Summary of "Digital Image Processing" Chapter 2: Digital Image Fundamentals

Chapter 2: Digital Image Fundamentals

2.1 A Simple Image Formation Model

　　As we know, a digital image can be represented as f(x,y), where x and y are the spatial coordinates. The function f(x,y) may be characterized by two components: (1) the amount of source illumination incident on the scene being viewed, and (2) the amount of illumination reflected by the objects in the scene. Thus we can write:

　　 f(x,y)=i(x,y)r(x,y), where i(x,y) is the illumination components and 0<i(x,y)<∞, and r(x,y) is the reflectance components and 0<r(x,y)<1.

2.2 Image Sampling and Quantization

　　Sampling is the process using a sensor to get images in discrete positions, while quantization, usually right behind sampling is the process to turn the data obtained in the sampling process into permissive discrete values.

　　In fact, here because we use digital variables, which is discrete, to represent the analogous information in the real world, we have to turn these analogous information into discrete form, and that's what sampling and quantization do. However, sampling turns the x and y coordinates to discrete, while quantization turns f to discrete.

2.3 Spatial and Intensity Resolution

　　Spatial resolution is a measure of the smallest discernible detail in an image, which can be stated quantitatively in a number of ways, with line pairs per unit distance and dots(pixels) per unit distance being amoung the most common measures. Dots per unit distance is a measure of image resolution used commonly in the printing and publishing industry, usually expressed as dots per inch(dpi). To be meaningful, measures of spatial resolution must be stated with respect to spatial units.

　　Intensity resolution similarly refers to the smallest discernible change in intensity level, and it is common practice to refer to the number of bits used to quantize intensity as intensity resolution.

2.4 Image Interpolation

　　Interpolation is the process of using known data to estimate values at unknown locations, which is a basic tool used extensively in tasks such as zooming, shrinking, rotating, and geometric corrections.

　　Nearest neighbor interpolation: looking for the nearest pixel in the original image and assign its intensity to the new pixel.

　　Bilinear interpolation: using four nearest neighbors to estimate the intensity at a given location. v(x,y)=ax+by+cxy+d, where the four coefficients are determined from the four equations in four unknowns that can be written using the four nearest neighbors of point (x,y).

　　Bicubic interpolation: using sixteen nearest neighbors. v(x,y)=∑i∑j a_ijxⁱy^j, where 0<= i,j <=3.

2.5 Some Basic Relationships between Pixels

Neighbors of a pixel

　　Four horizontal and verical neighbors of p(x,y) denoted by N₄(p): (x+1,y), (x-1,y), (x,y+1), (x,y-1).

　　Diagonal neighbors of p(x,y) denoted by N_D(p): (x+1,y+1), (x+1,y-1), (x-1,y+1), (x-1,y-1).

　　8-neighbors of p(x,y) denoted by N₈(p): N₄(p)+N_D(p).

Adjacency

　　Let V be the set of intensity values used to define adjacency, then p and q are

　　　　4-adjacency if their intensity are from V and q is in the set N₄(p).

　　　　8-adjacency if their intensity are from V and q is in the set N₈(p).

　　　　m-adjacency if their intensity are from V and (i)q is in N₄(p) or (ii)q is in N_D(p) and the set  N₄(p)∩N₄(q) has no pixels whose values are from V.

Path

　　A path from pixel with coordinates (x,y) to pixel with coordinates (s,t) is a sequence of distinct pixels with coordinates (x₀,y₀), (x₁,y₁), ..., (x_n,y_n) where (x₀,y₀)=(x,y) and (x_n,y_n)=(s,t), and here 1<=i<=n.

　　In this case, n is the length of the path. If (x₀,y₀)=(x_n,y_n), the path is a closed path. We can define 4-, 8-, or m-paths depending on the type of adjacency specified.

Connectivity

　　Let S represent a subset of pixels in an image, two pixels p and q are said to be connected in S if there exists a path between them.

　　For any pixel p in S, the set of pixels that are connected to it in S is called a connected component of S.

Regions

　　If a subset of pixels in an image has one connected component, then it is called a connected set, also called a region of the image.

　　Two regions are said to be adjacent if their union also forms a region. Regions that are not adjacent are said to be disjoint. For our definition to make sense, the type of adjacency used must be specified.

Boundaries

　　The boundary(also called the border or contour) of a region R is the set of points in R that are adjacent to points outside R in the image.

　　The preceding definition is sometimes referred to as the inner border of the region to distinguish it from the outer border, which is the corresponding border in the background.

Distance Measures

　　For pixels p, q, and z, with coordinates (x,y), (s,t), and (v,w), respectively, D is a distance function or metric if

　　　　(a) D(p,q)>=0 (D(p,q)=0 iff p=q),

　　　　(b) D(p,q)=D(q,p), and

　　　　(c) D(p,z)<=D(q,p)+D(q,z).

　　Euclidean distance D_e(p,q)=[(x-s)²+(y-t)²]^1/2.

　　D₄ distance, also called the city-block distance, D₄(p,q)=|x-s|+|y-t|.

　　D₈ distance, also called the chessboard distance, D₈(p,q)=max(|x-s|,|y-t|).

2.6 Spatial Operations - Geometric spatial transformations and image registration

　　Geometric transfromations modify the spatial relationship between pixels in an image, which consists of two operations: (1) a spatial transformation of coordinates and (2) intensity interpolation that assigns intensity values to the spatially transformed pixels.

　　One of the most commonly used spatial coordinate transformations is the affine transform(Wolberg[1990]), which has the general form 

　　　　[x y 1]=[v w 1]T,

which can scale, rotate, translate, or sheer a set of coordinate points, depending on the value chosen for the elements of matrix T. (Page 110 of the textbook shows the common affine matrix.)

　　Image registration is an important application of digital image processing used to align two or more images of the same scene. Usually includes the using of tie points (also called control points), which are corresponding points whose locations are known precisely in the input and reference images.