【正文】 s considered a random variable of two dimensions, the area function is proportional to its cumulative distribution function and the gray level histogram to its probability density density function. Figure 11 An image and its graylevel histogramFigure 12 Contour lines in an image For the case of discrete functions, we fix at unity, and E.(1) bees (2)The area function of a digital image is merely the number of pixels having gray level greater than or equal to D for any gray level D. The TwoDimensional Histogram Frequently, one finds it useful to construct histograms of higher dimension than one. This is particularly useful for color images [2],as discussed in Chapter 13 shows images digitized from a microscope field containing a white blood cell and several red blood cells. The field was digitized in white light and ,through colored filters, in red and blue light. At the lower right is the twodimensional redversusblue histogram of the latter two images.The twodimensional histogram is a function of two variables: gray level in the red image and gray level in the blue image. Its value at the coordinate is the number of corresponding pixel pairs having gray level in the red image and gray level in the blue image. Recall that a multispectral digital image such as this can be thought of as having a single pixel at each sample point, but each pixel has multiple values —in this case, two . The twodimensional histogram shows how the pixels are distributed among binations of two gray levels. If the red and blue ponent images were identical, the histogram would have zero value except on the diagonal. Pixels having higher red than blue gray level, and vice versa, contribute to the histogram above and below the diagonal line, respectively. In white light, the microscope field of Figure 13 shows considerable information in color. The red blood cells appear pinkish, Thus ,the red cells appear dark in blue light, which they absorb, and light in red light, which they transmit. Similarly, the nucleus is much denser in red light. The redversusblue histogram therefore has four distinct peaks, one each due to the background ,the red blood cells , and the nucleus and cytoplasm of the white cell. The analysis of twodimensional histograms is discussed further in Chapter 21. Propertise of the HistogramWhen an image is condensed into a histogram, all spatial information is discarded. The histogram specifies the number of pixels having each gray level, but gives no hint as to where those pixels are located within the image. Thus, the histogram is unique for any particular image, but the reverse is not true: Vastly different images could have identical histograms. Such operations as moving objects around within an image typically have no effect on the histogram. The histogram does. Nevertheless ,possess some useful properties.If we change variables in Fq.(1) and integrate both sides from D to infinity, we find that (3)The area function. If we then set ,assuming nonnegative gray levels, we obtain area of image (4) Figure 13 Example twodimensional histogram (a)whitelight image。(b)redlight image。(c)bluelight image。(d)redblue histogramOr ,in the discrete case, (5)Where NL and NS are the number of rows and columns, respectively,in the image.If an image contains a single uniformly gray object on a contrasting background, and we stipulate that the boundary of that object is the contour line defined by gray level ,then area of object (6)If the image contains multiple objects. All of whose boundaries are contour lines at gray level .Then Eq.(6) gives the aggregate area of all the objects.Normalizing the graylevel histogram by dividing by the area of the image produces the probability density function(PDF) of the image. A similar normalization of the area function produces the cumulative distribution function (CDF) of the image. These functions are useful in the statistical treatment of images. As illustrated in chapter 6.The histogram has another useful property,which follows directly from its definition as the number of pixels having each gray level: if an image consists of two disjoint regions. And the histogram of each region is known, then the histogram of the entire image is the sum of the two regional histograms. Clearly, this can be extended to any number of disjoint regions. USES OF THE HISTOGRAM Digitizing Parameters The histogram gives a simple visual indication as to whether or not an image is properly scaled within the available range of gray levels. Ordinarily, a digital image should make use of all or almost all of the available gray levels, as in Figure 11. Failure to do so increases the effective quantizing interval. Once the image has been digitized to fewer than 256 gray levels, the lost information cannot be restored without redigitizing.Likewise, if the image has a greater brightness range than the digitizeris set to handle, then the gray levels will be clipped at 0 and/or 255. Producing spikes at one or both ends of the histogram. It is a good practice routinely to review the histogram when digitizing. A quick check of the histogram can bring digitizing problems into the open before much time has been wasted. Boundary Threshold Selection As mentioned earlier,contour lines provide an effective way to establish the boundary of a simple object within an image. The technique of using contour lines as boundaries is called thresholding . The use of optimal techniques for selecting threshold gray levels is a subject of considerable discussion in the literature and is treated in Chapter 18.Suppose an image contains a dark object on a light background. Figure 14 illustrates the appearance of the histogram of such an image. The dark pixels inside the object produce the rightmost peak in the histogram. The leftmost peak is due to the large number of gray levels in t