Pixiq

This is the third (and final) post in a trilogy that started with Bokeh, delved into sharpness and now touches on resolution… It is opposite way to the way we usually encounter them but I have been pondering recently on things I am writing. After a while what seemed familiar and reasonable did not anymore…

 No-one needs to know about the functioning of integrated circuits to use an iPod – so does knowing about the theoretical background to photography help or hinder? For some it just gets in the way and for others well, knowing the background and the ‘why’ of things is part of the package – even if it is idle curiosity. 

When it comes to outdoor photography in general and the world at close quarters in particular, knowing a bit of background can certainly help those who want to experiment or who want to understand why their efforts have gone wrong and maybe how they can go right.

Sharpness and resolution: As you might expect, there will be differing views on exactly what constitutes sharpness in a digital image. It doesn’t seem like an objective judgment because a portion within a photo that is sharp enough for you may strike me as a bit soft, or vice versa. How do viewers decide exactly where something comes into or goes out of focus?

With human vision very small circles become indistinguishable from sharp points. Resolving power is a measure of the ability to distinguish between points/details very close together. Those who started their photography with film and prints will be aware of ‘line pairs’ used as a way of defining ‘resolution’ but it might be better in this digital age to start with ‘points’:

**Someone with normal vision cannot resolve objects separated by less than one minute of arc (one 60^th of a degree) so very small spots (blur circles) are perceived as sharp points. The smallest a circle can be and still seem a point is called the circle of confusion (CoC) – or circle of least confusion..

For someone with normal vision, a circle of 0.20mm is indistinguishable from a point on a 10in x8 in print viewed from about 10” away…in effect,  you could not separate details closer together than this since they would merge.

So what has this figure got to do with that figure of 0.03mm often quoted in relation to ‘circles of confusion’. This refers to detail on a 35mm film frame or an FX sensor and to  appreciate that this figure is not just plucked from the air we can to work backwards from a print – that ‘standard’ 10in x 8 in print, that owes its origins to  the ‘full plate camera’.

This size of print print is viewed at a ‘normal’ close viewing distance - taken to be 25cm (10in) – sometimes called the ‘near (or least) distance of distinct vision’. For a print of size 10’ x 8” this represents the so-called ‘comfortable’ viewing distance where the angle of view is 60° measured along the diagonal (let’s hear it for Pythagoras) - giving about 32 cm for this print

However, to produce a print like this from a 35mm film frame/ FX sensor takes an enlargement of about 7x…so the original circles of least confusion (on the sensor) would have had to be 0.2 ÷ 7 which is approx 0.03mm…the figure we came across before.

This might seem an esoteric idea but it has importance when we look closely at two things fundamental to Macro, for example: depth of field and lens resolution. The value of 0.03mm  is used by lens designers in computations for lens, though in the past they have used circles from 0.025mm to 0.033mm and with smaller sensors have to use values even less.

1. Depth of field.

When sharply focused on a point a lens images it as a point but move the focus slightly either side and the point becomes a tiny circle – the depth of field is distance between the limits either side of this sharp focus where the circle does not exceed 0.03mm. Within that range the subject appears sharp. The diagram used before for the article on ‘Bokeh’ shows why the depth of field increases as the lens aperture gets smaller. The value of 0.03mm is used to calculate the entries in depth of field tables.

2. Resolution of lenses and sensors

Consider what must happens to the circle of confusion as sensors get smaller - to make  that 10” x 8” print the image from the sensor has to get enlarged more.

The value of 0.03mm is accepted as the value of the circle of least confusion on an FX sensor/35mm film which has a diagonal of 43.4mm. The circle of confusion represents 1/1447^th of this…according to my on-screen calculator.

With a smaller sensor this value changes proportionally. For example consider the APS-C sensor used by  Nikon, Sony, Pentax: dimensions are 23.6 mm x 15.7mm with a diagonal of 28.3mm. the same 1/1447^th  fraction of the diagonal – the circle of confusion to use is 0.02mm and on the same basis, for a Canon APS-C sensor the CoC would be 0.019mm and for the 4/3 system it is 0.015mm.

**The smaller the sensor the smaller the CoC has to be

Small sensors make greater demands on lens design and the correction of defects. Another important factor is that as sensor sites gets smaller lenses need to have higher resolution to cope with them and get the best out of them in terms of image quality.

So why are there limits to what lenses and sensors can resolve and what are they?

The idea of a perfect lens free from all defects and aberrations might be the Holy Grail of lens manufacture and although designers get closer it would be easier to find the Holy Grail than lens perfection. However, wonderful a lens design one thing stands in the way - diffraction, the way light waves spread slightly when they pass through apertures: the smaller the holes or narrower the slits the more pronounced the effect which we photographers see as softening of image edges.

To get some numbers out of this needs we the answer to a question: Is a point a point. Or just how small an image circle can a lens produce? After all, we use the convenient idea of images built up from a myriad points without a second thought.

Answers to this were forthcoming in the early 19^th century when John Herschell (the son of Sir William Herschell discoverer of the planet Uranus) revealed that stars were not imaged as sharp points of light but as disks with a bright centre and a series of tight concentric rings around them getting fainter as they go outwards. It’s what is called a circular diffraction pattern and every ‘point’ image always has a bright centre called the ‘Airy Disk’ named after its discoverer George Biddell Airy who analysed the diffraction of light through small circular holes.

For astronomers this was important for they were looking at what they thought were points of light which when magnified sometimes turned out to be pairs of stars close together – binary pairs. They were up against the limits of their telescopes and wanted to know just when you could call these two stars ‘separate’ and not an aberration of the lenses used. These men did not go out and buy their kit they made it and ground their own lenses…

This idea of the points being two sets of rings gave a way of making a decision about when these images were separate ie: if the first maximum (centre of one bright Airy disk) coincides with the first minimum in the rings of the other they were considered separate.

 This is called the Rayleigh Criterion and works for very distant objects such as stars (ie when a telescope or some other lens system is focused on infinity – that is it produces parallel rays.

I realise that most people drift when faced with any sort of school math equation – I sympathise because some of those whom I love in my life (partner, daughter, friends…) do too.

If so...SKIP THIS.

It is expressed as: sin θ ≈ 1.22 λ/d  (for very small angles this becomes θ ≈ 1.22 λ/d)

 θ =  angular separation between points (measured in radians); λ =  wavelength of light

 d =  effective aperture diameter ; the figure 1.22 derives from the first order Bessel function used in the mathematics of the full analysis.

 So, what has this to do with photography. Well that angle ‘θ’ is not in degrees it is in circular measure or radians which is more useful in higher math.

 If a lens is focused on infinity it images a point at a distance called its focal length ‘f’

 If x is the distance apart of two points close together then θ = x/f

 So:     x/f = 1.22 λ/d 

 Or, rearranging x = 1.22 λf/d the useful thing about this is that f/d is the f-number of the lens

 So:        x = 1.22 λ multiplied by the (f-number)

Re-JOIN HERE

The smallest distance you can separate two details (x) depends on f-number, in fact the lower the f number (wider the lens aperture) the smaller the detail you can separate.

NB When we told f/8 is a ‘sweet spot’ – that is because there is a balance with other things such as aberrations in an imperfect lens.

To get some figures from this: suppose we set a lens at f/8 and use a typical wavelength of 550nm (nanometres) for green light in the region of the spectrum to which our eyes are most sensitive

Gives x = 1.22 x 8 x 550 x 10^-9   = 5.4 μm  ie 5.4 millionths of a metre (or microns - μm)  with about 4 μm and 6.5 μm for blue and red light respectively).

**Which implies that if a sensor site is made any smaller than 5.4 μm it does not increase resolution.

 Eg. For a Nikon D3  sensor sites are 8.45 μm across and for the D3x they are 5.94 μm  and for the D300 5.54 μm  getting close to a theoretical maximum resolution.

This is a good first approximation that gives a realistic idea of limits. In the world of sensors such things as pitch - the distance apart of sensor sites (pixel) centres is also important for and smaller sensors cram them in. This affects signal to noise ratio - also important when comparing sensor sizes and sensor site (pixel) sizes.

 **There are limits to how small sensor sites can be and offer higher resolution– not just the limits of manufacture but the limits of optics.

 Moral if you know the facts and how they arise you can counter the BS (taurean faecal material) don't get led by the nose (or any other part of the anatomy) about the number of pixels...biggest is not always best.