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The fractal dimension is a useful feature for texture characterization Estimation of D from an image is rather difficult because natural textures do not strictly follow the deterministic repetitive model of fractals assumed above, but have statistical variations

from crushed stone, gravel or air-cooled iron blast-furnace slag that is specifically processed to ensure suitable gradation Shape of the sand particles is important, because excessive flat and elongated sand particles result in a mortar with reduced workability Gradation of the sand is also important When the sand consists of particles of uniform size, whether that size is fine or coarse, the quantity of cementitious matrix required to coat each particle is greater than the quantity of cementitous matrix required to coat each particle of sand that is comprised of variable sized particles This phenomenon occurs because uniformly sized particles do not fit together as tightly as variable sized particles, as illustrated in Figure 22 The ASTM-required gradation of sand for use in masonry mortar is shown in Table 25 The standard explains that this gradation requirement may not be appropriate when the mortar joints exceed 1 2 inch in thickness In that case, coarser aggregate conforming to the fine aggregate in ASTM C 404 Standard Specification for Aggregate for Masonry Grout is more suitable If the sand that is proposed to be used does not meet the gradation requirements shown in Table 25, it may still be used provided that the mortar so prepared complies with the property requirements of the ASTM standard for mortar Those requirements

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Variations in the size, shape, and density of texture primitives provide clues for estimation of surface shape and orientation These are exploited in shapefrom-texture methods to recover 3-D information from 2-D images As an illustration, consider the regular ordered texture shown in Figure 73(a) slanted

at an angle a such that the top of the surface is farther away from the camera than the bottom; for simplicity, let us assume that all points along a given horizontal line are at the same depth from camera (ie, there is no tilt) This is illustrated in Figure 74(a) The corresponding image captured is shown in Figure 75 Note that the discs now appear as ellipses, which is a clue that the surface is not parallel to the image plane The sizes of these ellipses decrease as a function of y' in the image plane In other words, there are more ellipses for a unit area in the image plane near the top of the image compared with the center, resulting in a density gradient Furthermore, the aspect ratio (ratio of minor to major diameters of an ellipse) does not remain constant, resulting in an aspect ratio gradient [36] To show this, we first derive an expression for the major and minor diameters of the ellipse as a function of the slant angle and the position of the ellipse in the image plane Let the diameter of the disc be d Consider the disc at the image center The major diameter of the ellipse in the image plane corresponding to this disc is given by the perspective projection equation

(78)

39 310 311 312 313

where z is the distance of the disc from the camera center and f is the focal length of the camera The minor diameter of this ellipse is affected not only by the perspective projection but also by the foreshortening effect due to the slant angle a This is givenby the equation

dminor(O,O)

(79)

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