Getting Started

Here is a quick tour of selected functions and operators in the ImageProcessing package. For a detailed discussion and additional examples, follow the links at the end of each section to the complete Digital Image Processing User's Guide documentation. All the examples shown here are taken from the User's Guide.

Image Representation

A digital image is a two-dimensional (2D) discrete signal. Mathematically, such signals can be represented as functions of two independent variables, for example, a brightness function of two spatial variables. A monochrome digital image f(x,y) is a 2D array of luminance values,

with 0≤ f(x,y) ≤ L, and typically L=255. Each element of the array is called a pel (picture element), or more commonly a pixel. Typical image dimensions are M×N= 256×256 and 512×512.

This loads the package.

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A number of example images are included with the Digital Image Processing package. They are located in the Data directory of the root ImageProcessing directory. The location on any system is given by the system variable $ImageProcessingDataDirectory. In a typical Windows installation, the path may be as follows.

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When executing ImageRead, Mathematica tries to find the file in these directory paths. ImageRead also accepts absolute or relative file paths.

This loads an example image.

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This extracts a small 8 × 8 image segment.

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Small values represent dark areas of an image, while large values represent bright pixels.

A color digital image is typically represented by a triplet of values, one for each of the color channels, as in the frequently used RGB color scheme. The individual color values are almost universally 8-bit values, resulting in a total of 3 bytes (or 24 bits) per pixel. This yields a three-fold increase in the storage requirements for color versus monochrome images. Naturally, there are a number of alternative methods of storing the image data. Most widely used are the so-called pixel-interleaved (or meshed) and color-interleaved (or planar) formats. Less frequent are row-wise or column-wise interleaving methods. In a pixel-interleaved format, every image pixel is represented by a list of three values.

In the color-interleaved format, the color information is separated into three matrices, one for each of the three color channels.

Here we read a color example image.

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Here we extract a 4 × 4 region and display it in a meshed format.

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Here the same region of the color image is displayed in a planar format.

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The RGB color scheme is just one of many color representation methods used in practice. The letters R, G, and B stand for red, green, and blue, the three primary colors used to synthesize any one of 2²⁴ or approximately 16 million colors. Equal quantities of the three color values result in shades of gray in the range {0, 255}. Other supported color models include monochrome; HSV, which stands for hue, saturation, value; and CMYK, which stands for cyan, magenta, yellow, black. The latter has found application primarily in the printing and graphics markets. HSV is useful in color image processing since it separates the color information from brightness.

Further reading:

Getting Started: Example Images.

User's Guide: Sections 2.1, 2.2, 2.3.

Function Index: ImageRead, ImageTake, PlanarImageData, RawImageData.

Structure of ImageData

The color image data structure is encapsulated in a Mathematica ImageData expression. ImageData is a graphics primitive that facilitates the easy manipulation and handling of single-channel and multichannel images. This loads the package.

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This reads in a small example image in RGB color format.

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ImageData, like other graphics primitives, may be displayed using the Graphics command.

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Here is the input form of ImageData.

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This converts the image to a planar format.

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This returns a monochrome (i.e., single-channel) image.

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Further reading:

User's Guide: Section 2.4.

Function Index: ImageRead, ToGrayLevel, ToHSVColor, PlanarImageData.

Image Display

All image data imported into Mathematica using the ImageRead command may be displayed using Show. This loads the package.

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This reads an example image.

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This displays a meshed composite color image.

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Color image data in a planar format may be treated as three separate single-color images, one for each of the color channels. The default display form of a color image in planar format is GraphicsRow.

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Raw monochrome image data may be visualized in any number of ways, such as surface plots, contour plots, and intensity plots. Here are two examples.

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Further reading:

User's Guide: Section 2.5.

Function Index: ImageRead, ImageTake, ToChannels, ToGrayLevel, PlanarImageData.

Image Processing with Point Operations

Image enhancement refers to any technique that improves or modifies the image data, either for purposes of subsequent visual evaluation or for further numerical processing. Image enhancement techniques include gray level and contrast manipulation, noise reduction, edge sharpening, linear and nonlinear filtering, magnification, pseudocoelomate, and so on. One useful, broad categorization of enhancement techniques divides them into point- and region-based operations. Point operations modify pixels of an image based on the value of the pixel. These are also called zero-memory operations. By contrast, region-based operations calculate a new pixel value based on the values in a (typically small) local neighborhood.

This loads the package.

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This loads an example color image.

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Negating an image is a simple image operation. It changes large values to small and vice versa, according to

where x is a pixel value and n=8 for the typical 8-bits-per-pixel monochrome image. For color images the same transformation is applied to the individual color values. Here we show the effect of negation on both the color and grayscale images.

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A common approach to contrast modification is to use a power-law point transformation, where each pixel of the original image is raised to a specified exponent value. By selecting the exponent values appropriately, either high or low luminance values can be boosted. A simple yet useful contrast manipulation technique is to define a piecewise linear transformation to selectively stretch and/or compress a range of luminance values. The slope of the transformation is chosen to be greater than 1 in the region of stretch and less than 1 in the region of compression.

Examples of the effect of selected point operations on the color beans image are displayed here.

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The image histogram is an estimate of the probability density of the image pixels. As such, it measures the frequency of occurrence of the pixel luminance values. Many higher-level image processing tasks require the calculation of a histogram. Here we present the histograms of each of the three color channels in the beans image.

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Amplitude thresholding is one of many segmentation techniques. Two-level or binary thresholding changes a pixel value to one of two desired values (typically 0 and 1), depending on the setting of a threshold. Here is an image segmentation example where we extract the green beans by applying the threshold operation to the individual color channels. The respective thresholds were selected from an examination of the channel histograms.

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We now find all the image regions that do not have yellow pixels and set them to black.

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Here we display the original and the segmented images.

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Further processing with morphological filters (see User's Guide, Section 6.1) may be used to clean up the segmented image.

Further reading:

User's Guide: Sections 3.2, 3.3, 3.4, 7.2.

Function Index: ImageHistogram, PlanarImageData, RegionProcessing, ScaleLinear, Threshold, Where.

Image Processing Using LSI Filters

Many useful image operations are implemented with linear shift-invariant (LSI) filters. A smoothing operation is frequently a first step in operations such as noise reduction, edge detection, and interpolation. A commonly used smoothing filter has constant coefficients. Convolving an image with such a filter has the effect of smoothing or blurring the image. The second example demonstrates the use of convolution to sharpen an image by a method called unsharp masking. Here we load the package.

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This loads an example image.

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This shows the original image.

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Here we smooth and sharpen the example image.

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Here we display the two results.

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Edge detection is an important step in many shape-based recognition tasks. Edge detection is typically implemented as a convolution operation with appropriately chosen differentiating filters. Two examples of edge detection using two common edge filters, the Sobel gradient edge detector and the Laplacian-of-Gaussian edge detector, conclude this section.

This loads a grayscale example image.

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This shows the edges in the books image, with the gradient edge detector result on the left and the Laplacian-of-Gaussian on the right.

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Further reading:

User's Guide: Sections 5.2, 5.3, 7.3.

Function Index: DiscreteConvolve, EdgeMagnitude, ImagePlus, LoGFilter, SobelFilter, Threshold, ZeroCrossing.

Region-of-Interest Processing

It is sometimes of interest to process a single subregion of an image, leaving other regions unchanged. This is commonly referred to as region-of-interest (ROI) processing. Here we show an example of edge detection in a triangular ROI using a real image.

We first load the package.

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This loads an example image.

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Here we define a region-of-interest. Image regions may be conveniently selected using the mouse and keyboard. To select a particular region use any of the interactive functions introduced in Version 2.1.

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This shows the result of applying a Sobel edge detector to the region defined by roi.

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Further reading:

User's Guide: Sections 5.6, 7.3.

Function Index: ImageRead, EdgeMagnitude, RegionProcessing, SobelFilter, PolygonSelectGUI, RectangleSelectGUI, DiskSelectGUI.

Image Transforms: DCT

It is well known that LSI operators may be implemented in the Fourier transform domain, leading to computational efficiencies. The energy compaction property of transforms, such as the discrete Fourier transform (DFT), discrete cosine transform (DCT), and discrete wavelet transform (DWT), plays an important role in many image/video compression techniques. Here we demonstrate image compression using the DCT transform.

Here we load the package.

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Here we load an example image.

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Here we take the block cosine transform of the example image. The blocks are non-overlapping, with dimensions 8 × 8.

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Here we show a fragment of the example image and the DCT coefficients.

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We now retain cosine coefficients located in a low-frequency zone of each block. We then use the inverse DCT to calculate an approximation to the original image. Here is a typical zonal mask.

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This reconstructs an approximation of the original.

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Here is the processed image and the approximation error.

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The compression capabilities of the DCT are clearly visible. Using only 23% of the image's total energy, the reconstructed image (left) is a reasonable approximation of the original. The error signal is shown on the right. The approximation is guaranteed to improve as the number of coefficients is increased.

Further reading:

User's Guide: Sections 8.4, 8.7.

Function Index: InverseDiscreteCosineTransform, BlockProcessing, DiscreteCosineTransform, RawImageData.

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