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Davies: Computer Vision, 5th edition: Solutions to selected problems 34
© E. R. Davies 2017
This gives the disparity D:
D = x1 – x2 = bf/Z
Z = bf/D
16.7
From the main text in Chapter 16, the standard formulae for depth from disparity are:
f/Z = x1/(X + b/2) = x2/(X – b/2)
D = x1 – x2 = [(X + b/2) – (X – b/2)] f/Z = bf/Z
16.8
(a) A simple but clearly drawn figure will show immediately that the ordering A1, B1,
C1, D1 in the first image is the same A2, B2, C2, D2 as in the second image, and the same
A, B, C, D as on the object—assuming that all these points are visible in both images. For
(b) Such a response becomes increasingly likely as the number of observed features
increases. Obviously this is ultimately the same effect as parallax. Note that ordering is
sufficient to show what is going on, but measurement of disparity is still required to gauge
the relative depths.
Davies: Computer Vision, 5th edition: Solutions to selected problems 35
16.9
(a) These terms are bookwork (see the main text in Chapter 16). Ambiguities arise
because one must infer what’s happening in 3-D from limited 2-D data. This requires
assumptions which may not be justified.
Lambertian surfaces have reflectivities which depend only on the angle of incidence
i. Furthermore, the variation is as cos i. This is incredibly simple leading to the surface
A third cone is needed to make the surface normal completely unambiguous. If the
absolute reflectance R of the surface is unknown, it might reasonably be thought that the
cones must be grown until they lead to a single solution, as above. However, the first
solution need not be unique. To disambiguate the situation, one might think that at least
four cones would be needed, but this is not so: e.g., in the case of three lights in mutually
perpendicular directions, the sums of the squares of the cosines of the angles relative to
(b) Photometric stereo (as above) leads hopefully to a unique set of surface orientations.
These must be integrated over space to obtain depth maps. Hence binocular vision is to be
16.10
(a) Matte surfaces reflect light diffusely and ideally do not give any purely specular
components of reflection. The best model of matt surfaces is the Lambertian model,
(b) As the light reflected by a Lambertian surface depends only on cos i, for a given
reflected light intensity I, we will only know that the angle of incidence is given by
Davies: Computer Vision, 5th edition: Solutions to selected problems 36
(c) With three lights, each surface normal must lie in each of three cones having a
common apex: two cones intersect in two lines, and the third cone, in general position,
will narrow this down to a single line, giving a unique interpretation and estimate of the
direction of the surface normal. If any points are hidden from any of the lights, an element
Case of two lights: situation for three or four lights is similar.
(d) The surface map that is obtained is an orientation map, which distinguishes it from
that for binocular vision which leads to a depth map (though in principle integrating an
orientation map will yield a depth map, after much computation). The two approaches are
best applied to different situations: binocular vision is best applied to textured or highly
(e) Either of these approaches only provides information about surface shape.
Recognition requires further processing to identify likely candidates for specific objects:
