Figure 1 Conservatively smoothing a local pixel neighborhood. The central pixel of this figure contains an intensity spike (intensity value 150). In this case, conservative smoothing replaces it with the maximum intensity value (127) selected amongst those of its 8 nearest neighbors.

Figure 2 Six different structuring elements, for use in exercise 3. These local neighborhoods can be used in conservative smoothing by moving the central (white) portion of the structuring element over the image pixel of interest and then computing the maximum and minimum (and, hence the range of) intensities of the image pixels which are covered by the blackened portions of the structuring element. Using this range, a pixel can be conservatively smoothed as described in this worksheet.

Figure 1 1-D Gaussian distribution with mean 0 and =1

Figure 2 2-D Gaussian distribution with mean (0,0) and =1

Figure 3 Discrete approximation to Gaussian function with =1.4

Figure 4 One of the pair of 1-D convolution kernels used to calculate the full kernel shown in Figure 3 more quickly.

Figure 5 Frequency responses of Box (i.e. mean) filter (width 7 pixels) and Gaussian filter ( = 3 pixels). The spatial frequency axis is marked in cycles per pixel, and hence no value above 0.5 has a real meaning.

Figure 1 Three commonly used discrete approximations to the Laplacian filter. (Note, we have defined the Laplacian using a negative peak because this is more common; however, it is equally valid to use the opposite sign convention.)

Figure 2 The 2-D Laplacian of Gaussian (LoG) function. The x and y axes are marked in standard deviations ().

Figure 3 Discrete approximation to LoG function with Gaussian = 1.4

Figure 4 Response of 1-D LoG filter to a step edge. The left hand graph shows a 1-D image, 200 pixels long, containing a step edge. The right hand graph shows the response of a 1-D LoG filter with Gaussian = 3 pixels.

Figure 1 3×3 averaging kernel often used in mean filtering

Figure 1 Calculating the median value of a pixel neighborhood. As can be seen, the central pixel value of 150 is rather unrepresentative of the surrounding pixels and is replaced with the median value: 124. A 3×3 square neighborhood is used here --- larger neighborhoods will produce more severe smoothing.

Figure 1 Spatial sharpening.

Figure 2 Calculating an edge image for unsharp filtering.

Figure 3 Sharpening the original signal using the edge image.

Figure 4 The complete unsharp filtering operator.

Figure 5 Spatial sharpening, an alternative definition.

Figure 6 Three discrete approximations to the Laplacian filter.

Figure 7 Ringing effect introduced by the unsharp mask in the presence of a 2 pixel wide, high intensity stripe. (Gray levels: --1=Dark, 0=Gray, 1=Bright.) a) 1-D input intensity image slice. b) Corresponding 1-D slice through unsharp filter. c) 1-D output intensity image slice.

Figure 8 Sharpening filter.

Figure 9 Sharpening filter re-defined as eight edge directional kernels

Figure 10 Adaptive sharpening.