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Digital Image Processing (2000)

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3.3 Image Histogram

An important concept common to many image enhancement operations is that of a histogram, which is simply a count (or relative frequency, if normalized) of the gray levels in the image. Image enhancement by histogram modeling is a useful and powerful technique. Analysis of the histogram gives useful information about image contrast. Image histograms are important in many areas of image processing, most notably compression, segmentation, and thresholding.

Histogram functions.

This loads the package.
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This loads an example image.
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Here we calculate the histograms of the three color channels in the example image.
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This displays the histograms of each of the color channels using the default colors.
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The histogram is an approximation of the probability density function of a random variable whose realization is the particular set of pixel values found in the image. The corresponding cumulative probability distribution functions are given by the following.
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Second-order histograms.

Higher-order histograms have also been found useful in image analysis applications. A second-order histogram is an estimate of the joint probability density function of pairs of pixels. The pairs may originate from different locations in the same grayscale image or from the same location in two channels of a color or multispectral image. In the former case we get what is commonly called a co-occurrence matrix (also known as the gray-tone spatial dependence matrix), in the latter the joint histogram of a pair of color channels. In a co-occurrence matrix the value at the coordinate {i,j} is the number of pixel pairs having gray level i at position {x,y} and gray level j at position {x,y}+v, where v is some fixed displacement vector. In a joint histogram, the value at position i, j is the number of pixels with value i in one channel and value j at the same location in the other. For an 8-bit image (256 luminance levels), the dimensionality of the resulting histogram matrix is 256 × 256, which may be prohibitive for many applications. For this reason, prior to processing, pixel values are typically quantized to reduce the number of levels.
Here we show two concurrence matrices for an image with a strong diagonal pattern. This reads and quantizes the data.
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The separation vectors v1={3,3} and v2={3,-3} reveal the diagonal nature of the image by returning matrices with strong peaks on the main diagonals.
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Joint histograms have found many applications including image retrieval from image databanks, image registration, and color-based segmentation to name a few. Peaks in the histogram indicate color combinations that occur in large areas of the image. Analysis of such histograms may reveal information about dominant colors in the image and may serve as a starting point for color-based segmentation algorithms. Here we calculate the pairwise joint histograms of an example color image. First we load and quantize the image.
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This splits the image into individual channels.
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This calculates the joint histograms.
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This shows the result.
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As in the case of the first-order histogram, it is sometimes necessary to compute the distribution function given the density function. Here are the corresponding cumulative histograms, namely the estimates of the joint probability distributions of the pairs of colors in the image.
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