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(Extracted, with permission, from ASTM C 270 Standard Specification for Mortar for Unit Masonry, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428 A copy of the complete standard may be obtained from ASTM, phone: 610-832-9585, email: service@astmorg, website: wwwastmorg)

(933)

which says that the irradiance (brightness) at point (x, y) in the image plane is equal to the reflectance map value for the surface orientation p and q of the corresponding point on the scene surface For a Lambertian reflector and point light source, the reflectance map R(p, q) is given by Equation 932 and is shown in Figure 94

(934) (935)

The appropriate volume of aggregate, however, is based on the sum of the volumes of cement and lime (both are cementitous materials) and not on the unit volume of cement alone

The same energy is emitted in all directions; hence a surface appears to be brighter if it is slanted away from the viewer because the viewer is seeing more surface within a given angle of viewing direction

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Figure 94: Typical reflectance map, R(p, q), for a Lambertian surface illuminated by a point light source with Ps = 02 and qs = 04 Left: Gray level representation Right: Contour plot

.

Image intensity at a pixel as a function of the surface orientation of the corresponding scene point is captured in a reflectance map Thus, for fixed illumination and imaging conditions, and for a surface with known reflectance properties, changes in surface orientation translate into corresponding changes in image intensity The inverse problem of recovering surface shape from changes in image intensity is known as the shape from shading problem We now summarize the procedure for solving this problem using a surface smoothness constraint From the previous section, the relationship between image irradiance E(x, y) and the orientation (p, q) of the corresponding point on the surface is given by E(x, y) = R(p, q), (936) where R(p, q) is the reflectance map of the surface The goal is to recover surface shape by calculating the orientation (p, q) on the surface for each point (x, y) in the image Note that we have only one equation, but there are two unknowns, p and q Thus, this problem cannot be solved unless additional constraints are imposed A commonly imposed constraint is that

of surface smoothness We assume that objects are made up of piecewise smooth surfaces which depart from the smoothness constraint only along their edges A smooth surface is characterized by slowly varying gradients, p and q Thus, if Px, PY' qx, and qy represent the partial derivatives of p and q along x and y directions, we can specify the smoothness constraint as minimizing the integral of the sum of the squares of these partial derivatives as follows (937) Strictly speaking, we must minimize this integral subject to the constraint given in Equation 936 However, to account for noise which causes departure from the ideal, the problem is posed as that of minimizing total error e given by (938) where \ is a parameter which weighs the error in smoothness constraint

J J (E(x,

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