Figure 3: Example of effect of digital resolution on how a circle of diameter d is represented (top: pixel length = d/3, lower left: pixel length = d/7, lower right: pixel length = d/17).
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To our eye's perception, an artist painting a watercolor or oil picture makes an analog picture, although it is actually finely divided at the scale of individual, overlapping paint pigment particles. A digital image is a collection of individual, non-overlapping elements or pixels that have distinct intensities (gray scale or color) indicating the solid and pore phases of the material. The spatial resolution of the image indicates the size of the pixels, with high resolution meaning a small pixel is used. As the pixel size goes down, the number of pixels per unit length goes up, hence the designation "high." A digital image can be a gray scale image, where the intensity of each pixel ranges from black (0) to white (N). For many imaging systems (microscopes, etc.) N = 255, corresponding to 8 bits of intensity resolution.
A digital image can also be a color image, where each pixel contains three values, say from 0 to 255, for red, green, and blue, forming 24 bits of color resolution. For porous materials, if the solid part is a uniform material, all a digital image needs is 1 bit per pixel, where pore is black (0) and solid is white (1), or vice versa. The importance of digital images in science, as opposed to analog paintings, is that digital images allow quantitative analysis. Old-fashioned photographs and videos also have to be digitized before analysis. Modern digital cameras and scanning and transmission electron microscopes can produce digital images directly.
Usually a rectangular array of square pixels is used in 2-D, although other shapes, like a triangular lattice of hexagonal pixels, is also possible and can be useful for special applications [12]. Actually any area-filling collection of random shapes, on a random lattice, could be used to make a digital image. Requiring that the pixels have uniform shape restricts us to having them be a unit cell of one of the five Bravais lattices in 2-D [13]. The further requirement that the pixels be equilateral forces the choice of square lattices of square pixels and triangular lattices of hexagonal pixels. For the rest of this chapter we will discuss only digital images made from square pixels, and in 3-D, cubic lattices made up of cubic pixels.
In 2-D, digital images, at sufficient spatial resolution, portray areas well. Figure 3 shows the same physical size circle, but digitized at higher and higher resolutions. The real circle is centered on the middle of a pixel. If the circles were to be centered on a pixel corner, the digitized image would look slightly different, with no significant changes. The image appears more circular as the resolution increases. Simple calculations show that when 15 or more pixels are used per circle diameter, then the error in the area is always less than one percent [14].
Figure 3: Example of effect of digital resolution on how a circle of diameter d is represented (top: pixel length = d/3, lower left: pixel length = d/7, lower right: pixel length = d/17).
However, the perimeter of a curved surface is usually off by a large amount,
no matter
what the resolution. In Fig. 3, it is easy to see
that the perimeter of
a digital circle, P, obtained by counting pixel edges,
is given by P = 8 r, not P = 2 
r [15]. In the same
way, for a 3-D digital image, where the pixels (or voxels) are now cubes,
volumes are well-represented at high resolutions, but the surface area
of a sphere, obtained by
counting pixel faces, is always
approximately 6 π r2,
not 4 π r2.
These corrections must be kept
in mind when trying to analyze pore surfaces
based on digital images [16].
Another important issue in analyzing digital images of random porous materials is the ratio of image size to pore size. To get statistically meaningful results, the image must sample a representative area of the porous material. A more rigorous way of stating this can be formulated using the porosity. For a random porous material, the measured porosity will vary from image to image due to the randomness of the material. The smaller the image compared to the average length scale of the pores, the greater this fluctuation will be [17]. If the size of the images used is such that this fluctuation from image to image is small enough, then the image is considered to be large enough to be representative [17]. The terms "small enough" and "large enough" are defined for the application at hand. A rough rule of thumb is that the image should be 5-10 times the typical pore size.
Next: Geometrical and topological Up: Introduction to porous Previous: Porous materials