The Gray-Level Histogram
1.1 INTRODUCTIO N
One of the simplest and most uful tools in digital image processing is the gray-level histogram. The function summarizes the gray-level content of an image. While the histogram of any image contians considerable information,certain types of images are completely specified by their histograms. Computation of the histogram is simple and may be done at little apparent cost when an image is copied from one place to another.
1.1.1 Definition
The gray-level histogram is a function showing,for each gray level,the number of pixels in the image that have that gray level. The abscissa is gray level and the ordinate is frequency of occurrence(number of pixels). Figure 1-1 shows an example.
羽毛球用英语怎么说There is another way to define the gray-level histogram[1],and the following exerci yields insight into the ufulness of this function. Suppo we have a continuous image,defined by the function ()y x D ,,that varies smoothly from high gray level at the center to low gray level at the borders. We can lect 亚瑟连招>种植什么赚钱
some gray level 1D and define a t of contour lines connecting all points in the image with value 1D . The resulting contour lines from clod curves that surround regions in which the gray level is greater than or equal to 1D .
Figure 1-2 shows an image containing one contour line at the gram level 1D .A cond contour line has been drawn at a higher gray level 2D .
1A is the area of the region inside the first contour line, and similarly . 2A is the area inside the cond line.
The threshold area function ()D A of a continuous image is the area enclod by all contour lines of gray level D. Now the histogram may be defined as )(-ΔD ΔD)A(D -A(D)
lim )(0ΔD D A dD d
综合单价怎么算D H =+=→
(1)
Thus, the histogram of a continuous image is the negative of the derivatives of its
area function. The minus sign results from the fact that ()D
A decreas with increasing D. If the image is 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 1-1 An image and its gray-level histogram
Figure 1-2 Contour lines in an image
For the ca of discrete functions, we fix ΔD at unity , and E.(1) becomes ()1)()(+-=D A D A D H (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.
1.1.2 The Two-Dimensional Histogram
Frequently , one finds it uful to construct histograms of higher dimension than one. This is particularly uful for color images [2],as discusd in Chapter 21.figure 1-3 shows images digitized from a microscope field containing a white blood cell and veral 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 two-dimensional
red-versus-blue histogram of the latter two images.
The two-dimensional 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 ()B R D D , is the number of corresponding pixel pairs having gray level R D in the red image and gray level B D 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 ca, two . The two-dimensional histogram shows how the pixels are distributed among combinations of two gray levels. If the red and blue component images were identical, the histogram would have zero value except on the 。45 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 1-3 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 denr in red light. The red-versus-blue histogram therefore has four distinct peaks, one each due to the background ()B ,the red blood cells ()R , and the nucleus ()N and cytoplasm ()C of the white cell. The analysis of two-dimensional
histograms is discusd further in Chapter 21.
1.1.3 Properti of the Histogram
When an image is condend 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 tho pixels are located within the image. Thus, the histogram is unique for any particular image, but the rever 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 ,posss some uful properties.
来约我吧If we change variables in Fq.(1) and integrate both sides from D to infinity , we find that
⎰∞
威海旅行∞=-=0)()]([)(D A P A dP P H D
(3) The area function. If we then t 0=D ,assuming nonnegative gray levels, we obtain
=⎰∞
dP P H 0)(area of image
(4)
Figure 1-3 Example two-dimensional histogram
4k美女壁纸(a)white-light image;(b)red-light image;(c)blue-light image;(d)red-blue histogram Or ,in the discrete ca,
∑=⨯=2550
)(D NS NL D H
(5) Where NL and NS are the number of rows and columns, respectively ,in the image.
野生眉图片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 1D ,then
