Conservative Smoothing

Common Names: Conservative Smoothing

Brief Description

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.

How It Works

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.

Guidelines for Use

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.

Interactive Experimentation

You can interactively experiment with this operator by clicking here.

Exercises

Explore the effects of conservative smoothing on images corrupted by increasing amounts of Gaussian noise. At what point does the algorithm become incapable of producing significant noise suppression?
Compare conservative smoothing with Crimmins Speckle Removal on an image which is corrupted by low levels (e.g. 0.1%) of salt and pepper noise. Use the image

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.)

When low-pass filtering (e.g. by smoothing with a mean filter), the magnitudes of intensity gradients in the original image decrease as the size of the kernel increases. Consider the effects of increasing the neighborhood size used by the conservative smoothing algorithm. Does this trend exist? Could repeated calls to the conservative smoothing operator yield increased smoothing?
Conservative smoothing is a morphological operator. Viewed as such, we can define other neighborhoods (or structuring elements) besides the square configurations used in the examples. Consider the effects of conservatively smoothing an edge (of different orientations) using the structuring elements from Figure 2.

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.

References

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.

Local Information

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.