GAUSSIAN SMOOTHING in Software

Printer GS1 - 12 in Software GAUSSIAN SMOOTHING

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Figure 410: The results of a 3 x 3, 5 x 5, and 7 x 7 median filter on the noisy images from Figure 45

06 04 02

than in another The property of rotational symmetry implies that a Gaussian smoothing filter will not bias subsequent edge detection in any particular direction 2 The Gaussian function has a single lobe This means that a Gaussian filter smooths by replacing each image pixel with a weighted average of the neighboring pixels such that the weight given to a neighbor decreases monotonically with distance from the central pixel This property is important since an edge is a local feature in an image, and a smoothing operation that gives more significance to pixels farther away will distort the features 3 The Fourier transform of a Gaussian has a single lobe in the frequency spectrum This property is a straightforward corollary of the fact that the Fourier transform of a Gaussian is itself a Gaussian, as will be shown below Images are often corrupted by undesirable high-frequency signals (noise and fine texture) The desirable image features, such as edges, will have components at both low and high frequencies The single lobe in the Fourier transform of a Gaussian means that the smoothed image will not be corrupted by contributions from unwanted high-frequency signals, while most of the desirable signals will be retained

4 The width, and hence the degree of smoothing, of a Gaussian filter is parameterized by (j, and the relationship between (j and the degree of smoothing is very simple A larger (j implies a wider Gaussian filter and greater smoothing Engineers can adjust the degree of smoothing to achieve a compromise between excessive blur of the desired image features (too much smoothing) and excessive undesired variation in the smoothed image due to noise and fine texture (too little smoothing) 5 Large Gaussian filters can be implemented very efficiently because Gaussian functions are separable Two-dimensional Gaussian convolution can be performed by convolving the image with a one-dimensional Gaussian and then convolving the result with the same one-dimensional filter oriented orthogonal to the Gaussian used in the first stage Thus, the amount of computation required for a 2-D Gaussian filter grows linearly in the width of the filter mask instead of growing quadratically

The rotational symmetry of the Gaussian function can be shown by converting the function from rectangular to polar coordinates Remember the two-dimensional Gaussian function (413)

= i2 + j2,

that the Gaussian function in polar coordinates, (414) does not depend on the angle () and consequently is rotationally symmetric It is also possible to construct rotationally nonsymmetric Gaussian functions if they are required for an application where it is known in advance that more smoothing must be done in some specified direction Formulas for rotationally nonsymmetric Gaussian functions are provided by Wozencraft and Jacobs [257, pp 148-171], where they are used in the probabilistic analysis of communications channels

The Gaussian function has the interesting property that its Fourier transform is also a Gaussian function Since the Fourier transform of a Gaussian is a real function, the Fourier transform is its own magnitude The Fourier transform of a Gaussian is computed by F{g(x)}

(415)

1-00

00 00

(416) (417)

1 e-2,;7 -00

1-00 e-2,;7

(418)

The Gaussian is a symmetric function and the sine function is antisymmetric, so the integrand in the second integral is antisymmetric Therefore, the integral must be zero, and the Fourier transform simplifies to:

F{g(x)} =

1-00 e-2,;7

y'2;(7e-~,

(419)

1 v2 = (7 "2 (420)

The spatial frequency parameter is w, and the spread of the Gaussian in the frequency domain is controlled by v, which is the reciprocal of the spread parameter (7 in the spatial domain This means that a narrower Gaussian function in the spatial domain has a wider spectrum, and a wider Gaussian function in the spatial domain has a narrower spectrum This property relates to the noise suppression ability of a Gaussian filter A narrowspatial-domain Gaussian does less smoothing, and in the frequency domain its spectrum has more bandwidth and passes more of the high-frequency noise and texture As the width of a Gaussian in the spatial domain is increased, the amount of smoothing that the Gaussian performs is increased, and in the frequency domain the Gaussian becomes narrower and passes less high-frequency noise and texture This simple relationship between spatialdomain Gaussian width and frequency-domain spectral width enhances the ease of use of the Gaussian filter in practical design situations The Fourier