Low Pass Filtering Spatial frequency is a measure of how rapidly brightness or colour varies as we traverse an image. Figure 7.11a shows that an image containing low frequency in which grey level varies slowly and smoothly are characterised solely by components with low spatial frequencies. Figure 7.11b is a noisy images with high frequency components. It contains sudden grey level transitions & fine detail or strong texture with high spatial frequencies.

Fig Grey level variations & spatial frequency. (a) Image containing low frequency variation only. (b) Noisy image with high frequency component. © Grey level profile along the white marks in (a) & (b) (a)(b)

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A low pass filter allows low spatial frequencies to pass unchanged, but suppresses high frequencies. The low pass filter smoothes or blurs the image. This tends to reduce noise, but also obscures fine detail. High pass filter preserves sudden variations in grey level, such as those that occur at the boundaries of objects, but suppresses the more gradual variations. It can have the adverse effect of making noise more prominent, because noise has a strong high frequency component.

Any kernel whose coefficients are all positive will act as a low pass filter. For example, the 3 x 3 and 5 x 5 kernels

Now it becomes clear what these kernels do; pixel values from the neighbourhood are summed, and the sum is divided by the number of pixels in the neighbourhood. Convolution with these kernels is therefore equivalent to computing the mean grey level over the neighbourhood defined by the kernel. For this reason, these kernels are sometimes described as mean filters.

An example of noise reduction by low pass filtering is given in Figure The filtering operation has suppressed, but has not eliminated, the noise. It has also blurred the objects of interest, making their edges less well defined.

Fig Noise reduction by low pass filtering. (a) Image corrupted by 1% impulse noise. (b) Result of applying 5 x 5 mean filter (a)(b)

Figure 7.13 gives another example of low pass filtering. Here, we see that larger kernels produce more pronounced smoothing. A high degree of smoothing can also be achieved through repeated application of a small kernel to an image.

(a) (b) © Fig Effect of kernel size on smoothing. (a) Original image. (b) Result of applying a 5 x 5 mean filter. © Result of applying a 15 x 15 mean filter.

High Pass Filtering High pass filtering is accomplished using a kernel containing a mixture of positive and negative coefficients. An omnidirectional high pass filter ‑ that is, one whose response is the same, whatever the direction in which grey level varies ‑ should have positive coefficients near its centre and negative coefficients in the periphery of the kernel.

The classic 3 x 3 high pass filter kernel is

The sum of the coefficients in this kernel is zero. This means that, when the kernel is over an area of constant or slowly varying grey level, the result of convolution is zero or some very small number. However, when grey level is varying rapidly within the neighbourhood, the result of convolution can be a large number. This number can be positive or negative, because the kernel contains both positive and negative coefficients. We therefore need to choose an output image representation that supports negative numbers.

If we wish to display or print the filtered image, we must map the pixel values onto a 0 ‑ 255 range. This is usually done in such a way that a filter response of 0 maps onto the middle of the range. Thus, negative filter responses will show up as dark tones, whereas positive responses will be represented by light tones. This can be seen in Figure 7.15.

(a) (b) Fig High Pass Filtering. (a) Original image. (b) Result of high pass filtering with 3 x 3 kernel.

High-boost filtering We can compute a weighted sum of the original image and the output from a high pass filter. The result is an image in which high spatial frequencies are emphasised relative to lower frequencies. The degree of emphasis achieved depends on the weighting given to the original and high pass filtered images. This 'high boost filter' can be used to sharpen an image.Note that we can perform high boost filtering in a single convolution operation, using the kernel

When the central coefficient, c, is large, convolution will have little effect on an image. As c gets closer to 8, however, the degree of sharpening increases. If c = 8 the kernel becomes the high pass filter described earlier. Figure 7.16 gives examples of high boost filtering with two different values of c.

(a) (b) Fig.7.16 High boost filtering of the image in Fig 7.13(a). (a) Result of filtering with central coefficient c = 12. (b) Result of filtering with c = 9