Friday 19 June 2015

Forensics - Lambert's Cosine Law & Related Factors

While recently posting on Grey Scales and Gulls I referred to an earlier posting which I had made on Lighting and Perspective.  Writing on these kinds of topics I had arrived at the conclusion fairly quickly that lighting in a scene is influenced by the varying angles created between the light source, subjects and camera. I had already demonstrated this through experimentation using identical targets on a grid.  I hadn't actually explored some of the science behind it before now, and it is complex.  The principals are explained in part in Illumination Fundamentals, by the Lighting Research Center. 

Obviously birds are highly variable but occasionally we may be interested in comparing specific tones between multiple birds in the same image.  There are a few things to consider.  Firstly we have Lambert's Cosine Law which states that the illuminance (surface illumination, lumens/m2) falling on any surface depends on the cosine of the light's angle of incidence.  I have already used this principal in another posting establishing the direction to a light source in a photograph (HERE).  Because surface illuminance is highest when a surface faces the light source if we find the brightest point on a surface of our subject and draw a perpendicular line from that point (called a surface normal) the direction of that line indicates the direction to the light source (all else being equal).  For more on that subject please refer to that post.

However our digital image is formed not by the light illuminating the surface but by the light reflecting from it and entering our camera.  So we need to talk about reflectance.  Light hitting a surface is either reflected perfectly in one direction (specular reflection i.e. a mirror image) or it is scattered in numerous directions (Diffuse Reflection, i.e. most surfaces).  In the case of a perfect diffuser, or Lambertian Surface light is scattered equally in all directions.  Ideally our subjects would all have Lambertian surfaces.  That way we wouldn't have to concern ourselves with the relative angles between our subjects as they would look much the same from any angle.  But things obviously are not so simple.

While birds feathers tend to be quite matt in surface texture they are not Lambertian and therefore tones are influenced by both incident light direction and the angle of reflectance to the camera.  Beaks are often glossy in texture and even less ideally suited for direct comparison purposes.

Due to these principals we need subjects to adopt similar posture relative to the sun and the camera in order for us to make a meaningful comparison between them.  This doesn't happen very often in the field but gulls and other birds often face the same direction in a breeze at a tidal roost giving us some chance.  We still have a problem aligning our relative angles but there is a solution to this.

We can improve our chances of comparing surfaces meaningfully by closing the angles between our subjects.  As we get closer to our subjects our angle of view is increased, and conversely, the further away we are the narrower the angle of view.  Hence the digiscoped image of gulls above with it's extremely low angle of view (<<1 degree) offers a better comparison of mantle colour tones than an equivalent image taken with a 50mm lens kneeling right beside these birds.

Calculating angle of view is not difficult provided the lens is rectilinear (i.e. not spatially distorted, such as a macro or fish-eye lens).  The equation is angle of view = 2 arctan d/2f where d is a dimension of the film sensor (vertical or horizontal) and f is the effective focal length of the lens.

I worked out these values for my camera and lens configuration and created a map for the angle of view which I can slot an image into, so now I can roughly gauge the angular distance between birds in my images.  The Laughing Gull (Larus atricilla) below is just off centre.  The Herring Gull (Larus argentatus argenteus) to it's left is approx. half a degree separated and the Black-headed Gull (Larus ridibundus) flying to it's right is 1 degree separated.  The adult Herring Gull standing on the harbour wall, at a different angle relative to the camera is just under 1.5 degrees separated from the Laughing Gull.  Obviously this tool is only really possible with a fixed lens as it can be hard to calibrate with a zoom lens or a digiscoping arrangement where both the zoom setting of the camera and scope are liable to change.

One further note of caution with all of this of course is vignetting.  Vignetting includes the 'cosine fourth' law (natural vignetting).  It also includes for instance to lens design (optical vignetting), often a feature of digiscoping as in the image above.  Vignetting adds to the difficulty and complexity of lighting in digital images.

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