## Pixel Operations

In this section we present methods of image enhancement that depend only upon the pixel gray level and do not take into account the pixel neighborhood or whole-image characteristics.

3.1 Compensation for Nonlinear Characteristics of Display or Print Media

Digital images are generally displayed on cathode ray tube (CRT) type display systems or printed using some type of photographic emulsion. Most display mechanisms have nonlinear intensity characteristics that result in a nonlinear intensity profile of the image when it is observed on the display. This effect can be described succinctly by the equation e(m, «) = C(/(m, «)), where /(m,«) is the acquired intensity image, e(m,«) represents the actual intensity output by the display system, and C() is a nonlinear display system operator. In order to correct for the nonlinear characteristics of the display, a transform that is the inverse of the display's nonlinearity must be applied [14,16].

g(m, n) =f (m, n), where T() is a nonlinear operator which is approximately equal to C(), the inverse of the display system operator, and g(m, n) is the output image.

Determination of the characteristics of the nonlinearity could be difficult in practice. In general, if a linear intensity wedge is imaged, one can obtain a test image that captures the complete intensity scale of the image acquisition system. However, an intensity measurement device that is linear is then required to assess the output of the display system, in order to determine its actual nonlinear characteristics.

A slightly exaggerated example of this type of a transform is presented in Fig.1. Figure 1a presents a simulated CRT display with a logarithmic characteristic. This characteristic tends to suppress the dynamic range of the image decreasing the contrast. Figure 1b presents the same image after an inverse transformation to correct for the display nonlinearity. Although these operations do in principle correct for the display, the primary mechanism for review and analysis of image information is the human visual system, which is fundamentally a nonlinear reception system and adapts locally to the intensities presented.

### 3.2 Intensity Scaling

Intensity scaling is a method of image enhancement that can be used when the dynamic range of the acquired image data significantly exceeds the characteristics of the display system, or vice versa. It may also be the case that image information is present in specific narrow intensity bands that may be of special interest to the observer. Intensity scaling allows the observer to focus on specific intensity bands in the image by modifying the image such that the intensity band of interest spans the dynamic range of the display [14, 16]. For example, if f1 and f2 are known to define the intensity band of interest, a scaling transformation may be defined as e = I f A < f < f2

where e is an intermediate image, g is the output image, and fmax is the maximum intensity of the display.

These operations may be seen through the images in Fig. 2. Figure 2a presents an image with detail in the intensity band from 90 to 170 that may be of interest to, for example a gum specialist. The image, however, is displayed such that all gray levels in the range 0 to 255 are seen. Figure 2b shows the histogram of the input image and Fig. 2c presents the same image with the 90-to-170 intensity band stretched across the output band of the display. Figure 2d shows the histogram of the output image with the intensities that were initially between 90 and 170, but are now stretched over the range 0 to 255. The detail in the narrow band is now easily perceived; however, details outside the band are completely suppressed.

FIGURE 1 (a) Original image as seen on a poor-quality CRT-type display. This image has poor contrast, and details are difficult to perceive — especially in the brighter parts of the image such as in areas with high tooth density or near filling material. (b) The nonlinearity of the display is reversed by the transformation, and structural details become more visible. Details within the image such as the location of amalgam, the cavity preparation liner, tooth structures, and bony structures are better visualized.

FIGURE 2 (a) Input image where details of interest are in the 90-to-170 gray level band. This intensity band identifies the bony structures in this image and provides an example of a feature that may be of dental interest. (b) Histogram of the input image in (a). (c) This output image selectively shows the intensity band of interest stretched over the entire dynamic range of the display. This specific enhancement may be potentially useful in highlighting features or characteristics of bony tissue in dental X-ray imagery. This technique may be also effective in focusing attention on other image features such as bony lamina dura or recurrent caries. (d) Histogram of the output image in (c). This histogram shows the gray levels in the original image in the 90-to-170 intensity band stretched over 0 to 255.

FIGURE 2 (a) Input image where details of interest are in the 90-to-170 gray level band. This intensity band identifies the bony structures in this image and provides an example of a feature that may be of dental interest. (b) Histogram of the input image in (a). (c) This output image selectively shows the intensity band of interest stretched over the entire dynamic range of the display. This specific enhancement may be potentially useful in highlighting features or characteristics of bony tissue in dental X-ray imagery. This technique may be also effective in focusing attention on other image features such as bony lamina dura or recurrent caries. (d) Histogram of the output image in (c). This histogram shows the gray levels in the original image in the 90-to-170 intensity band stretched over 0 to 255.

### 3.3 Histogram Equalization

Although intensity scaling can be very effective in enhancing image information present in specific intensity bands, often information is not available a priori to identify the useful intensity bands. In such cases, it may be more useful to maximize the information conveyed from the image to the user by distributing the intensity information in the image as uniformly as possible over the available intensity band [3, 6, 7].

This approach is based on an approximate realization of an information-theoretic approach in which the normalized histogram of the image is interpreted as the probability density function of the intensity of the image. In histogram equalization, the histogram of the input image is mapped to a new maximally-flat histogram.

As indicated in Section 2, the histogram is defined as h(i), with 0 to P — 1 gray levels in the image. The total number of pixels in the image, M*N, is also the sum of all the values in h(i). Thus, in order to distribute most uniformly the intensity profile of the image, each bin of the histogram should have a pixel count of (M * N)/P.

It is, in general, possible to move the pixels with a given intensity to another intensity, resulting in an increase in the pixel count in the new intensity bin. On the other hand, there is no acceptable way to reduce or divide the pixel count at a specific intensity in order to reduce the pixel count to the desired (M * N)/P. In order to achieve approximate uniformity, the average value of the pixel count over a number of pixel values can be made close to the uniform level.

A simple and readily available procedure for redistribution of the pixels in the image is based on the normalized cumulative histogram, defined as

The normalized cumulative histogram can be used as a mapping between the original gray levels in the image and the new gray levels required for enhancement. The enhanced image g(m, n) will have a maximally uniform histogram if it is defined as g(m, n) = (P — 1) • H(f (m, n)).

Figure 3 a presents an original dental image where the gray levels are not uniformly distributed, while the associated histogram and cumulative histogram are shown in Figs 3b and 3c, respectively. The cumulative histogram is then used to map the gray levels of the input images to the output image shown in Fig. 3d. Figure 3e presents the histogram of Fig. 3d, and Fig. 3f shows the corresponding cumulative histogram. Figure 3f should ideally be a straight line from (0,0) to (P — 1, P — 1), but in fact only approximates this line to the extent possible given the initial distribution of gray levels. Figure 3g through 1 show the enhancement of a brain MRI image with the same steps as above.

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