Common Names: Conservative Smoothing
Conservative smoothing is a
noise reduction
technique that derives
its name from the fact that it employs a simple, fast filtering
algorithm that sacrifices noise suppression power in order to
preserve the high spatial frequency detail (e.g. sharp edges) in an
image. It is explicitly designed to remove noise spikes
--- i.e. isolated pixels of exceptionally low or high pixel intensity
(e.g. salt and pepper noise) and is, therefore, less
effective at removing additive noise (e.g. Gaussian
noise) from an image.
Like most noise filters, conservative smoothing operates on the assumption that noise has a high spatial frequency and, therefore, can be attenuated by a local operation which makes each pixel's intensity roughly consistent with those of its nearest neighbors. However, whereas mean filtering accomplishes this by averaging local intensities and median filtering by a non-linear rank selection technique, conservative smoothing simply ensures that each pixel's intensity is bounded within the range of intensities defined by its neighbors.
This is accomplished by a procedure which first finds the minimum and maximum intensity values of all the pixels within a windowed region around the pixel in question. If the intensity of the central pixel lies within the intensity range spread of its neighbors, it is passed on to the output image unchanged. However, if the central pixel intensity is greater than the maximum value, it is set equal to the maximum value; if the central pixel intensity is less than the minimum value, it is set equal to the minimum value. Figure 1 illustrates this idea.
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.
If we compare the result of conservative smoothing on the image segment of Figure 1 with the result obtained by mean filtering and median filtering, we see that it produces a more subtle effect than both the former (whose central pixel value would become 125) and the latter (124). Furthermore, conservative smoothing is less corrupting at image edges than either of these noise suppression filters.
Images are often corrupted by noise from several sources, the most frequent of which are additive noise (e.g. Gaussian noise) and impulse noise (e.g. salt and pepper noise). Linear filters, such as the mean filter, are the primary tool for smoothing digital images degraded by additive noise. For example, consider the image
which has been corrupted with Gaussian noise with mean 0 and deviation 13. The image
is the result after mean filtering with a 3×3 kernel. Comparing this result with the original image
it is obvious that in suppressing the noise, edges were blurred and detail was lost.
This example illustrates a major limitation of linear filtering, namely that a weighted average smoothing process tends to reduce the magnitude of an intensity gradient. Rather than employing a filter which inserts intermediate intensity values between high contrast neighboring pixels, we can employ a non-linear noise suppression technique, such as the median filtering or conservative smoothing, to preserve spatial resolution by re-using pixel intensity values already in the original image. For example, consider
which is the Gaussian noise corrupted image considered above passed through a median filter with a 3×3 kernel. Here, noise is dealt with less effectively, but detail is better preserved than in the case of mean filtering.
If we classify smoothing filters along this Noise Suppression vs Detail Preservation continuum, conservative smoothing would be rated near the tail end of the former category. The image
shows the same image conservatively smoothed, using a 3×3 neighborhood. Maximum high spatial frequency detail is preserved, but at the price of noise suppression. Conservative smoothing is unable to reduce much Gaussian noise as individual noisy pixel values do not vary much from their neighbors.
The real utility of conservative smoothing (and median filtering) is in suppressing salt and pepper, or impulse, noise. A linear filter cannot totally eliminate impulse noise, as a single pixel which acts as an intensity spike can contribute significantly to the weighted average of the filter. Non-linear filters can be robust to this type of noise because single outlier pixel intensities can be eliminated entirely.
For example, consider
which has been corrupted by 1% salt and pepper noise (i.e. bits have been flipped with probability 1%). After mean filtering, the image is still noisy, as shown in
After median filtering, all noise is suppressed, as shown in
Conservative smoothing produces an image which still contains some noise in places where the pixel neighborhoods were contaminated by more than one intensity spike
However, no image detail has been lost; e.g. notice how conservative smoothing is the only operator which preserved the reflection in the subject's eye.
Conservative smoothing works well for low levels of salt and pepper noise. However, when the image has been corrupted such that more than 1 pixel in the local neighborhood has been effected, conservative smoothing is less successful. For example, smoothing the image
which has been infected with 5% salt and pepper noise (i.e. bits flipped with probability 5%), yields
The original image is
Compare this result to that achieved by smoothing with a 3×3 median filter
You may also compare the result achieved by conservative smoothing to that obtained with 10 iterations of the Crimmins Speckle Removal algorithm
Notice that although the latter is effective at noise removal, it smoothes away so much detail that it is of little more general utility than the conservative smoothing operator on images badly corrupted by noise.
You can interactively experiment with this operator by clicking here.
and use the original
as a benchmark for assessing which algorithm reduces the most noise while preserving image detail. (Note, you should not need more than 8 iterations of Crimmins to clean up this image.)
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.
R. Boyle and R. Thomas Computer Vision: A First Course, Blackwell Scientific Publications, 1988, pp 32 - 34.
A. Jain Fundamentals of Digital Image Processing, Prentice-Hall, 1986, Chap. 7.
D. Vernon Machine Vision, Prentice-Hall, 1991, Chap. 4.
Specific information about this operator may be found here.
More general advice about the local HIPR installation is available in the Local Information introductory section.
©2003 R. Fisher, S. Perkins,
A. Walker and E. Wolfart.
