A Shadow's Penumbra

Go outside and stand with your back towards the sun and look at your shadow that is cast onto the sidewalk. You will notice that the shadow near your feet has a sharp boundary, while the shadow near your head is fuzzy. This simple observation regarding the expected shape of a shadow can be used to detect fake shadows in an image.

Shown below is a scene in which a cylinder is being illuminated by a spotlight and is casting a shadow onto the ground. The middle part of the shadow which receives no light from the spotlight is called the umbra. The boundary of the shadow which receives a little bit of light from the spotlight, but is still partially in shadow, is called the penumbra. Notice how the width of the penumbra near the base of the cylinder is narrow and that it continues to widen proportional to the distance from the cylinder base.

The reason for this difference in the shadow’s penumbra is illustrated on the right. The illustration shown below is a top view of the scene shown above, where the white semi-circle is the spotlight illuminating the scene. For two points on the penumbra, one near and one far from the cylinder base, I’ve drawn in yellow the portion of the light that is illuminating the penumbra. Far from the base quite a bit of light is reaching the penumbra as compared to near the base. This difference is what causes the change in the width of the penumbra.

The width of the penumbra also depends on the size of the illuminating light. A small point light source (a flash) will not cause much change in the penumbra. The larger the light source the larger the change in the penumbra — this is why the effect is particularly striking outdoors where objects are illuminated by the sun.

The angle at which you view the surface onto which the shadow was cast will of course also impact the shadow’s appearance — if possible, perform planar rectification in order to remove this perspective distortion.

When an object is inserted into an image, it is often necessary to generate a shadow. Careful attention must be paid to creating a shadow with a consistent shape and color. Most editors will mistakenly create the shadow’s penumbra with a fixed width, particularly if the shadow is created by just darkening an existing surface. 

In an image forensic setting, the shape of the penumbra can be examined to make sure that it increases in width proportional to the distance from the base of the object, that it is consistent with other objects in the image, and that it is consistent with the expected size of the light source.