Wednesday, 24 June 2015

Forensics - 'Cosine Fourth' Law of Illumination Falloff

Vignetting is the falloff of illumination in an optical system towards the edge of the image.  Birders who digiscope would be very familiar with Optical Vignetting which is caused when optical elements obstruct one another along the path to the sensor, and it can be very obvious for instance as a dark surround to the image if we fail to properly align the camera or phone with the scope eyepiece.   There are other causes of vignetting and one of them, referred to as Natural Vignetting or natural illumination falloff, is caused by a property of light and optics referred to as the 'Cosine Fourth' Law of Illumination Falloff.  

I have already explored Lambert's Cosine Law which makes the direct correlation between illuminance at a point on a surface and the angle of incident light hitting that point.  The cosine fourth law applies a similar principal and states that as the angle of light hitting a sensor or film plane diverges from the centre outwards (i.e. diverging from perpendicular to the plane) illuminance is reduced by a certain factor.  That factor is the fourth power of the cosine of the angle that the light makes to the perpendicular.

This has important considerations for photography, because unlike the retina of the human eye which is convex and doesn't suffer from natural vignetting, the film plane or sensor is flat, so potentially can suffer due to this problem.  As one might expect super wide-angle lenses are most prone to this problem because there is a relationship here with Angle of View.  Surprisingly the widest-angle lenses of all, fish-eye are immune to the problem due to curvilinear distortion (the same principal that applies to the human retina).

Of course like a lot of laws in optics this principal applies to a perfect system, but introduce other variables and real optical systems may behave differently.  There are various papers discussing these other variables including at THIS LINK.  In many cases it seems the degree of falloff observed in real optical systems may be more than predicted by the cosine fourth law alone.

For birders, the cosine fourth law is of virtually no significance because our lenses, be it binoculars, scope or camera all tend to have a relatively high magnification and therefore a low angle of view.  Considering that most DSLRs in mainstream birding are not full-frame cameras and therefore crop the edges of the image, the likely impact of this law is even lower.  Nonetheless it is important to be aware that the illumination of images is never totally uniform, especially when trying to carefully calibrate, measure and compare subtle tones.

Below I have mapped the angle of view of my Canon D70 and 300mm lens and overlayed the map on an image containing various gulls.

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