WAVELET-BASED METHOD FOR IMAGE FILTERING USING SCALE-SPACE CONTINUITY J UNG C.R. , S CHARCANSKI J.
UFRGS–Universidade Federal do Rio Grande do Sul Instituto de Inform´atica Av. Bento Gonc¸alves, 9500. Porto Alegre, RS, Brasil 91501-970
[email protected] [email protected] UNISINOS - Universidade do Vale do Rio dos Sinos Departamento de Matem´atica Av. UNISINOS, 950. S˜ao Leopoldo, RS, Brasil, 93022-000
[email protected] Abstract. This paper proposes a novel technique to reduce noise whilst preserving edge sharpness during image filtering. This method is based on the image multi-resolution decomposition by a discrete wavelet transform, given a proper wavelet basis. In the transform space, edges are implicitly located and preserved, at the same time that image noise is filtered out. At each resolution level, geometric continuity is used to preserve edges that are not isolated. Finally, we compare consecutive levels to preserve edges having continuity along scales. As a result, the proposed technique produces a filtered version of the original image, where homogeneous regions appear separated by well-defined edges. Possible applications include image pre-segmentation, and image denoising. Keywords: image pre-segmentation, edge detection, filtering, wavelets, multi- resolution
1 Introduction A difficult problem for image analysis researchers is how to remove noise from images without blurring the edges. In general, filtering methods that remove the image high-frequency components are utilized. Some of the removed high-frequency components are in fact related to the edges, which causes edge blurring. A solution to this problem could be to detect the edges first, and then filter the noise, except across the edges previously identified. However, to detect the edges correctly we need a filtered image. Should the first step, therefore, be edge detection or image filtering? We argue that both can be done at the same time. The method proposed by Mallat and Hwang [1] estimates local regularity of the image by calculating the Lipschitz exponents. Coefficients with low Lipschitz exponent values are removed, and the image is reconstructed with the remaining coefficients (actually, only the local maxima are used). The reconstruction process is based on an interactive projection procedure, which is computationally demanding. 1
Other researchers have proposed edge detection using wavelets [2]. In their approach, local maxima are tracked in scale-space, and a tree structure is used to detect and preserve edges. This approach has limitations in complex noisy images, where local maxima are dense. In fact, most proposed methods use thresholds, coefficients less than a certain value being set to zero. However, if the image is noisy, some edges will be lost, and filtering may not then be efficient. Xu and collaborators [3] used the correlation of wavelet coefficients between consecutive scales to distinguish noise from meaningful data. Wavelet coefficients related to noise have much smaller correlation than coefficients of a noiseless image. If the correlation is smaller than a threshold, the coefficient is set to zero. To determine a proper threshold, a noise power estimate is needed for this technique, which may be difficult to obtain for some images. Malfait and Roose [4] developed a filtering technique that takes into account two measures for image filtering. The first is a measure of local regularity of the image through the H¨older exponent, and the second takes into account geometric constraints. These two measures are combined in a Bayesian probabilistic formulation, and implemented as a Markov random field model. The signal-to-noise ratio (SNR) gain achieved by this method is significant, but the stochastic sampling procedure needed for the probabilities calculation is computationally demanding. Chang et. al. [5] proposed using adaptive wavelet thresholding for image denoising. Each wavelet coefficient is modelled as a Generalized Gaussian random variable, whose parameters are estimated locally (within a given neighborhood). An adaptive threshold based on these parameters is calculated, and applied to the wavelet coefficients, with good qualitative results. A drawback of such an approach is that noise variance must be estimated from the image, which can be troublesome. Also, consistency along scales was not explored, indicating that this method may not be efficient for very noisy images. In fact, not many examples were provided in [5]. Pizurica and collaborators [6] proposed a computationally efficient method for image filtering, that utilizes local noise measure and geometrical constraints in the wavelet domain. A shrinkage function based on the two measures is used to modify the wavelet coefficients, and the image is reconstructed with the updated wavelet coefficients. Although this method is fast, it does not take into account the evolution of wavelet coefficients along scales. The present paper proposes a new method based on wavelets, that both filters the image and preserves edges at the same time. For that purpose, we derive geometrical constraints based on the observation that edges appear in contour lines in real images, and are not isolated. At each wavelet decomposition level, an adaptive threshold is used, and a shrinkage function assuming values between “0” and “1” is assembled. The shrinkage functions for consecutive levels are then combined to preserve edges that are persistent in scale-space (i.e., appear in several consecutive levels). The next section gives a brief description of the wavelets framework, and the section that follows describes the new method. Section 4 presents some experimental results of our approach, and a comparison with other wavelet based methods. The last section contains the conclusions.
2 The Wavelet Transform 2.1 One Dimension Initially, the wavelet decomposition process for continuous time signals is briefly described for one with zero-mean, i.e., dimension. Consider a function 2
(1)
If satisfies certain admissibility conditions, then is a wavelet. Here, the hat operator , there is a scaling function denotes the Fourier Transform. Associated with the wavelet that satisfies :
Also, define :
and
(2)
and . , at the scale , and a position Now, introduce the wavelet transform of a function
!
by :
, given
%$
#"
(3)
(4)
where the symbol " denotes the convolution operation :
&"('
) ' &*+) ,)-
(5)
The function is a dilation of scaled by . Equation (5) indicates that the convolution of , with the function , at the scale . Let us denote produces a smoothed version of by : . this smoothed version of
/
#"
%
(6)
The evaluation of the wavelet transform at dyadic scales transform : ! 4
#"
Under some conditions, there exists a dual wavelet 5
76 !
4 2
.
4
1032
, leads to the dyadic wavelet
%
that reconstructs the signal "(5 4
(7) as follows:
%
(8) .
! . There 9 4 ; 8 : It can be shown that the level is a linear . combination of and 498;: # =< fore, considering that is the representation of! at the finest resolution, the BA 2DC E E E F .
4 ? > original function could be calculated from and @ . ! It should be noticed that filter, and is a band-pass filter [7]. Thus, is the decomposition of isa low-pass is the 4 into its low frequency components, and 4 into its octave band frequency components, both at the scale G0H2 . If decomposition of , as different scales are considered, we obtain the multi-resolution representation of the signal .
4
proposed by Mallat [7]. Next, edge detection using the wavelet transform will be discussed. 3
2.1.1 Edge Detection in One Dimension
Mallat and Zhong [8] proposed that the wavelet be designed using a smoothing function such that the conditions in Equation (2) are satisfied :
%$
Using the wavelet in Equation (9), the wavelet transform of ! !
, at scale
#" %
correspond to edges of the signal The local maxima of scale [9]. A cubic spline with compact support is chosen for . These functions are shown in Figure 1. wavelet
, (9)
is : (10)
, smoothed by , at the , and its derivative for the
1.4 1.2 1 0.8 0.6 0.4 0.2 0 −1.5
−1
−0.5
0
0.5
1
1.5
−1
−0.5
0
0.5
1
1.5
1
0.5
0
−0.5
−1 −1.5
%
Figure 1: Top:
Bottom:
The discrete version of the wavelet transform is used when we have a discrete-time signal , . In this case, four discrete filters are used: and for instead of the continuous-time signal the signal decomposition, which have the impulse responses and ' , respectively; and the filters and for the signal reconstruction process, with impulse responses and . The discrete filters and are, respectively, the low-pass and band-pass analogous to the functions and used in the continuous domain. Define ' 2 as the filter obtained by placing 0 2 *+ . ! zeros between adjacent coefficients of ' ; also, define 2 in a similar way, with respect to .
4
4 . Then, and can be . computed recursively, as follows : ! 498;: 498;:
.
4 4
" 2 "(' 2
(11)
and , are respectively filters and ' , corresponding to the functions 3The 0 $B $B $B 30 ?* 0 $%0 and ' . Further details on discrete wavelet transforms and their inverses are given in [8]. 4
2.2 Wavelet Transform in Two Dimensions The wavelet-based method described in the last subsection can be extended to two-dimensions. In this paper, the 2-D wavelet decomposition uses only two detail images (horizontal and vertical details) [8], instead of the classical approach where three detail images (horizontal, vertical and diagonal details) are used This two-dimensional wavelet transform requires two wavelets, $ ) and $ [7]. ) namely . At a particular scale we have :
The dyadic wavelet transform
$ ) /
, at a scale
$ )
!
4
$ ) /
0 2
4
#"
%$
) $
;$%0
(12)
has two components given by :
$ ) %$
;$%0
.
(13)
/
(14)
The multi-resolution wavelet coefficients are therefore:
4
!
$ ) /
4
$ ) %$
!
4
$ )
Extending the one dimensional wavelet transform to two dimensions, for signal reconstruction we $ ) and 5 $ ) , and the original signal $ ) is then represented obtain two dual wavelets 5 by :
!
$ ) /6
4
$ )
"(5 4
2
$ )
!
To build a multi-scale representation, a scaling function the plane equals one, and tends to zero at infinity. Thus, 0 2 associated component at the scale . is : .
The. component ? > $ ) , for nents 4 $ ) to ?> .
$ )
4
$ )
#" 4
4
$ )
"(5 4
$ )
(15)
$ ) whose integral over $ ) isisaneeded, smoothing function, and its
$ ) %
(16)
. , and the can be interpreted as a smoothed version of compo;$ $ $ ) < , as the image details lost by smoothing going from
2.2.1 Edge Detection in Two Dimensions
$ )
$ )
It is now necessary to find a wavelet basis such that its components 4 are related to the $ ) is 0 2 edges at the scale . Extending Equation (2) to two dimensions, a smoothing function selected such that its integral over the plane is equal to one, and it tends to zero at infinity. We then define :
$ ) /
$ )
Note that the gradient of the smoothed version of
$ )
!
!
$ ) $ )
and
)
$ ) /
(17)
at the scale is:
#" #" 5
$ )
$ ) $ )
#"
$ ) %$
(18)
Observing that an edge can be defined as a local maximum of the gradient modulus along the $ ) gradient direction [9], we can detect the edges at the scale from . $ As in the one-dimensional case, because we are interested in digital images , the $ ) $ ) wavelet transform is made discrete. We will deal with wavelets and ) $ that $ can be written as separable products of functions in and . Initially, let us define " ' $ as the separable convolution of the rows of with the filter , and the columns of $ with the filter ' . It can be shown that the discrete wavelet coefficients can be calculated recursively from [8]: . .
$$ 498;: $ 498;:
! 498;: !
where
.
.
4
$$ " 2 $ $ 2 4 " ' 2 $
$ 4 " ' 2
$
(19)
is the discrete unit impulse function.
3 Our Approach for Image Filtering using Scale-Space Continuity ! ! Given an image . , we first apply the redundant wavelet transform using only two detail images, as $ 4 proposed by Mallat and Zhong [8]. As a result, we obtain the detail images 4 and the 0 2
4 smoothed image at each level . The edge magnitudes can be calculated as follows:
4
!
4
!
$
4
(20)
and the edge orientation is given by the gradient direction, which is expressed by: 4
!
!
4 4
(21)
Therefore, we now have the edge magnitudes and their corresponding orientations. Due to noise, some coefficients of 4 associated with uniform regions may be misinterpreted as edges. We basically need a technique that allows us to discriminate responses due to noise, from responses due to actual edges. To do so, we rank coefficients according to their magnitudes, geometric continuity and consistency along scales, as described next.
3.1 Geometric continuity Usually, edges do not appear isolated in an image. They form contour lines, which we assume to $ has a higher likelihood of being be piecewise linear. In our approach, a coefficient 4 an edge if its neighbors along the local contour direction also have large magnitudes. To detect $ $; or . The this kind of behavior, we first quantize the gradient directions 4 into contour lines are orthogonal to the gradient direction at each edge element, so we can estimate $ the contour direction from 4 . We then add up the magnitudes 4 along the contour 6
direction, according to the following updating rule:
4
D$ $
if the local contour direction is
4
D$ $
if the local contour direction is
6
C 6
$
4
C
6
C 6
where 0
C
$
$
4
4
D$1* $
if the local contour direction is if the local contour direction is
$
$
(22)
; $
$
is the number of adjacent pixels that should be aligned for geometric continuity, and is a window that allows neighboring pixels to be weighted differently, according to their $ distance from the pixel under consideration. After updating the magnitudes 4 , coefficients with large magnitude neighbors aligned with the local contour direction will be strengthened, while coefficients with no geometric continuity will not. This approach has limitations close to corners and junctions, where two or more different local contour directions arise. If the image is sufficiently sampled, high curvature points should also be enhanced. In the presence of noise, randomly aligned coefficients occur, and also would be strengthened. To attenuate this problem, we compare the contour direction in two consecutive levels. It is expected that contours would be aligned along the same direction in two consecutive levels (it is the same contour at different resolutions), but responses due to noise should not be aligned (gradients associated to noise will be randomly oriented at all resolutions). Therefore, a second updating rule is applied to the magnitude coefficients 4 . At each resolution, the updated local gradient coefficients may be interpreted as vectors with magnitude 4 and argument 4 . The second updating rule takes into account the normalized inner product of corresponding vectors in consecutive resolutions:
Notice that the factor
4
$
*
4 $
$
*
498;:
$
(23)
really provides a measure for direction continuity. It has value unity if the same direction occurs in two consecutive levels, and value zero ; if the orientations differ by (i.e., are orthogonal). In Figure 2, the magnitude images before and after applying the local geometrical constraints are shown. 4
$
$
4
498;:
3.2 Adaptive Threshold and Shrinkage After applying the geometrical constraints, a set of geometrically consistent edges, at different resolutions, has been enhanced, but some coefficients related to noise still remain. Typically, the magnitudes of these coefficients are smaller than the magnitude of edges. To eliminate (or just 2 reduce) noisy coefficients, an adaptive threshold is2 calculated from the edge magnitudes at each 0 2 2 $ 2 !level ! . Based on this threshold , an interval is chosen, and the wavelet coefficients 4 and 4 are updated according the following rule: ! !
4
$
4
$
' 2 7
4
$
%$
for
;$%0
(24)
(a)
Figure 2: (a) Coefficients
The function ' if ' 2
2
4
(b)
. (b) Coefficients
4
, after geometric constraints were applied.
is a non-negative non-decreasing function such that ' 2 2 was chosen: . In this paper, a piecewise linear function ' 2
$
if
if
2
and
2
(25) if Coefficients whose magnitudes are greater than are kept unchanged (so that they are likely to be related to edges), and coefficients whose magnitudes are smaller than are set to zero (coefficients likely to be related to noise are removed). The interval is considered as ' 2
4 ;$
:
4
:
4
$
2
if
2
2
2
2
2 $
2
a “transition band”, where a coefficient may be either related to noise or to meaningful image structures. Within this “transition band”, magnitudes may be shrunk to zero 2 according to the $ 2 2 function ' , or left unchanged, as described. The choice of the interval depends on the amount of noise in the image. For noisier images, coefficients related to noise have larger 2 2 and should also be larger. We now describe how the interval magnitudes, and the values 2 $ 2 can be chosen, based on the histogram of the coefficient magnitudes. ! ! If the original image is contaminated by additive white noise, the corresponding coefficients 4 and ! 4 can be !considered Gaussian [10], with standard deviation 2 . As a consequence,
4
4
4
will be a Rayleigh process [11] with probability density given by:
Notice that the mode of a Rayleigh distribution occurs at
2
/
4
(26)
2
2
. As a consequence, noisier images will have a higher mode value (i.e., it will be further away from the origin). Our goal is to choose the transition band without knowing any a priori estimate of the noise. For that purpose, we calculate the mode and the standard deviation of the histograms of 4 at 0 2 each level It should be observed that, if an image is only white noise, the histogram of 4 may be modelled by a Rayleigh distribution. In real images, some coefficients of 4 represent edges, and the other ones are associated with noise. If the image signal-to-noise ratio (SNR) is small, noisy coefficients will dominate over “clean” ones in the image, and the histogram of 4 will
8
be closer to a Rayleigh distribution. On the other hand, if the image SNR is high, the amount of noise will be small, and the histogram of 4 will vary according to the edge information in the image. For this type of images, the histogram of 4 will typically have the largest peak close to the origin, corresponding to the homogeneous regions (i.e., the corresponding coefficients 4 will be low). Consequently, the mode of these images will be located close to the origin. By computing the mode 2 of the histogram we can estimate the noise in the image. In our2 approach, the mode of the histogram is used as a reference point to find the transition band 2 0 2 $ , along with the standard deviation 2 As the level increases, noise is smoothed out, and the coefficients 4 are less contaminated. In fact, the noise standard deviation is reduced to half 0 2 0 2 $ as the level increases from to as observed by Donoho [10]. The most reliable estimate for the noise standard deviation is thus obtained at the highest resolution level D: . At this stage, edge related coefficients have been enhanced by the geometrical consistency procedure. However, noise related coefficients are not enhanced as much, and we assume that Donoho’s considerations are valid for the updated coefficients D: as well. 0 (corresponding to the first level ): as the standard deviation of the coefficients We define 2 0 2 D: 4 : , since the standard deviation is reduced to . For the other levels ( ), we choose 2 2 $ half as the level increases. To find the transition band limits and we combine the information 2 provided by and the mode 2 . The value 2 is the most frequent magnitude in the image. Usually, in noisy images this peak is associated with noisy coefficients, which indicates that noisy coefficient magnitudes are located 2 represents the variability of the near the value 2 in the histogram. The standard deviation 2 coefficient magnitudes. If is small, the magnitude range for noisy coefficients is also small, 2 indicating that edge and noise related magnitudes are well separated. On the other hand, when is large, magnitudes related to edge and noise are closer to2 each other, requiring a wider transition 2 band. To take into account these considerations, and are chosen as follows:
2 2
2
2
2 $ 2 $
(27)
$
and are supplied where and are parameters dependent on image content satisfying by the user. The selection of the parameters and allows the user to control the transition band. If large values for and are chosen, noise removal will be increased, but some edges may be lost. On the other hand, the use of smaller values for these parameters results in less filtering and more edge preservation. However, our experiments showed that default values for and produced good results for most images (further details will be given in Section 4). Notice
2 , we have a hard threshold given by 2 that, when . At the end of this process, we have a shrinkage function ' 2 for each level . In this work, we used the piecewise linear function (25), although other functions can also be used. Next we apply consistency along different scales to further improve discrimination of the coefficients due to noise, from those that originate at edges.
3.3 Consistency Along Scales 032
It is known that coefficients associated with noise vanish as the level increases, while coefficients associated with edges are preserved when increases. In [1] and [4] the H¨older exponent was 9
calculated in order to explore this property. We analyze the consistency of the wavelet coefficients 0 2 along scales differently, by combining the shrinkage functions ' 2 at various levels . 0 2 4 $ may be interpreted as a confidence measure that For each level , the value ' 2 $ 4 $ is close to the coefficient 4 is in fact associated to an edge. If the value ' 2 0 2 $ unity for several consecutive levels , it is more likely that 4 is associated with an edge. $ 4 On the other hand, if ' 2 decreases as increases, it is more likely that 4 $ is actually associated with noise. 0 2 For each level , we use the information provided by the function ' 2 and also by the functions ;$%0 $ $ ' 2 , for , where is the number of consecutive levels that will be taken into consideration for the consistency along scales. As observed by Xu et. al. [3], it seems that using 0 or consecutive levels is better than using more consecutive levels, because the positions of the $ may change as increases. local maxima of 4 $ $ $ is approximately Thus, we need to find a function such that $ $ $ unity if all the are close to one, and must be close to zero if any of the is close to zero. There are many functions satisfying this property, and we chose the harmonic mean:
$ For the level
0 2
$ $
/
' !
!
4
: 2 new
, the updated function ' 2 new
4
(28)
8;:
is given by:
498;:
498
$
(29) .
and the coefficients and are modified according to Equation (24), using the function instead of ' 2 ! . The ' 2 new image! is then reconstructed using the low-pass image > and the 4 4 , through the iterative procedure given by Equation (19). updated coefficients and Typically, three levels in the decomposition process produce acceptable visual results and high SNR gains.
4
3.4 Overview of the Proposed Method .
Below, we present a schematic overview of our method: !
2. Calculate the edge magnitudes
4
!
4
!
$
4
1. Compute the wavelet transform, obtaining the coefficients
!
4
and 4 ;
;
4
3. Apply geometrical constraints (contour continuity and orientation continuity along consecu $
4 and obtain the updated coefficients 4 ; tive levels) to 4. For each level, compute the mode and standard deviation of 2 $ 2 ; and the shrinkage functions ' 2 band”
4
, obtaining the “transition
5. Combine the shrinkage functions in consecutive scales, obtaining the updated shrinkage ' 2
new : : : functions ' 2 is the number of consecutive levels
4 498;: 498 , where to be analyzed; 10
!
6. Modify the coefficients ;$%0 for ;
!
4
and
4
, obtaining .
! 7. Apply the inverse wavelet transform with 4 , obtaining the filtered image.
?>
!
4
$
!
4
$
'
2 new
!
and the modified coefficients
4
$
%$
!
4
and
4 Experimental Results We applied our technique to synthetic and natural images, and compared the results with those obtained by the software wave2 c , which is an implementation of the method described in [1]. The chosen parameter values for the software are the same as those used by Malfait and Roose [4]. To evaluate the performance of the method, both visual quality and SNR gain are utilized. was used for geometric continuity in Equation (22), In our experiments, the value and the window is a Gaussian (so that larger weights are assigned to the nearest neighbors). 0 ; 0 Default values for and were, respectively, and . For each image, fine tuning of the parameters and is required to obtain the optimal performance of the method (when and are not specified in an example, the default values were used). Figure 3 shows, on the top line, the original house image [1] and its noisy version (SNR = 0 0 pixels. On the bottom line, the results of the software wave2 c applied 2.88 dB), both to the noisy image is shown, followed by the results that were obtained applying our technique. Quantitatively, filtering by software wave2 c resulted in a SNR of 12.83 dB, while filtering with our technique resulted in a SNR of 14.88 dB. A similar image (noisy house image with SNR = 3 dB) was used in [4], and the resulting filtered image achieved SNR = 14.86 dB. Qualitatively, it is possible to see that visually relevant edges are preserved, while noise is efficiently filtered out in the image processed with our method. A quantitative analysis for the house image is shown in Figure 4. Versions of the house image with different signal-to-noise ratios were created, and the output SNR gains obtained by two different denoising methods are compared (i.e. our technique and the standard Wiener filtering). Some window sizes were tested in the Wiener filtering, but the window showed the best results. As the window size is increased, denoising is improved, but the edges become more blurred. Our method showed a better performance than Wiener filtering, specially for input images with low SNR. The peppers test image was also used to test the performance of our method. Figure 5 shows the original peppers image on the left. The middle image is the noisy version (SNR = 3 dB), and the rightmost image is the result of our method (SNR = 13.55 dB). All images have a resolution of 0 0 pixels. The image processed with the new technique has a visually acceptable quality, and the SNR gain achieved is considerable. In [4], the peppers image with added noise (also SNR = 3 dB) was also analyzed, and the filtering method only achieved SNR = 12.36 dB. Figure 6 shows, on the left, a synthetic image of a square and a circle. The middle Figure shows the same image contaminated with noise (SNR = 1 dB). The image on the right shows the result of our method (SNR = 16.9 dB), using four wavelet levels. The SNR gain achieved was very good, and object borders were relatively well preserved, despite some blurring at the corners of the square. We also used images of natural scenes in our experiments. Figure 7(a) shows the original chemical plant image. Figures 7(b) and 7(c) show, respectively, the filtered images using ( 0 $ ; 0 $ $ ) and ( ). Figure 7(b) shows that some details (as the crop field on the top and some low contrast details in the buildings) are lost. By decreasing the values of
11
(a)
(b)
(c)
(d)
Figure 3: (a) Original house image. (b) Noisy house image (SNR = 2.88 dB). (c) Result obtained by software wave2 c (SNR = 12.83 dB). (d) Result obtained with the new technique (SNR = 14.86 dB).
and , these details are better preserved, as shown in Figure 7(c). In both filtered images the background noise was efficiently removed. 0 pixels) is shown in Figure 8 (top image). The result of our Another realistic image ( method is shown at the bottom, using and . The noise seems to be effectively suppressed, while image details are preserved in most structures. The texture of the trees was smoothed out, since the method was unable to distinguish this subtle texture from the background noise. Our method was also applied to real satellite (SAR) images containing a large amount of , 0 SAR image of a rice crop in Rio Grande do Sul, speckle noise. Figure 9(a) shows a Brazil, and Figure 9(b) shows the filtered image using our method. Figures 9(c) and 9(d) show, respectively, the histograms of these images. It is possible to see that background noise was effi12
16
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output SNR gain (dB)
12
10
8
6
4
2 −4
−2
0
2
4 6 original SNR (dB)
8
10
12
14
Figure 4: Comparative results: noise removal by Wiener filtering (’x’) and results obtained with method (’o’). (a)
(b)
(c)
Figure 5: (a) Original peppers image. (b) Noisy peppers image (SNR = 3 dB). (c) Result obtained with the new technique (SNR = 13.55 dB). ciently attenuated, while crop regions were enhanced. The bimodal form of the histogram shown in 9(d) indicates mainly two distinct types of regions in the filtered image, that may be segmented using a simple threshold (such image could be used for crop area monitoring). Figure 10 shows the segmentation result of Figure 9(b), using as the threshold (morphological opening was applied to remove small artifacts). Our algorithm was implemented in MATLAB, running on a 300 MHz Pentium II personal 0 0 computer, with 64 MB RAM. Typical execution time for a image, using 3 dyadic scales, is about 8 seconds. A more efficient implementation on a compiled language is expected to improve the execution time.
13
Figure 6: Left: synthetic image of a square and circle. Middle: image with noise. Right: result of noise suppression, with the new method. (a)
(b)
Figure 7: (a) Original chemical plant image. (b) Filtered image, using Filtered image, using and .
(c)
0
and
; 0
. (c)
Therefore, based on our experiments, the proposed method can obtain a performance comparable to, or even better than, other methods proposed in the literature, with reduced computational complexity, and less control parameters. 5 Concluding Remarks In the proposed filtering technique, three features were used for noise removal. First, geometrical constraints were applied to enhance edges appearing as contours, and therefore connected. Second, an adaptive transition band was chosen, and shrinkage functions for each wavelet level were assigned. Finally, these functions were combined in consecutive resolution levels to explore continuity along scales. The experimental results obtained are considered good, both quantitatively and qualitatively. From this point of view, the new method is comparable to, or better than, other wavelet-based methods, with the advantage of having only three user-defined parameters (i.e. and , corresponding to the transition band, and , corresponding to the neighborhood size used for geomet-
14
Figure 8: Top: original aerial image. Bottom: Result of noise suppression, using .
and
rical consistency). No estimate of the noise is needed, as opposed to [4]. The proposed algorithm is also computationally efficient. It does not require the iterative projections needed in the recon15
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Figure 9: (a) Original SAR image. (b) Filtered SAR image. (c) Histogram of image (a). (d) Histogram of image (b). struction procedure for the method described in [1], nor the time-consuming sampling procedure needed for the probabilities calculation in the method proposed in [4]. Future work will concentrate on improving our method using adaptive noise model estimation, distinct choices of shrinkage functions and multi-scale consistency. Also, we intend to investigate the application of our method to edge enhancement in noisy images. References [1] S. G. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inform. Theory 38(2), 617–643 (1992). [2] J. Lu, J. B. Weaver, D. M. Healy, and Y. Xu, “Noise reduction with multiscale edge representation and perceptual criteria,” in Proc. of IEEE-SP Intl. Symp. on Time-Frequency and Time-Scale Analysis, (Victoria, BC), 555–558 (1992). [3] Y. Xu, J. B. Weaver, D. M. Healy, and J. Lu, “Wavelet transform domain filters: A spatially selective noise filtration technique,” IEEE Trans. Image Processing 3(6),747–758 (1994). 16
Figure 10: Segmentation of the filtered SAR image. [4] M. Malfait and D. Roose, “Wavelet based image denoising using a Markov Random Field a priori model,” IEEE Transactions on Image Processing 6(4), 549–565 (1997). [5] S. G. Chang, B. Yu and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” in 1998 International Conference on Image Processing (ICIP98), 535–539 (1998). [6] A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, “Image de-noising in the wavelet domain using prior spatial constraints,” in Seventh International Conference on Image Processing and its Applications, (Manchester, England), 216–219 (1999). [7] S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. on PAMI 2(7), 674–693 (1989). [8] S. G. Mallat and S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. on PAMI 14(7), 710–732 (1992). [9] J. Canny, “A computational approach to edge detection,” IEEE Transactions on PAMI 8, 679–698 (1986). [10] D. L. Donoho, “Wavelet shrinkage and w.v.d.: A 10-minute tour,” Progress in Wavelet Analysis and Applications (1993). [11] H. J. Larson and B. O. Shubert, Probabilistic Models in Engineering Sciences, vol. I. John Wiley & Sons (1979).
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