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5.3. VECTOR REPRESENTATION OF IMAGES

Lecture



In ch. 1 brightness, color coordinate, or some other suitable parameter describing the image, seemed to be a continuous function   5.3.  VECTOR REPRESENTATION OF IMAGES . With the help of image sampling methods, discussed in Ch. 4, a continuous image recorded at some point in time can be represented as an array of samples   5.3.  VECTOR REPRESENTATION OF IMAGES in some rectangular area   5.3.  VECTOR REPRESENTATION OF IMAGES . Often it is useful to consider this array as a matrix with   5.3.  VECTOR REPRESENTATION OF IMAGES elements:

  5.3.  VECTOR REPRESENTATION OF IMAGES , (5.3.1)

Where   5.3.  VECTOR REPRESENTATION OF IMAGES , and the indices of counts are renumbered as is customary in the theory of matrices.

To facilitate the analysis, it is convenient to go from the matrix representation of the image to the vector one, collecting the elements of the columns (or rows) of the matrix   5.3.  VECTOR REPRESENTATION OF IMAGES in one long vector [9]. Formally, this operation can be represented by an auxiliary vector.   5.3.  VECTOR REPRESENTATION OF IMAGES size   5.3.  VECTOR REPRESENTATION OF IMAGES and matrices   5.3.  VECTOR REPRESENTATION OF IMAGES defined as follows:

  5.3.  VECTOR REPRESENTATION OF IMAGES   5.3.  VECTOR REPRESENTATION OF IMAGES (5.3.2)

In this case, the matrix   5.3.  VECTOR REPRESENTATION OF IMAGES will be presented in vector form using the ordering operation

  5.3.  VECTOR REPRESENTATION OF IMAGES (5.3.3)

Vector   5.3.  VECTOR REPRESENTATION OF IMAGES highlights   5.3.  VECTOR REPRESENTATION OF IMAGES matrix column   5.3.  VECTOR REPRESENTATION OF IMAGES and matrix   5.3.  VECTOR REPRESENTATION OF IMAGES puts this column in the space reserved for   5.3.  VECTOR REPRESENTATION OF IMAGES of the vector segment   5.3.  VECTOR REPRESENTATION OF IMAGES . So the vector   5.3.  VECTOR REPRESENTATION OF IMAGES contains all matrix elements   5.3.  VECTOR REPRESENTATION OF IMAGES consecutively read in columns. Inverse vector transform operation   5.3.  VECTOR REPRESENTATION OF IMAGES to matrix   5.3.  VECTOR REPRESENTATION OF IMAGES described by the ratio

  5.3.  VECTOR REPRESENTATION OF IMAGES . (5.3.4)

Using formulas (5.3.3) and (5.3.4), it is easy to establish a connection between the matrix and vector representations of a two-dimensional array. The advantages of representing the image in a vector form are the greater compactness of the notation and the possibility of directly using the methods developed for processing one-dimensional signals. It should be noted that expressions (5.3.3) and (5.3.4) not only describe the lexicographic connection between the matrix and the vector, but also define some operators that can be used in mathematical analysis. The following sections provide numerous examples of the use of these ordering operators.


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Digital image processing

Terms: Digital image processing