Figure A6. Three Levels of a Pyramid.

64 x 64

128 x 128

256 x 256

A7.5. Sharpening

The purpose of sharpening an image is to emphasize edges or areas of intensity discontinuities. Since the goal of this operation can be considered opposite to that of smoothing, the techniques involved in sharpening images are theoretically opposite to those used in smoothing: smoothing entails integration and sharpening entails differentiation. Two types of image sharpening techniques are discussed in this section: differentiation and high-pass filtering.

A7.5.1. Differentiation

The computation of gradients is the most common method of differentiation of an image. The gradient operation not only extracts the local edges which represent areas in the image where grey levels are changing rapidly, but also can be used to provide information about the direction and the magnitude of the rate of increase of intensity at each point on the edge. This filtering method is described in detail in Appendix B, section B1.1.

A7.5.2. High-Pass Filtering

The use of high-pass filters is based on the distribution of information in the frequency domain as computed by the Fourier transform. Edges and other sudden changes in grey level are associated with high frequency components. Thus image sharpening involves "attenuating the low frequency components in the Fourier transform without disturbing high frequency information" [GONZA77], and in effect removing contrast information of the image while emphasizing edges.

The Laplacian operator (fig. A7) is an example of a high-pass filter applied in the spatial

domain. The result of convolving an image with the Laplacian mask is that areas of the

original image containing edge information will map into a bright and a dark edge adjacent to each other in the filtered image, while low frequency information is mapped into a mid-grey intensity [ROSEN82].

Unsharp masking is a local technique for sharpening an image by subtracting its Laplacian transformation from the original blurred image. This operation emphasizes edges while preserving the grey level information in the non-edge portions of the image.

8. Appendix B: Segmentation Techniques

Segmentation of an image occurs in the sensory processing module of Level 1. The image data is broken up into components or features which classify pixels by distinct categories [JARVIS84]. The features are extracted from filtered or enhanced images and classify pixels in an image based on similarities or differences. The classification of data points produces compressed information; the information reduction process is irreversible [BROWN86]. Two basic approaches to segmentation accomplish boundary (gradient) extraction and surface patch (region) extraction. The goal at this level in the sensory processing system is to operate directly on pixel data to measure important spatial or spectral properties in the image.

B8.1. Boundary Extraction

Methods for extracting boundaries in an image rely on detection of discontinuities in intensity. The grey level at an edge changes abruptly at the border of two adjacent regions. A local edge operator measures this change detects this change over a small spatial extent using a mathematical operation. There are basically three main classes of edge operators: mathematical gradient operators, template matching, or parametric model fitting. These boundary features take the form of either edges or corners. Corner detection will be discussed in the Level 2 Perception Processing document. The following sections provide more detail about these methods.

B8.1.1. Mathematical Gradient Operators

Gradient operators respond strongly to places in an image where the grey level changes rapidly. Digital approximations made to either the first or second partial derivatives respond numerically to intensity changes. The first order partial is a directional derivative which enables calculation of magnitude and direction of the change. The second order partial is not sensitive to direction and also responds to corners as well as edges.

The first order partial derivatives, df/dx and df/d8y, measure the rate of change of a function f in perpendicular directions. The direction of the rate of change is a linear function given

These continuous operations can be approximated using discrete difference operations. They measure horizontal and vertical changes in f across a pixel located at (x,y) in the image by:

(4 ̧ f)(x,y) = f(x+1, y+1) - f(x,y)

(4,f)(x,y) = f(x,y+1) - f(x+1,y).

[4]

[5]

Some of the most historical edge operators are numerical masks that are convolved with the image such as:

Using a 3x3 operator instead of a 2x2 operator enables greater local averaging to reduce noise. The Sobel operator includes a weighted average to combine the pixel values, which increases the response of sharp edges.

Because these gradient operators measure a response across multiple pixels, the edge detection process produces responses on both sides of the edges even if the edge is perfectly sharp. Since edges are usually slightly blurred, the operator generally produces responses with a thickness of several pixels in the gradient direction. In subsequent applications, it is often necessary to have one pixel wide edges, so non-maximum suppression is used to eliminate multiple responses in the gradient direction. Edge responses are quantized into one of

Each of these directions implies checking the edge response in a different direction within a local neighborhood to see if its gradient is a maximum. For example, an edge whose gradient direction is 45 degrees is retained if its gradient magnitude is a maximum among itself, its northeast and its southwest neighbors in an 8-connected neighborhood of pixels.

After passing an image through a bandpass filter at multiple resolutions, described by Crowley and Parker [CROWL84], a difference of low-pass transform images are formed. Peaks and rides are detected in the resulting images; a peak corresponds to a local positive maxima or negative minima in a two dimensional 8-connected neighborhood of pixels, and a ridge is similarly a maxima or minima in one dimension.

Canny [CANNY86] presents similar measures for good edge detection. He defines detection and localization criteria for edges and derives mathematical forms for these criteria. In addition, he adds the constraint that the operator must provide a single response across the width of a single edge. Good detection of a noisy step edge correspond to low probabilities of either failing to mark an existing edge point or falsely marking a non-existent edge point. This criterion is met by maximizing the signal to noise ratio:

where A is the amplitude of the input step edge. The localization of the marked edge points to their true position is given by:

To meet both objectives, the two functions are multiplied together and maximized. The probability of marking multiple edges is reduced by constraining the distance between adjacent