be viewed at the same distance and same display dimensions.

Recall that in order to quantify observer RP in photography, the ability of the eye to resolve detail is defined using a pattern of line pairs.¹⁰ Each line pair consists of a vertical black stripe and vertical white stripe of equal width. As the width of the lines decreases, the stripes eventually become indistinguishable from a gray block. The least resolvable separation (LRS) measured in mm per line pair is the minimum distance between the centres of neighboring white stripes or neighboring black stripes when the pattern can still just be resolved by the eye. Observer RP is the reciprocal of the LRS and is measured in line pairs per mm ($\mathrm{lp}/\mathrm{mm}$). Observer RP depends upon the viewing distance. At the least distance of distinct vision $D_v$, which is defined as 10 in. or 25 cm, a value of around $5\mathrm{lp}/\mathrm{mm}$ is often assumed.¹¹ However, RP varies considerably depending on the ambient conditions and the visual acuity of the individual.

The limits of near peripheral vision are defined by the 60-deg cone of vision, as illustrated in Fig. 4. If it is assumed that the width of the viewed photo just fits within the limits of near peripheral vision when viewed from $D_v$, then the enlargement factor $x$ from a 35-mm full-frame sensor will be 8. In this case, the observer RP projected down to the sensor dimensions becomes $40\mathrm{lp}/\mathrm{mm}$. Mathematically,

Fig. 4

A viewing circle of $\sim 60\mathrm{deg}$ diameter defines the limits of near peripheral vision. This is shown in relation to a $3\times 2$ image (green) printed on A4 ($210\times 297\mathrm{mm}$) paper (black border) viewed at the least distance of distinct vision, $D_{\mathrm{v}}$. The cone of vision roughly forms a circle of diameter $D_{\mathrm{v}}\tan 30°=288\mathrm{mm}$. Figure reproduced from Ref. 5 courtesy of IOP Publishing Ltd.

An equivalent viewpoint is that the value RP (on sensor) affords a certain amount of defocus blur that is undetectable to the observer to be present in the print. The allowed defocus blur can be treated rigorously by calculating the defocus point-spread function (PSF) using wave optics. However, for simple photographic calculations, the defocus blur is instead treated in a purely geometrical manner by assuming that the blur will be uniform over a circle that approximates the shape of the lens aperture. (A Gaussian function would be a more realistic model of the true geometric PSF.⁹) The size of the circle restricts the blur radius, and convolving this blur circle with the optical image formed on the sensor yields a blurred optical image. The circle on the sensor that affords the largest amount of undetectable blur in the print when it is viewed from a specified distance and at specified display dimensions, is known as the acceptable CoC, or simply the CoC.⁸

By treating the CoC as a geometrical PSF and then calculating the cut-off frequency, the relationship between the value RP(sensor dimensions) defined by Eq. (21) and the corresponding CoC diameter $c$ can be shown as follows:⁵

This is illustrated graphically in Fig. 5. If RP (on sensor) = $40\mathrm{lp}/\mathrm{mm}$ for a 35-mm full-frame camera, then the corresponding CoC diameter will be $c=0.030\mathrm{mm}$. Smaller or larger sensors require a smaller or larger diameter $c$, respectively, because the enlargement factor in Eq. (21) will change accordingly. To see this, consider format 1 with a sensor diagonal $d$ and format 2 with a sensor diagonal $d/R$, where $R$ is the equivalence ratio. Provided equivalent photos obtained from these formats are viewed from the same distance and at the same display dimensions, the enlargement factor appearing in Eq. (21) will be a factor $R$ larger for format 2. Now, combining Eqs. (21) and (22) for both formats yields the following important result:

Fig. 5

The required CoC diameter on the sensor is slightly wider than the LRS between neighboring like stripes. A convolution of the CoC with the line pattern renders the stripes unresolvable to an observer of the print under the specified viewing conditions. In other words, the CoC describes the amount of defocus blur that can be tolerated before details in the print begin to appear “out of focus” to the observer. Figure reproduced from Ref. 5 courtesy of IOP Publishing Ltd.

Table 3

Example CoC diameters for various sensor formats.

It should be noted that a photographer can define a custom CoC based upon a known viewing distance $L$ and enlargement $x$ rather than those assumed by the manufacturer. The custom CoC diameter is obtained as follows:

2.4.

Same Depth-of-Field: Equivalent f-Numbers

Recall the standard DoF equations:

For completeness, a derivation of these formulae is given in Sec. 9 as an appendix. In the above, $c$ is the CoC diameter described in the previous section, $m$ is the magnification, $D=f/N$ is the entrance pupil diameter, and $s_{\mathrm{ep}}=(1-m_{\mathrm{p}}^{-1})f$ is the separation between the first principal plane and the entrance pupil defined by Eq. (64) of Sec. 9 and illustrated in Fig. 3. This means that $s-s_{\mathrm{ep}}$ is the object-plane distance measured from the entrance pupil rather than the first principal plane. Equations (26) and (27) can be applied for object-plane distances $s-s_{\mathrm{ep}}<\mathcal{H}$, where $\mathcal{H}$ is the hyperfocal distance defined by Eq. (76) of Sec. 9. At $\mathcal{H}$ and beyond, the far DoF and total DoF are both infinite.

Now, consider format 1 with sensor diagonal $d$, front effective focal length $f_1$, entrance pupil diameter $D_1$, CoC diameter $c_1$, and consider an object plane positioned a distance $s_1-s_{\mathrm{ep},1}$ away from the entrance pupil. The total DoF is given by

Now, consider format 2 with a smaller sensor diagonal $d/R$, front effective focal length $f_2$, entrance pupil diameter $D_2$, CoC diameter $c_2$, and consider the same object plane positioned a distance $s_2-s_{\mathrm{ep},2}$ from the entrance pupil. The total DoF is given by

However, according to Eq. (5) of Sec. 2.1, Eq. (12) of Sec. 2.2, Eq. (16) of Sec. 2.2.1, and Eq. (23) of Sec. 2.3, respectively, the following relationships hold when equivalent photos are taken:

Combining the second and third equations above also shows that $|m_2|=|m_1|/R$. Substituting these relations into Eq. (29) yields the following result:

By comparison with Eq. (28), it can be concluded that ${\mathrm{DoF}}_2={\mathrm{DoF}}_1$ provided the following condition is satisfied:

In other words, a necessary condition for producing equivalent photos is the use of the same entrance pupil diameter on each format. As discussed in Sec. 1.3, in photographic situations where the smaller format cannot provide an entrance pupil diameter that matches that of the larger format, the smaller format will be unable to produce an equivalent photo. Since $N_1=f_1/D_1$, $N_2=f_2/D_2$, and $f_2=f_1/R_{\mathrm{w}}$, it now follows that

When focus is set on an object plane positioned closer than infinity, the above result reveals that the equivalence ratio $R$ must formally be replaced by the working equivalence ratio $R_{\mathrm{w}}$ when relating equivalent $f$-numbers as well as equivalent focal lengths.

2.5.

Same Shutter Speed

Equivalent photos taken by cameras based on different formats must be produced in the presence of the same amount of subject motion blur. This is defined as blur that occurs due to objects moving in the scene during the camera exposure. Since equivalent photos have the same perspective and framing, it follows that equivalent photos must be taken using the same exposure duration (shutter speed):

This requirement does not specify an appropriate shutter speed but merely states that it must be the same for each camera format. The required shutter speed depends upon the nature of the scene luminance distribution and the exposure strategy of the photographer.

2.6.

Same Lightness: Equivalent ISO Settings

Even though equivalent photos are taken using the same shutter speed, the resulting lightness of the images will not be the same unless equivalent ISO settings are used rather than the same ISO settings. This is because the ISO setting determines the sensitivity of the camera digital output to incident photometric exposure, and different formats receive different levels of photometric exposure when equivalent photos are taken. The ISO equivalence relationship is derived in this section.

First, recall from Eq. (32) that the entrance pupil diameters on each format are the same when equivalent photos are taken. Since the shutter speeds are also required to be the same, it follows that equivalent photos are produced using the same total amount of light, as previously discussed in Sec. 1.3. More specifically, the total luminous energy $Q$ incident at the sensor plane of both formats will be the same:

In order to rigorously derive this result, consider the photometric exposure at an infinitesimal area element on the sensor plane. This is defined by the well-known camera equation:

Now, consider the larger format labelled format 1 and the smaller format labelled format 2. From Eq. (36), the photometric exposure $H_1$ at an infinitesimal area element $\mathrm{d}{A}_1$ on the larger format sensor with focus set on a specified object plane is given by

Analogously, the exposure $H_2$ at an infinitesimal area element $\mathrm{d}{A}_2$ on the smaller format sensor with focus set on the same object plane is given by

The working $f$-numbers for these systems are defined by

Substituting these into Eq. (33) and then utilizing Eq. (13) yields

This result shows that the “working” $f$-numbers are always directly related through the equivalence ratio $R$ when equivalent photos are taken with focus set at any chosen object-plane distance.

As illustrated in Fig. 3, the luminance and cosine fourth terms appearing in Eqs. (37) and (38) will be the same for both formats at the same perspective and framing (AFoV). It must also be assumed that the lens transmission factors are the same. (Lens transmission factors depend upon the underlying lens technology; the ISO 12232 standard assumes a standard value $T=0.9$ when ISO settings are measured, along with $\phi =10\mathrm{deg}$ and an infinite object distance.) Since the shutter speeds are also the same, combining Eqs. (37), (38), and (40) yields the following relationship:

The exposure at an infinitesimal area element on the smaller sensor is therefore a factor $R^2$ greater than the exposure at an infinitesimal area element on the larger sensor when equivalent photos are taken. The total luminous energy $Q_1$ and $Q_2$ incident at the sensor plane of the larger format and smaller format, respectively, during the camera exposure are given by integrating the photometric exposure over the corresponding sensor areas:

However, the area $A_2$ of the smaller format sensor is a factor $R^2$ smaller than the larger sensor format area $A_1$, as shown in Fig. 6. Therefore,

Fig. 6

(a) The area of sensor format 1 is $d_x{d}_y$ and (b) the area of sensor format 2 is $(d_x/R)(d_y/R)$. Format 2 is therefore a factor $R^2$ smaller than format 1.

Substituting Eqs. (41) and (43) into Eq. (42) shows that the total luminous energy incident at the sensor plane of both camera formats is equal when equivalent photos are taken, thus proving Eq. (35).

Now, consider the arithmetic average photometric exposures $⟨H_1⟩$ and $⟨H_2⟩$ for both camera formats. These are defined as follows:

Utilizing Eqs. (35) and (43) yields

In traditional exposure strategy, the product of the arithmetic average exposure with the exposure index (ISO setting) $S$ defines a photographic constant $P$, which is independent of sensor format. (The ISO 12232 standard⁷ uses $P=10$, which indirectly implies an average scene luminance of $˜18\%$ for a typical photographic scene.⁵) This means that

Now, combining Eqs. (45) and (46) yields the final result:

The required ISO setting on the smaller format is therefore a factor $R^2$ lower than the required ISO setting on the larger format when equivalent photos are taken. Equation (47) holds when focus is set at any chosen object-plane distance.

Brightness is commonly used to describe the nonlinear perceptual response of the human visual system to luminance. Lightness can be thought of as brightness defined relative to a reference white. While brightness as a descriptor ranges from “dim” to “bright,” lightness ranges from “dark” to “light.” The lightness function $L^{*}$ defined by the CIE is illustrated in Fig. 7 and is specified by the following formula:

Fig. 7

18% relative luminance corresponds to 50% lightness (middle gray) on the CIE lightness curve used by the CIE LAB color space. The corresponding 8-bit DOL in the sRGB color space is 118. This DOL is $˜46\%$ of the maximum DOL since the sRGB gamma curve is not identical to the CIE lightness curve.

Since 2004, Japanese camera manufacturers have been required by CIPA to use the SOS method⁶ (or the related REI method) to determine camera ISO settings. The SOS method is based on a measurement of the photometric exposure required to map 18% relative luminance to a standard DOL in the output JPEG file. For an 8-bit JPEG file encoded using the sRGB color space, CIPA chose 118 as the standard DOL because this value corresponds with middle gray (50% lightness) on the standard gamma curve of the sRGB color space, as shown in Fig. 7. In other words, 18% relative luminance will always correspond with 50% lightness in the output JPEG file, irrespective of the shape of the JPEG tone curve used by the camera. Camera ISO settings will be discussed further in Sec. 4.

As already mentioned, the value of the photographic constant $P=10$ corresponds to assuming that the average scene luminance will be $~18\%$ of the maximum for a typical photographic scene. Provided the scene is typical and traditional metering based on average photometry is used, the SOS method described above ensures that the average scene luminance will map to a standard lightness in the output JPEG file. In other words, the same camera exposures used on the same format will map the average scene luminance to the standard lightness. When equivalent photos are taken using different formats, the same applies to the equivalent camera exposures.

In summary:

2.7.

Equation Summary

3.1.

Portrait Photography

Now, consider focus set on an object plane positioned 5 m from the entrance pupil of the 100 mm lens on format 1. In this case, the ratio $s/f_1=50$, which lies in the portrait photography regime, and the magnification is found to be $|m|\approx 0.02$ by using the formula $|m|=f_1/(s-f_1)$. The equivalent camera exposures listed in Table 5 for a selection of format 2 sizes produce equivalent photos. This data confirms the trend shown in Fig. 8; it is clear that there is negligible difference from the infinity focus results. Evidently, $R_{\mathrm{w}}$ is very well-approximated by $R$ in the portrait regime and beyond toward infinity.

Table 5

Example equivalent camera exposures at portrait object-plane distances.

3.2.

Macro Photography

Macro object distances are traditionally defined as 1:1 magnification ($|m|=1$) or larger. Using the formula $|m|=f_1/(s-f_1)$, when $|m|=1$, the ratio $s/f_1=2$. From Fig. 8, it is evident that the equivalence correction terms are important in this regime, particularly for very small formats. For example, the ratio $R_{\mathrm{w}}/R\approx 0.58$ when $|m|=1$ for the 1/2.5 in. format.

For the same example 100 mm lens on format 1, the equivalent camera exposures listed in Table 6 produce equivalent photos at 1:1 magnification. The numerical differences with the portrait and infinity focus cases are large. Significantly, larger equivalent focal lengths and equivalent $f$-numbers are required on smaller formats than would be expected using the traditional infinity-focus formulae. Although the pupil magnification $m_{\mathrm{p}}$ has been set to unity in the above example, a nonunity pupil magnification will also have a large effect in the macro regime and should be included in real-world calculations.

Table 6

Example equivalent camera exposures at macro object-plane distances.

In conclusion, the equivalence ratio $R_{\mathrm{w}}$ is important for macro photography. However, $R_{\mathrm{w}}$ is well-approximated by the equivalence ratio $R$ in general photographic situations, and the simplified equivalence equations can be used.

4.1.

ISO Sensitivity Settings

Prior to 2004, camera manufacturers typically used the saturation-based method from ISO 12232⁷ for determining camera ISO settings. This method involved a measurement of the photometric exposure required to saturate the JPEG output for a known scene luminance. (For an 8-bit JPEG file, saturation corresponds to a DOL of 255.) However, a drawback of this method was that the actual shape of the JPEG tone curve below saturation was not taken into account. This meant that different camera models could produce images with middle gray (18% relative luminance or 50% lightness) mapped to a different DOL for the same exposure settings. In other words, 18% relative luminance did not necessarily map to 50% lightness in the output JPEG file, and so photographers were faced with the unsatisfactory situation that certain cameras would produce images that appeared darker or lighter than those from other cameras when using the same exposure settings.

In order to address the above issue, CIPA introduced the SOS method⁶ in 2004, and this was subsequently incorporated into the ISO 12232 standard. Since that time, Japanese camera manufacturers have been required by CIPA to use the SOS method (or the related REI method) to determine camera ISO settings.

As briefly described in Sec. 2.6 earlier, the SOS method is based upon a measurement of the photometric exposure required to map 18% relative luminance to a standard DOL in the output JPEG file. For an 8-bit JPEG file encoded using the sRGB color space, CIPA chose 118 as the standard DOL because this value corresponds with middle gray (50% lightness) on the standard gamma curve of the sRGB color space, as shown in Fig. 7 of Sec. 2.6. In other words, 18% relative luminance will always correspond with 50% lightness in the output JPEG file, irrespective of the shape of the JPEG tone curve used by the camera. If the camera manufacturer alters the shape of the JPEG tone curve, then the required exposure duration and hence measured ISO value will adjust accordingly in order to ensure that 18% relative luminance maps to the standard DOL. This can affect IQ. For example, camera manufacturers can sacrifice SNR for extra highlight headroom in the JPEG output.¹²

Once the camera ISO settings have been determined, it is instructive to consider their use as part of a practical exposure strategy for a real-world scene. As mentioned in Sec. 2.6 earlier, a “typical” photographic scene is assumed to have an average relative luminance of 18%, i.e., an average scene luminance that is 18% of the maximum. (This follows from the value of the photographic constant $P=10$.) For such a scene, the use of traditional metering based on average photometry will analogously ensure that the average scene luminance will map to 50% lightness in the output JPEG file. If the scene is not typical, then the photographer can either use exposure compensation to adjust the metered average luminance, use a “matrix” metering mode, or a combination of both.

4.2.

Noise

Although SNR is a useful quantifiable measure of IQ, it is also useful to gain a visual impression of the image noise to be expected at a given ISO setting for a typical photographic scene.

For photographers who primarily use JPEG output from the camera, camera reviewers may, for example, compare (or provide a comparison tool for) images obtained using the camera default JPEG settings. In this case, the quality of the internal camera signal processing and noise filtering will be taken into account. For photographers that primarily process RAW output themselves, camera reviewers may instead provide images that have been processed from the RAW files using a standard workflow with noise filtering disabled.

In order to demonstrate equivalence theory using the visual impression of image noise, consider the cameras listed in Table 7 all set at the same camera exposure. These cameras are based on the 35-mm full frame, APS-C, micro four thirds, and 1 in. formats, respectively. Lenses with equivalent focal lengths have been used so that the AFoV is the same in all cases. However, the images are not equivalent because the total light collected and DoF are different in all cases.

Table 7

From top to bottom, cameras based on the 35-mm full frame, APS-C, micro four thirds, and 1 in. formats, respectively, all set at the same camera exposure.

The left column of Fig. 9 shows corresponding images reproduced from Ref. 1. Images (a)–(d) correspond to the Canon, Fujifilm, Panasonic, and Nikon cameras, respectively. The images were obtained by processing the RAW files using Adobe^® Camera Raw with the Adobe^® Standard color profile and noise filtering minimized. The images were subsequently downsampled to a common pixel count. Note that the Adobe Standard color profile aims to give a consistent contrast response across cameras but is calibrated to respect the manufacturer’s JPEG ISO implementation. In other words, the images all appear equally as light as each other.

Fig. 9

(a)–(d) Noise appearance for cameras using the same camera exposure corresponding to Table 7. (e)–(h) Same as left column but using equivalent camera exposures corresponding to Table 8. The figure is best viewed on a computer monitor. Figure derived from Ref. 1 courtesy of www.dpreview.com.

Table 8

From top to bottom, cameras based on the 35-mm full frame, APS-C, micro four thirds, and 1 in. formats, respectively, all set at equivalent camera exposures. The nearest exposure variables available on the camera for equivalence to hold are listed, with the ideal equivalent values given in brackets.

The noise clearly becomes more apparent as the format size decreases. This is to be expected because the largest contribution to the image noise is photon shot noise. As mentioned in Sec. 1.1 earlier, photon shot noise scales as the square root of the total amount of light collected (or more precisely, the number of photoelectrons generated) and so overall, SNR is higher as the format size increases.

Now, consider the same cameras all set at the “equivalent” camera exposures listed in Table 8. The right column of Fig. 9 again shows corresponding images reproduced from Ref. 1 using the same RAW workflow as above. Slight lightness adjustments have been made to compensate for the use of any nonideal equivalent settings. Since the images are equivalent, it is clear that the level of visual noise is now very similar for all cameras.

Remaining differences in noise are due to differences in the underlying camera and lens technology, for example:

A higher quantum efficiency enables more photoelectrons to be produced for a given level of photometric exposure. This increases the sensitivity of the camera digital output to incident photometric exposure and raises the ISO value (both JPEG and RAW) corresponding to the selected analog gain getting. In other words, more light can be collected and a higher SNR can be achieved at a given ISO value compared to a camera with lower quantum efficiency. When comparing equivalent photos, the same applies to the equivalent ISO settings because in this case, the total incident light is the same rather than the level of incident photometric exposure.

The noise floor or read noise is another factor, which will become apparent in darker regions of the image. When producing equivalent photos, the value of the equivalent ISO setting is lower on a smaller format because a smaller format requires less gain to achieve the same DOL when equivalent photos are taken, as proven in Sec. 2.6 earlier. In principle, this compensates for the format size difference in terms of read noise. In practice, there will be a dependence upon signal processing technology. There will also be a dependence upon sensor pixel count because the total read noise for an aggregate of smaller pixels is generally different to that of a larger pixel of equal area. More formally, it is actually differences in the read noise per percentage sensor area at equivalent ISO settings that matters when comparing equivalent photos. This is discussed further in Sec. 4.3 below, which covers SNR in more detail.

Sensor pixel count also plays another role, which primarily affects photon shot noise when comparing equivalent photos. Because equivalent photos are required to be viewed at the same display dimensions, different pixel counts require different amounts of resampling in order to match the required display resolution in pixels per inch. Downsampling reduces noise at the expense of resolution, whereas upsampling preserves any resolution advantage without reducing noise. As discussed in Sec. 4.3 below, sensor pixel count can be automatically taken into account when measuring SNR by using a metric such as SNR per percentage sensor area.

Since the camera ISO settings are based upon the JPEG output and are determined using the SOS method described in the previous section, camera manufacturers can alter the shape of the JPEG tone curve in order to position middle gray ($\mathrm{DOL}=118$ in the JPEG output) at any desired position on the sensor response curve. For example, shifting $\mathrm{DOL}=118$ to correspond with a lower RAW value can increase the highlight headroom in the JPEG output while maintaining the standard mid-tone lightness that defines the ISO setting.¹² However, this will lower the SNR because digital gain is effectively being applied to the mid-tone region. Camera manufacturers will choose to balance such trade-offs differently, and this will have an effect on the level of noise seen in the JPEG output.

Finally, the $f$-number and ISO setting used on the full-frame format in the example given in Table 8 are both relatively high. This means that there is a large equivalence overlap with the smaller formats. A major step-up in noise performance can be achieved using a larger format only when its extra photographic capability is utilized. As discussed in Sec. 1.3 earlier, this requires the use of a low ISO setting and/or $f$-number that do not have equivalents on the smaller format.

4.3.

Signal-to-Noise Ratio

SNR is a more precise way of quantifying camera capability in terms of image noise. Typically, SNR is measured using RAW data and so “RAW” ISO settings should ideally be used. If the camera ISO settings are used instead, the results will favor cameras that place middle gray ($\mathrm{DOL}=118$ in the JPEG output) higher on the sensor response curve.

Photon transfer curves are plots of noise (or alternatively SNR) as a function of normalized exposure value (EV), and the curves can be plotted at various ISO settings. The normalised EV is simply the number of stops and can be calculated based on input-referred units (photoelectron count) or output-referred units (RAW values). The normalized EV based on input-referred units does not take ISO gain into account and so the speed value is not included. On the other hand, output-referred units do take into account the ISO gain and so the speed value is included in the number of stops.

When performing cross-format comparisons using equivalence theory, the total light received by each format must be the same. It is therefore most convenient to use output-referred units when specifying the normalised EV (i.e., the base 2 logarithm of the RAW value specified as a digital number or analog-to-digital unit.) Cross-format comparisons can then be performed simply by comparing curves at equivalent ISO settings.

SNR itself is typically specified as a measure per sensor pixel (photosite). However, per photosite measurements are not particularly useful when comparing sensors with different photosite sizes, the reason being that SNR is dependent upon the area over which it is measured.¹⁴ For example, consider two sensors that are identical other than photosite count, one having four times as many photosites as the other. The sensor with the larger photosites will have a greater SNR per photosite because the light-gathering area per photosite is larger. However, this does not mean that the higher-resolution sensor is noisier; the same SNR per photosite could in principle be achieved using the higher-resolution sensor simply by binning every group of four photosites together, either on the sensor or by subsequent image resampling. In other words, a more appropriate measure of SNR when comparing sensors of the same format is SNR per percentage sensor area.¹⁴

When performing cross-format comparisons, SNR should be specified in a way that corresponds to a comparison of equivalent photos, i.e., the use of equivalent camera exposure settings. Since the display dimensions of equivalent photos must be the same, the different enlargement factors from the sensor areas to the viewed output image dimensions must be taken into account. It turns out that SNR per percentage sensor area is again the appropriate measure to use. In this case, it takes into account both sensor pixel size and format size. Although the measurement area scales in direct proportion with the format size, the level of photometric exposure is greater as the format size decreases because the equivalent ISO setting is lower for the same image lightness. In other words, the measure is based upon the same amount of incident light when comparing each format, at least over the equivalence overlap between them. Photon transfer curves can be conveniently compared by plotting SNR per percentage sensor area as a function of normalized EV based on RAW value, and then comparing the curves at equivalent ISO settings. The main advantage of a larger format is the extra photographic capability afforded by low ISO settings that have no equivalent on the smaller format.

Finally, it should be mentioned that FWC per percentage sensor area is an important sensor characteristic that can increase the maximum achievable SNR and in turn lead to greater dynamic range.

4.4.

Dynamic Range

Camera reviewers may measure “highlight” and “shadow” dynamic range. These are per-pixel metrics that are only valid with JPEG output and are intended to give information about the nature of the JPEG tone curve used by the camera manufacturer. Highlight dynamic range is a measure of the number of exposure stops needed to increase middle gray ($\mathrm{DOL}=118$) to saturation ($\mathrm{DOL}=255$). Analogously, shadow dynamic range is a measure of the number of exposure stops needed to increase $\mathrm{DOL}=1$ to $\mathrm{DOL}=118$.

However, it is the dynamic range in the RAW data that is more important in terms of camera capability. Dynamic range is defined in engineering as the ratio of the maximum signal to the minimum usable signal, with the latter defined as the signal corresponding to SNR = 1. This engineering definition of dynamic range is commonly used to quantify the dynamic range of digital cameras, with SNR typically calculated as a per sensor pixel (photosite) value using RAW data.

However, as discussed in the previous section, it is much more appropriate to use SNR per percentage sensor area when comparing digital cameras. This measure takes into account sensor pixel count when comparing the same formats, and both sensor pixel count and format size when performing cross-format comparisons using equivalent camera exposure settings. In order to illustrate the former case, consider the common misconception that larger photosites automatically provide higher dynamic range due to a higher FWC. This is not the case in practice because larger photosites also have a larger surface area for collecting light, and so the ratio between the maximum and minimum signal (photoelectron count) remains the same provided the same sensor technology is used. It is actually FWC per percentage sensor area which matters and this could favor either larger or smaller photosites depending on the technology used. On the other hand, read noise does have a dependence on sensor pixel count because the total read noise for an aggregate of smaller photosites is generally greater than that of a larger photosite of equal area, as already mentioned in Sec. 4.2. It follows that larger photosites may provide an SNR advantage at high ISO settings, where the contribution to the read noise upstream from the ISO gain amplifier dominates,¹⁴ and this can lead to a greater DR because of a lower “usable” signal.

In conclusion, SNR should ideally be normalized by percentage sensor area when calculating the dynamic range of digital cameras. Nevertheless, one issue that arises is which actual percentage value to choose as this will affect the absolute dynamic range values. A way forward is provided by the CoC described in Sec. 2.3. Since the CoC area scales in direct proportion with format size, SNR per CoC area is an equally valid measure that can be used instead of SNR per percentage sensor area. It can be calculated as follows:

Table 9 lists the PDR values for the same cameras and equivalent ISO settings used in the noise comparison of Sec. 4.2 with the addition of the full-frame Canon 5D mk4 camera. The PDR values are seen to be in very close agreement. Figure 10 shows that this equivalence behavior is maintained over the equivalence overlap between the cameras, at least for the Fujifilm, Panasonic, Nikon, and Sony cameras. At lower ISO settings, the extra photographic capability is evident as the format size increases. For example, the maximum PDR of the Panasonic GH4 occurs at ISO 100, however, the Nikon 1 V3 is unable to produce a PDR value of the same order of magnitude because its lowest ISO setting is 160. The equivalent ISO setting, ISO 54, does not exist and so, an equivalent photo cannot be produced. The camera is unable to use the same exposure duration as the Panasonic camera without overexposing the photo. Interestingly, the full-frame Canon 1D-X does not appear to take advantage of the extra photographic capability that its sensor size affords because the PDR curve levels off at low ISO settings. On the other hand, the more recent full-frame Canon 5D mk4 camera does take advantage of its extra photographic capability at low ISO settings. The long exposure duration available at these ISO settings enables the camera to achieve a higher SNR, which in turn leads to higher PDR. Furthermore, the Canon 5D mk4 has a very low base saturation-based RAW ISO setting. In a modern camera with high quantum efficiency, this is characteristic of a sensor with a high FWC per percentage sensor area.

Fig. 10

Photographic dynamic range as a function of ISO setting for a selection of cameras. The data is not available at all ISO settings. Data sourced from Ref. 16.

Table 9

Photographic dynamic range (PDR) measured in stops at the equivalent ISO settings listed in Table 8. Data sourced from Ref. 16.

4.5.

Resolving Power

System RP describes the full capability of a camera and lens system to resolve detail. It is determined by the system MTF cut-off frequency. Many component contributions to the system MTF can be defined, and typically, an effective system cut-off frequency is used, which is defined by the spatial frequency at which the system MTF drops to a small percentage value, such as 10%. The system cut-off frequency is typically limited by either the sensor Nyquist frequency or the lens cut-off frequency, whichever is lower at the selected lens $f$-number. When the system RP is not limited by the lens, an optical low-pass filter may be required to lower the system cut-off frequency down to the sensor Nyquist frequency in order to prevent aliasing.

When performing cross-format comparisons, spatial frequencies $\mu$ at the sensor plane scale in proportion with the equivalence ratio $R$ between the formats being compared, i.e.,

For example, consider the lens cut-off frequency for an ideal aberration-free diffraction-limited lens, i.e., the diffraction cut-off frequency. This is given by

The diffraction cut-off frequency is reduced as the $f$-number increases. Since equivalent photos are produced using equivalent $f$-numbers according to $N_2=N_1/R$, the diffraction cut-off frequency increases in proportion with $R$, which is consistent with the above analysis.

Fortunately, there is an alternative spatial frequency unit that is very convenient when performing cross-format comparisons, line pairs per picture height (lp/ph). This is related to lp/mm in the following way:

In the present context, picture height refers to the short edge of the imaging sensor, which is proportional to the sensor diagonal. This means that the unit is format-independent and equivalent spatial frequencies are automatically accounted for provided the formats have the same aspect ratios. In particular, the sensor Nyquist frequency expressed in lp/ph units depends only on sensor pixel count (and fill-factor) and is independent of the actual pixel pitch.

It should be apparent that when performing cross-format system capability comparisons, the larger format can in principle gain a RP advantage when the system cut-off frequency is limited by the lens and an equivalent $f$-number on the smaller format does not exist. For example, when $N=2.8$ on the 35-mm full-frame format, an equivalent $f$-number does not exist on the small 1/2.5 in. format for which $R=6.02$. However, real-world comparisons are strongly affected by the nature of the lens aberrations, the presence/absence of a low-pass filter, pixel count, and other optical phenomena that can profoundly affect the effective system cut-off frequency.

On the 35-mm full-frame format, the lens cut-off frequency typically reaches its maximum at an $f$-number a stop or two higher than the lowest available because of geometric lens aberrations that dominate at the maximum entrance pupil diameter. As the format size decreases, there is less need to stop down to alleviate the effect of aberrations because the maximum achievable entrance pupil diameter is smaller.

4.6.

Sharpness

The standard viewing conditions assumed by the camera or lens manufacturer typically involve an enlargement factor of 8 on the 35-mm full-frame format.¹¹ As discussed in Sec. 2.3, this means that a full-frame camera does not actually need to resolve spatial frequencies greater than $40\mathrm{lp}/\mathrm{mm}$ on the sensor plane because further detail cannot be resolved by an observer of the output image under these viewing conditions. In other words, the RP of a camera system only becomes important when the output image is viewed under more extreme conditions (such as a greater enlargement or closer viewing distance) than the standard viewing conditions assumed by the camera or lens manufacturer.

Under the standard viewing conditions described above, the nature of the system MTF curves between 10 and $40\mathrm{lp}/\mathrm{mm}$ on the full-frame format is much more significant in terms of IQ. When combined with the contrast sensitivity function of the human visual system, it is the system MTF values at these spatial frequencies that primarily determine perceived image sharpness. Several sharpness metrics exist, in particular, subjective quality factor¹⁷ has found application in photography.

From the discussion of spatial frequencies in the previous section, it follows that the nature of the MTF curves between $10R$ and $40R\mathrm{lp}/\mathrm{mm}$ (where $R$ is the equivalence ratio) are most important for determining image sharpness under standard viewing conditions when performing cross-format capability comparisons between the full-frame format and a smaller format. For example, system MTF at $20\mathrm{lp}/\mathrm{mm}$ on the 35-mm full-frame format should be compared with $30\mathrm{lp}/\mathrm{mm}$ on APS-C and $40\mathrm{lp}/\mathrm{mm}$ on micro four thirds. This means that the smaller format needs to match the system MTF of the larger format for all relevant equivalent spatial frequencies in order to achieve the same perceived sharpness. Naturally, the component lens MTF curves should be based on equivalent focal lengths and equivalent $f$-numbers. In practice it is more convenient to use the line pairs per picture height (lp/ph) spatial frequency unit defined in the previous section.

An important contribution to perceived image sharpness is lens diffraction softening. This arises due to the fact that the diffraction cut-off frequency scales with f-number according to Eq. (53). Equivalent photos suffer the same level of diffraction softening. This is because blur due to diffraction softening will generally not become visible until the Airy disk diameter $d_{\mathrm{Airy}}=2.44\lambda N$ (corresponding to the diameter of the diffraction PSF) approaches the CoC diameter $c$. Since the equivalent $f$-numbers and CoC diameters scale in proportion with $R$ according to Eqs. (2) and (23), the net result is that equivalent $f$-numbers on different formats lead to the same level of diffraction softening when the equivalent output images are viewed at the same distance and display dimensions. For the $c$ value corresponding to standard viewing conditions, diffraction softening generally becomes noticeable at $N=16$ on the full-frame format, $N=10.5$ on the APS-C format, and $N=8$ on micro four thirds.

Sensor pixel count is another important contribution to perceived image sharpness. Although a higher sensor pixel count only has a negligible effect on the system RP when the system is limited by the lens, a higher sensor pixel count can improve perceived images sharpness irrespective of the nature of the limiting component contribution to the system MTF. This can be seen by considering the detector-aperture contribution to the system MTF:

5.1.

Display Size

The fundamental reason that images appear to be less noisy when viewed at smaller display sizes relates to the fact that the CoC diameter increases as the display dimensions decrease according to Eq. (22). For example, consider a mobile phone with a 1/2.5 in. sensor and 5 in. screen in landscape orientation. At the least distance of distinct vision ($~25\mathrm{cm}$), the CoC diameter will be doubled from the 0.005 mm value given in Table 3 since the enlargement factor from the sensor dimensions to the mobile phone screen dimensions will be approximately half that to the A4 paper size used to calculate Table 3. In this case, the CoC area on the sensor will be four times as large.

In Secs. 4.3 and 4.4, it was explained that SNR is dependent on the size of the sensor area over which it is measured. In the present context, the SNR per CoC on the sensor is a measure that corresponds with the minimum area on the viewed output image over which detail and noise can be perceived by a human observer. According to Eq. (51), the SNR per CoC will be doubled when viewing an image on the mobile phone compared to viewing at A4 paper size, and so perceived noise will be lower. Although the lower noise occurs at the expense of perceived scene resolution due to the limited RP of the human visual system, mobile phone camera JPEG engines typically apply sharpening optimized for mobile phone display sizes in order to improve perceived sharpness. Various other image processing algorithms including noise reduction will also be applied.

Since the CoC diameter is the limiting factor in determining perceived noise and resolution, downsampling an image to a lower pixel count such as the mobile phone screen pixel count will not have a noticeable effect on IQ under the viewing conditions described above. However, downsampling can improve noise when “zooming in” to the image, albeit at the expense of captured scene resolution. In this case, the CoC is temporarily reduced to a small size and the SNR associated with individual pixels becomes more important. To see that downsampling can improve the SNR associated with individual pixels, consider a crude downsampling filter that averages over every block of four pixels in order to downsample an image to 25% of its original pixel count while maintaining the same image dimensions. Although this process discards captured scene resolution, temporal noise adds in quadrature and so each “larger” pixel in the downsampled image is associated with an SNR of up to twice that of any of the individual pixels in the original image when only temporal noise is considered.¹⁴ This higher SNR is maintained when the image dimensions are subsequently reduced.

5.2.

Equivalence Overlap Example

As an example of the equivalence overlap, consider the Google Pixel 2 smartphone, which has a 1/2.6 in. sensor ($\sim 5.5\mathrm{mm}\times 4.1\mathrm{mm}$), a base ISO setting $S=50$, and a lens with focal length $f=4.45\mathrm{mm}$ and fixed $f$-number $N=1.8$. Also, consider a camera based on the micro four thirds format ($17.3\mathrm{mm}\times 13.0\mathrm{mm}$), which has an equivalence ratio $R=2$ with 35-mm full frame. The Google Pixel 2 and micro four thirds sensors both have a 4:3 aspect ratio and the equivalence ratio between them is $R\approx 3.15$. Therefore, the maximum photographic capability of the Google Pixel 2 corresponds to using $S\approx 500$ and $N\approx 5.6$ on the micro four thirds format, along with a lens focal length $f\approx 14\mathrm{mm}$ in order to produce an equivalent photo.

In order to use equivalence theory as a framework for comparing the IQ of the above camera formats, it is instructive to group the camera exposures used on the micro four thirds format into the three possible scenarios listed below.

Comparison between any other mobile phone sensor format and larger camera sensor format can be achieved by replacing the $S$ and $N$ limits above with the appropriate values.

As discussed further in the section below, computational photography features available on modern smartphones such as the Google Pixel 2 must be disabled in order to apply equivalence theory in the manner above. With such features disabled, Fig. 11 shows a crude visual noise comparison between the Google Pixel 2 and the Olympus E-M1 camera, which is based on the micro four thirds format. Equivalent camera exposure settings were used; $S=80$ and $N=1.8$ on the Google Pixel 2 (left diagram), and $S=800$ and $N=5.6$ on the Olympus E-M1 (middle diagram). The same exposure duration $t=1/2\mathrm{s}$ was used in both cases along with equivalent focal lengths. In order to eliminate the different JPEG processing by the manufacturer from the comparison, the images were obtained from the RAW data by processing Adobe^® DNG files using default settings and noise reduction minimized. The Google Pixel 2 image was subsequently upsampled to match the sensor pixel count of the Olympus E-M1 and crops from the images were then taken. Evidently, visual noise is of the same order of magnitude for both cameras when equivalent photos are taken.

Fig. 11

(Left) Google pixel 2; $S=80$, $N=1.8$, $t=1/2\mathrm{s}$. (Middle) Olympus E-M1; $S=800$, $N=5.6$, $t=1/2\mathrm{s}$. (Right) Olympus E-M1; $S=200$, $N=2.8$, $t=1/2\mathrm{s}$.

The diagram on the right shows the same result for the Olympus E-M1 when the shutter speed was maintained at $t=1/2\mathrm{s}$ but the ISO setting was lowered to $S=200$ and the $f$-number lowered to $N=2.8$ in order to widen the aperture and let in two extra stops of light. Consequently, the image is less noisy. Equivalent settings do not exist on the Google Pixel 2, however, such limitations can be overcome by using computational photography features such as HDR+ discussed below.

5.3.

Computational Photography

Due to the small equivalence overlap between a mobile phone camera sensor and a larger format sensor described above, the mobile phone camera is unable to receive as much light as a camera based on a larger format in most photographic situations. This places limitations on SNR, dynamic range, RP, and possible shallow DoF. Recent smartphones now employ computational photography techniques in an attempt to overcome such limitations, and this means that equivalence theory cannot always be applied.

Many computational photography techniques are based upon multiple camera exposures or even multiple cameras. For example, the high dynamic range (HDR) capture mode is sometimes the default mode on modern smartphones. While traditional HDR imaging works by combining frames from multiple camera exposures with differing exposure durations, alternative techniques have been developed, such as the HDR+ technology¹⁹ available on the Google Pixel/Nexus smartphones. This works by taking multiple (typically 10) underexposed frames, each using the same exposure duration. Underexposing each frame ensures that highlight data is preserved and the shorter exposure duration required by each frame minimizes camera shake. The frames are subsequently aligned using the lucky imaging technique and then averaged. Averaging frames averages the temporal noise and therefore increases SNR and dynamic range, a technique often used in scientific photography and astrophotography. The shadows are then lifted and the image adjusted, the result of which is a photo with improved IQ and an attractive appearance. Other computational photography techniques have been developed for smartphones that have dual cameras on the rear. For example, dual cameras can compute a stereo depth map that enables images with synthetic shallow DoF to be produced, mimicking the shallow DoF produced by larger-format cameras at wide apertures (low $f$-numbers). The portrait mode found on the Google Pixel 2 camera is able to achieve a similar effect by using only one camera; a convolutional neural network is used to compute a segmentation mask for the subject, and information from phase-detect autofocus is used to compute the depth map.

Equivalence theory cannot be easily applied when techniques such as those described above are employed, either in terms of translating camera exposure settings between different format sizes or when evaluating IQ. Although it is possible to disable such features (as done in the crude noise comparison of Fig. 11), doing so will not utilize the full capability of the smartphone in terms of photographic output. Instead, metrics such as those developed by DxOMark^®20 can be useful when comparing the IQ between different smartphone cameras.