Note: The Effects of Sensor Size

Understanding the practical effects of a different sensor size

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The purpose of this note is to help understand the practical effects of a different sensor size.


The relevant question for the photographer is "What are the changes that a change in sensor size induces in photographic practice?"


This question arises particularly when purchasing equipment. For example, a photographer equipped with fullframe wants (for example because of weight or ergonomics but it doesn't matter) to buy an APS-C. For example, with his fullframe he uses an 85mmf1.8 usually at full aperture to take portraits with a shallow depth of field and he is looking for the equivalent lens for the APS-C.


Some proposals are scientifically accurate but are at the very least misleading from a practical point of view for the photographer. So it is with the famous "A 50mm remains a 50mm regardless of the size of the sensor." This is perfectly accurate: using an APS-C sensor is like cutting into what a fullframe sensor would have picked up.  But, the photograph would not be the same.

The question can be restated "What are the changes need to be made to get the same picture when the sensor size changes?" 

In the following, k is the "crop factor" defined as the ratio of the diagonal of the sensor to that of a full-format "FF" (1). For a Fujifilm APSC, the crop factor is 1.5.

REMINDER

Concerning exposure, but to see also the point 4, the number f does not depend on the size of the sensor (2). If a light meter indicates f/4, it is f/4.

1/ POINT OF VIEW

Perspective depends only on the point of view, not the focal length. The same applies to distortions of perspective as far as this makes sense.
To have the same perspective, i.e. the same relationships between the fields covered at different distances to the device:

• The point of view must remain the same.

2/ EFFECT ON FOCAL LENGTH

Having an identical framing, i.e. having the same angle of field, then requires a change of focal length, otherwise the APS-C sensor only returns part of the FF image.

• The focal length on the APSC must be equal (3) to : Focal length FF/ k  (4)

3/ EFFECT ON DEPTH OF FIELD
The changes in sensor size and focal length (above 2/) must now be compensated to obtain the same depth of field.

• The aperture to the APS-C must be set (5) to :  Aperture FF/ k

4/ EFFECT ON EXPOSURE

The change in the aperture introduced gives a more exposed image.
To restore the same exposure, it is therefore needed to reduce the exposure time, or sensitivity as you wish. Since the exposure time has an effect on the motion blur, the change in sensitivity is retained here.
Since the photometric aperture has changed, the sensitivity must be reduced.

• The sensitivity on the APS-C can be set to : Sensitivity FF/ k²
  
  

APPLICATION

With a Fuji X APS-C with a k of 1.5, to take a photograph corresponding to that which would be taken with a FF 85mm F4 ISO 800 with a focusing distance of 10m, it is necessary to:

1. keep the same distance 
2. use a 56mm (85/1.5)
3. open to F2.8 (4/1.5  = 2.67)
4. set theoretically ISO 350 (800/1.5^2 = 355,5) or in practice ISO 400 as the aperture has been rounded to f2.8 
   

OTHER POINTS

Three other points must be taken into account: diffraction, noise and camera shake blur.

Diffraction

The effect at the sensor level of the diffraction (6) is proportional to the aperture, the modification made at point 2 leads to an aperture Ok = O/k, but this effect on the final image is  proportional to the ratio between the sensor sizes (k). Thus the effect of the diffraction on the final image is identical for both sensor sizes.   

Noise

Regarding the noise, even more important as the size of the pixels of the sensor is small, if we have, as indicated in point 4 decreased the sensitivity, the result for the sensor of reduced size is more or less the same as for a full full frame camera (7).

Camera shake blur

Camera shake blur is mainly a factor of an angular modification of the view axe. Hence the comon rule giving the lowest speed (SL) to take an image without camera shake with a lens of focal length f :

SL = 1/f  must be modified in SL_k  = 1 / (f k).

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Notes :

1) The reference to a full size sensor is not mandatory but is generally retained.

In addition, attention must be paid to the length to width ratios of the two sensors of both devices. If these ratios are not identical, the photographs will not be similar unless one is reframed during postprocessing or at printing.  It will therefore be necessary to determine the useful "crop factor" to take the dimensions of the part of the sensor that will correspond to the final image.

2) The geometric exposure (see below in this same note) is a function of the size of the entrance pupil of the lens. This size is set by the diaphragm. The f-number is equal to the ratio between the focal length and the diameter of the entrance pupil : N = f / d, with: focal (mm): f, aperture: N , entrance pupil diameter (mm): d.

The 'f' numbers were 'invented' to avoid having to take into account the entrance pupil and focal length.

The photometric exposure (T stop) also takes into account the losses related to the presence of lenses in the lens and their surface treatments. The T-number is indicated on the cinema lenses but also on some photographic lenses like the Fujifilm XF 56mm APD on which both data are indicated because of the presence of the apodization filter.   

3) The same angle of view implies that F/L=f/l, with F and L the focal length and width of the sensor for FF and f and l the focal length and length of the sensor for APSC.

4) k being the ratio of length of the sensor FF / length of the sensor APS-C. The same reasoning can also be applied to 4/3 or medium sensors, but it should be noted that these sensors do not have the same width/length ratio as fullframe or APS-C sensors.

5) In the following ;
- the spot of the real point is the zone in the plane of the point of the space of the shooting for which the points will merge;
- values with a _k index correspond to the case of a crop factor k;
- links, unless otherwise indicated, refer to Wikipedia

With: 
- Focal (mm): f
- Aperture: N 
- Entrance pupil diameter (mm): d
- Circle of confusion (mm): c
- Distance to the subject (mm): D

The size of the spot of the real point is a function of d (see Depth of Field Outside the Box by Richard F. Lyon), to have the same blur effect it is thus necessary that d_k = d.

From N = f / d, it follows that N_k = f_k / d_k = (f / k) / d = (f / d) / k = N / k.

To be convinced of this, one can make the calculation in the formulas for the limit sharp plans in standard cases, i.e. outside of macro photography:
- Front Sharp Plan (mm) : FSP = D H / (H + D)
- Back Sharp Plan (mm) : BSP = D H / (H - D)
and
- Hyperfocal (mm) : H = f²/(c N)

In order to maintain the size of the object point during visualization, the relationship between the confusion circles c and c_k must be such that c_k = c / k

It follows that:
H_k = f_k² / (c_k N_k) = (f / k)² / ((c / k) (N / k)) = f² / (c N) = H

QED

6) See What Is Diffraction In Photography? by  Alik Griffin

7) Signal to noise (SNR) is defined as µ / σ where µ = mean pixel value and σ standard deviation of pixel value.

It can be shown  (Hamamatsu: Calculating SNR) that 
SNR = QE * S  / 5qrt(Fn² * QE * (S+Ib) + (Nr/M)²)
where :
- QE = Quantum efficiency
- S = Photons/pixel
- Fn = Noise factor
- Ib = Background
- Nr = Readout noise
- M = EM gain
As S_k =  S / k², then SNR_k > SNR / k