Double-end filter applied to grayscale and color image bilateral filtering for gray and color Images_ two-port filters

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Bilateral filtering for Gray and Color Images

Introduction the idea the Gaussian case experiments with black-and-white Images experiments with Color Images References I Ntroduction

Filtering is perhaps the most fundamental operation of image processing and computer vision. In the broadest sense of the term ' filtering ', the value of the filtered image at a given location is a function of the VA Lues of the input image in a small neighborhood of the same location. For example, Gaussian low-pass filtering computes a weighted average of pixel values into the neighborhood, in which the Wei Ghts decrease with distance from the neighborhood center. Although formal and quantitative explanations of this weight fall-off can be given, the intuition was that images Vary slowly over spaces, so near pixels are likely have values, and it are similar therefore to appropriate average Together. The noise values that corrupt this nearby pixels are mutually less correlated the than values, so signal is noise D away while signal is preserved.
The assumption of slow spatial variations fails at edges, which-are consequently blurred by linear low-pass. How can we prevent averaging across edges, while still averaging within, smooth? Many efforts have been devoted to reducing this undesired effect. Bilateral filtering is a simple, non-iterative scheme for edge-preserving smoothing.

Back to Index the idea

The basic idea underlying bilateral filtering are to does in the range of a image what traditional filters do in its domain. Two pixels can be closeto one another, which, occupy nearby spatial location, or they can be similar to one another, th At are, have nearby values, possibly in a perceptually meaningful fashion.
Consider a shift-invariant low-pass domain filter applied to an image:

The bold font for F and H emphasizes the fact so both input and output images May is multi-band. In order to preserve the DC component, it must is

Range filtering is similarly defined:

In this case, the kernel measures the photometric similarity between pixels. The normalization constant in this case is

The spatial distribution of the image intensities plays no in range filtering taken by itself. Combining intensities from the entire image, however, makes little sense, since the distribution of image values far From X ought affect the final value at x. In addition, one can show this range filtering without domain filtering merely changes the color map of an image, and is T Herefore of little use. The appropriate solution is to combine domain and range filtering, thereby enforcing, both geometric and photometric locali Ty. Combined filtering can be described as follows:

With the normalization

Combined domain and range filtering'll be denoted as bilateral filtering. It replaces the pixel value in X with a average of similar and nearby pixel values. In smooth regions, pixel values in a small neighborhood are similar to all other, and the bilateral filter acts Ly as a standard domain filter, averaging away the small, weakly correlated differences between pixel values caused by Noi SE. Consider now a sharp boundary between a dark and a bright region, as in Figure 1 (a).

(a)

(b)

(c)

Figure 1


When the bilateral filter is centered, say, on a pixel on the bright side of the boundary, the similarity function  s assumes values close to one for pixels on the same side, and values close to zero for pixels on the dark Side . The similarity function is shown in Figure 1 (b) for a 23x23 filter support centered two pixels to the right of the "step" in Figure 1 (a). The normalization term k (x)  ensures the weights for all of the pixels add to one. As a result, the filter replaces the "bright pixel at the" center by a average of the bright pixels in its vicinity, and ES Sentially ignores the dark pixels. Conversely, when the "filter is centered on a dark pixel, the bright pixels are ignored. Thus, as shown in Figure 1 (c), good filtering behavior are achieved at the boundaries, to the domain component of th E filter, and crisp edges are preserved at the same time, the "to" range component.

Back to Index the Gaussian case

A Simple and important case of bilateral filtering are shift-invariant Gaussian filtering, in which both the closeness F Unction c and the similarity function s are Gaussian functions of the Euclidean distance between R arguments. More Specifically, c is radially symmetric:

where

is the Euclidean distance. The similarity function s is perfectly analogous to c :

where

is a suitable measure O f Distance in intensity spaces. In the scalar case, this may is simply the absolute difference of the pixel difference or, since noise increases with imag E intensity, an intensity-dependent version of it. Just as this form's domain filtering is shift-invariant, the Gaussian range filter introduced above be insensitive to Ove Rall additive changes of image intensity. Of course, the range filter is shift-invariant as.

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experiments with black-and-white Images

Figure 2 (a) and (b) show the potential of bilateral filtering for the removal of texture. The picture, "simplification" illustrated by Figure 2 (b) can is useful for data reduction without loss of overall shape FE Atures in applications such as image transmission, the picture editing and manipulation, the image description for retrieval.

(a)

(b)

Figure 2


Bilateral filtering with parameters SD =3 pixels and SR =50 intensity values are applied to the image in Figure 3 (a) to Yi Eld the image in Figure 3 (b). Notice that most of the the fine texture has been filtered away, and yet all contours are as the crisp image. Figure 3 (c) shows a detail of Figure 3 (a), and Figure 3 (d) shows the corresponding filtered version. The two onions have assumed a graphics-like appearance, and the fine texture. However, the overall shading is preserved, because it is very-within the band of the domain filter and is almost UNAFFECTE D by the range filter. Also, the boundaries of the onions are preserved.

(a)

(b)

(c)

(d)

Figure 3


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Experiments with Color Images

For black-and-white images, intensities between any two gray levels are. As a consequence, when smoothing black-and-white images with a standard low-pass filter, intermediate levels of gray are p Roduced across edges, thereby producing blurred. With color images, a additional complication arises from the fact of that between any two colors there, are, often ER different colors. For instance, between blue and red there are various shades of pink and purple. Thus, disturbing color bands may is produced when smoothing across color edges. The smoothed image does not just look blurred, it also exhibits odd-looking, colored auras, around.

(a)

(b)

(c)

(d)

Figure 4


Figure 4 (a) shows a detail from a and a red jacket against a blue sky. Even in this unblurred the picture, a thin pink-purple line is visible, and are caused by a combination of lens blurring and pi Xel averaging. In fact, pixels along the boundary, when projected to the scene, intersect both red jacket and blue sky, and the RE Sulting color is the pink average of red and blue. When smoothing, this effect was emphasized, as the broad, blurred pink-purple area in Figure 4 (b) shows.
To address this difficulty, edge-preserving smoothing could is applied to the red, green, and blue components of the image Separately. However, the intensity profiles across the edge in the three color bands the are. Smoothing the three color bands separately results in a even more pronounced pink and purple band than in the original, a s shown in Figure 4 (c). The Pink-purple band, however, is isn't widened as in the standard-blurred version of Figure 4 (b).
A Much better result can be obtained with bilateral filtering. In fact, a bilateral filter allows combining three color bands appropriately, and measuring photometric distances Een pixels in the combined. Moreover, this combined distance can is made to correspond closely to perceived dissimilarity by using Euclidean distance In The cie-lab color spaces. This color the based on a large the body of psychophysical data concerning color-matching experiments performed by human Observers. In this spaces, small Euclidean distances are designed to correlate strongly with the perception of color discrepancy as ex Perienced by a "average" COLOR-NORMAL human observer. Thus, in a sense, bilateral filtering performed in the Cie-lab color spaces are the most natural type of filtering for color Images:only perceptually Similar colors are averaged, and only together perceptually. Important edges are. Figure 4 (d) shows the image resulting from bilateral smoothing of the ImagE in Figure 4 (a). The pink band has shrunk considerably, and no extraneous colors.

(a)

(b)

(c)

Figure 5


Figure 5 (c) shows the result of five iterations of bilateral filtering of the "image in Figure 5 (a). While a single iteration produces a much cleaner image (Figure 5 (b)) than the original, and are probably sufficient for MO St Image processing needs, multiple iterations have the effect of flattening the colors in a image considerably, but with Out blurring edges. The resulting image has a much smaller-color map, and the effects of bilateral filtering-are to-when easier On a printed page. Notice the cartoon-like appearance of Figure 5 (c). All shadows and edges are preserved, but most of the shading are gone, and no "new" colors are by introduced.

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References

[1] C. Tomasi and R. Manduchi, "Bilateral filtering for Gray and Color Images", Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India.
[2] T. Boult, R.A Melter, F. Skorina, and I. Stojmenovic, "G-neighbors", Proceedings of the SPIE Conference on Vision Etry II, Pages 96-109, 1993.
[3] r.t Chin and C.L. Yeh, "Quantitative evaluation of some edge-preserving-noise-smoothing techniques", Computer Vision, Graphics, and Image processing, 23:67-91, 1983.
[4] l.s Davis and A. Rosenfeld, "noise cleaning by iterated local averaging", IEEE transactions on Systems, Mans, and Cybe Rnetics, 8:705-710, 1978.
[5] R.E. Graham, "snow-removal-a noise-stripping process for picture signals", IRE transactions on information theory, 8 : 129-144, 1961.
[6] N. Himayat and S.A. Kassam, "Approximate performance analysis of the Edge preserving filters", IEEE transactions on Signal Processing, 41 (9): 2764-77, 1993.
[7] T.s Huang, G.j Yang, and g.y Tang, "A fast two-dimensional median filtering algorithm", IEEE transactions on Acoust ICS, Speech, and Signal processing, 27 (1): 13-18, 1979.
[8] J.s Lee, "Digital image enhancement and noise filtering by use of the local statistics", IEEE transactions in pattern Ana Lysis and Machine Intelligence, 2 (2): 165-168, 1980.
[9] M. Nagao and T. Matsuyama, "Edge preserving smoothing", Computer Graphics and Image processing, 9:394-407, 1979.
[Ten] p.m. Narendra, "A separable median filter for image noise smoothing", IEEE transactions on mode analysis and Machi Ne Intelligence, 3 (1): 20-29, 1981.
[One] K.J Overton and T.E Weymouth, "A Noise reducing Preprocessing algorithm", Proceedings of the IEEE Computer science C Onference on recognition and Image processing, pages 498-507, Chicago, IL, 1979.
[P. Perona and J. Malik, "scale-space and edge detection using anisotropic diffusion", IEEE transactions on Alysis and Machine Intelligence, 12 (7): 629-639, 1990.
[G] G. Ramponi, "A rational edge-preserving Smoother", Proceedings of International Conference on Image Processing, V Olume 1, pages 151-154, Washington, DC, 1995.
[G] G. Sapiro and D.l. Ringach, "Anisotropic diffusion of Color Images", Proceedings of the SPIE, Volume 2657, pages 471- 382, 1996.
[D.C.C] Wang, A.H Vagnucci, and c.c. Li, "A gradient inverse weighted scheme and the smoothing of its evaluation Ormance ", Computer Vision, Graphics, and Image processing, 15:167-181, 1981.
[S] G. Wyszecki and W. Styles, Color science:concepts and Methods, quantitative Data and formulae, John Wiley and Son S, New York, NY, 1982.
L. Yin, R. Yang, M. Gabbouj, and Y. Neuvo, "Weighted median filters:a tutorial", IEEE Transactions on circuits and Sy Stems Ii:analog and Digital Signal processing, 43 (3): 155-192, 1996.


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