数字图像平滑和伽马滤波

Digital Image Smoothing and the Sigma Filter
数字图像平滑和伽马滤波

1.Introduction
引言

Generally,digital image smoothing techniques fall into two categories.In the first category,the noisy image is processed globally in the sense that the whole or a large section of a noisy image is correlated to obtain a smoothed image. Techniques in the transform domain using Wiener or leaset squrares filtering and techniques applying one-dimensional or two-dimensional Kalman filter are in this category. Statistical models for the signal(noise free image)and the noise are requried for the implementation fo these techniques.Unfortunately,the statistical model for most images is either unknown or impossible to describe adequately with a simple random process. The smoothed images display bllurred edges and conceal subtle details.In addition, the techniques are computationally costly. In the second category local operators are applied to noisy images. Only those pixels in a small neighborhood of the concerned pixel are involved in the computation, The immediate advantage of these techniques is their efficiency. They have great potential for real-time or near real-time implementation,because several pixels can be processed in parallel without waiting for their neighboring pixels to be processed. Recent research in image smoothing and segmentation favors the local techniques.
一般说来，数字图像平滑分为两大类别。在第一个类别中，是通过对噪声图像的全部或者很大一部分进行分析处理，得到平滑后图像。在技术实现上，一般采用维纳滤波和最小方差值滤波，或者使用一维或二维的卡尔曼滤波。这些方法的实现都需要知道无噪声图像信号及噪声信号的统计模型，但是显示，对于大多数图像，统计模型是不可能仅通过一个简单的随机函数等进行恰当描述的。使用这些方法得到的图像模糊，而且把细微细节都隐去了。另外，这些方法也是很耗时的。在第二种类别里，对噪声图像使用局部描述符进行描述。只对落在感兴趣像素领域里的像素进行计算，而不是针对全部。这种方法直接的好处就是很高效。因为像素之间的计算可以并行，这使得他们有很大的潜在可能性适用于实时或接近实时实现，近来的在图像平滑以及图像分割等领域的研究也支持了这种方法。

There are many local smoothing methods. The well-known median filter in one or two dimensions has attracted much attention. The edge preserving smoothing of Nagao and Masuyama, the gradient inverse weighting scheme of Wang et al., the box filtering algorithm, and the local statistics method of Lee are just a few other altorithms in this category. Obviously, it is nearly impossible to rank them, because an algorithm may be effective for a class of images, but ineffective for others. In this paper a new class of local smoothing schemes is introduced. It is motivated by the sigma probability of the Gaussian distribution. The basic idea is to replace the pixel to be processed by the average of only those neighboring pixels having their intensity within a fixed sigma range of the center pixel. Replacing the center pixel by the average of selected neighboring pixels has been explored by many algorithms. Nagao's filter replaces the center pixel by the average of a subregion which has minimum variance. Lee in his refined local statistics method selected the region by using gradient information. Graham and Prewitt replace a pixel by the average of the surrounding area if the absolute value of their difference is smaller than some threshold. Rosenfeld in his region growing and tracking algorithm excludes high contrast edges, lines,and points from the average by judging extended box-filtering algorithm restricts the average to only neighbouring pixels with a fixed intensity range. The main differencis between the box filter and the sigma filter of this paper is that the former has the intensity range fixed throughout the center pixel. The advantages are numerous:(1)noise near edge areas will be smoothed without blurring the edge because only pixels on one side of the edge are included in the average;(2)subtle details of several pixel clusters and linear features of one to three pixels in width will be preserved since only those pixels and not the background are included in the average; (3)it will not create artifacts and will retain shapes, because no directional masks are used, unlike the algorithms of Nagao and Lee; it is computationally efficient, since only simple compare and fixed point add instructions are involved.
有很多种局部平滑的方法，最为大家熟知的当属一维或二维的中值滤波，还有Ngao和Masuyama的边界保留平滑，Wang et al的灰度反转权重模式，箱式滤波算法（http://tech-algorithm.com/articles/boxfiltering/），以及Lee的局部统计方法。显然，我们无法对这几种方法分个三六九等，因为一种算法可能对某一类的图像处理效果较好，但对某些图像效果不好，在这篇文章中，我们提出一种新的局部平滑算法，它是基于高斯分布的伽马概率。基本的思想是把待处理的像素用与中心像素相距固定伽马参数值的像素的平均值来代替（翻译的不是很好，这句），用选中区域像素的均值来代替待处理像素，这种方法在很多算法都可以见到。Nagao的滤波器是具用最小变动值的像素的平均值来代替，Lee在其改进的算法中使用灰度信息来选取像素区域，Graham和Prewitt定义了一个阀值，如果选中像素与邻边像素值的差的绝对值在阀值内，就取这些满足要求的像素值的均值，来代替选中像素。Rosenfeld在他的算法把高亮度的边界，线以及点排除使用。。。方法。箱滤波与本文的伽马滤波方法的主要区别在于箱滤波将围绕中心像素的像素灰度值固定，伽马滤波的优点有几点：（1）在不平滑边界的同时，边界区域附近的噪声将被平滑，因为只有边界一边的像素将在取均值的时候包含，而另一边对均值没有贡献；（2）一些像素分类的细节及宽度在一到三个像素的线性特征将被保留，因为在对均值的计算过程中，只有这些像素而不包括背景将会包括；（3）不会产生虚假的信息，不会对图像进行改动，因为没有用到掩码，不像Nagao和Lee的算法；另外，它计算起来比较高效，因为只涉及简单的比较和固定点数的增加操作。

A comparison of the sigma filter,the gradient inverse filter,Nagao's filter,and the median filter are conducted in this paper. Comparisons are based on the following criteria:(1)effectiveness in smoothing noise;(2)preservation of suble details and linear features;(3)immunity from shape distortion;(4)retention of step edges and sharpening of ramp edges;(5)removal of high-contrast spot noise;(6)computational efficiency. For the smoothing algorithms to be effective they are applied iteratively three times to test images of dimension 128*128. In many respects the sigma filter performs better than other algorithms except as regards the ability to remove sharp spot noise. Some methods of reducing this deficiency are presented.
对于伽马滤波，灰度反转滤波，Nagao滤波，还有中值滤波的比较，在本篇论文中均有涉及。对这些方法的比较主要基于以下几点：（1）在噪声平滑上的高效性;(2)对局部细节及线性特征的保留；（3）形状不变形；（4）保留step边界，锐化ramp边界；（5）能够去除高亮度噪声；(6)运算快速；为使算法更高效，我们选取128*128的图像进行测试，并且迭代3次。结果显示在很多方法伽马滤波表现性能更优异，除了在去除某些尖锐的噪声外。弥补这一不足，有一些方法，在本文中也有所展现。

2.The Sigma Filter
伽马滤波器

The noise in an image is generally considered as spatially uncorrelated and with continuous intensity spectrum. White Gaussian noise is an example. We shall regard as noise any random clutter of the size of three or fewer pixels. It is well known that that the "straight" averaging filter will smooth noise at the expense of blurring edges and smearing subtle details. An indiscriminate average of pixels in a window is the cause of the problem. As mentioned in Section 1 many schemes have been developed to overcome this problem. The merits of these algorithms will be explored in more detail in the next section. In this section, a conceptually simple algorithm is developed which easily excludes significantly different pixels from the average.
普遍认为一幅图像里的噪声在空间上是不相干的，并且在灰度范围上是连续的。白色高斯噪声就是一个例子。我们将把任意的分成三个或更小的尺寸的分类认为是噪声，我们大家都知道，如果直接地运用中值滤波将不光会平滑噪声，还会模糊边缘，并且丢失细节，为什么会这样呢，对像素不加区别地进行中值滤波是这一现象的直接原因。就像第1节中提到了很多算法都想法克服这种问题。在下一节里我们将对这些算法进行详述。在本节中，我们提出一种很简单的算法，可以很容易地将那些与中值差别的像素剔除。

Most image noise is Gaussian in distribution. The two-sigma probability is defined as the probability of a random variabe being within two standard deviations of its mean. The two-sigma probability for a one-dimensional Gaussian distribution is 0.955. This can be interpreted as meaning that 95.5% of random samples lie within the range of two standard deviations. In image smoothing, any pixel outside the two-sigma range most likely comes from a different population and, therefore, should be excluded from the average. If we assume that the a priori mean is the gray level of the pixel to be smoothed, we can establish a two-sigma range from the gray level and include in the average only those pixels within the two-sigma intensity. Let x(i,j) be the intensity or gray level of pixel(i,j), and x'(i,j)be the smoothed pixel(i,j). Also we assume that the noise is additive with zero mean and standard deviation theta. The sigma filter procedure is then described as follows:
大多数的噪声都是依照高斯分布的。2伽马概率被定义为中值的两个标准偏差的随机概率，对于一维的高斯分布，2伽马概率值是0.955，这也就是说有百分之95.5%的随机噪声存在于两个标准偏差之间。在图像平滑的过程中，那些不在2伽马概率值之间的像素，都可被去除。假定我们是对像素的灰度进行平滑处理，我们可以建立一个2值的伽马灰度区间，而只将在灰度区间里的像素进行求均值。用x(i,j)来代表像素(i,j)的亮度或灰度值，x'(i,j)代表是平滑后的像素(i,j）值。同时我们假定噪声对零均值及标准偏差theta是垒加的。伽马滤波过程描述如下：

(1)Establish an intensity range (x(i,j)+p,x(i,j)-p),where p=2*delta;
(2)Sum all pixels wich lie within the intensity range in a (2n+1,2m+1) window.
(3)Compute average by dividing the sum by the number of pixels in the sum.
(4)Then x'(x,j)=the average. (To reduce sharp spot noise, step(4) will be modified later in this section.)
Or, mathematically, let
if(（x(i,j)-p)<=x(k,l)&&（x(i,j)+p)>=x(k,l)) s(k,l)=1 else s(k,l)=0;
Then
temp1(i,j)=0;
temp2(i,j)=0;
for k=i-n:n+i
for l=j-m:m+j
temp1(i,j)+=s(k,l)*x(k,l);
for k=i-n:n＋i
for l=j-m:m+j
temp2(i,j)+=s(k,l);
x'(i,j)=temp1(i,j)/temp2(i,j);
（1）第一步，先建立一个灰度区间（x(i,j)+p,x(i,j)-p），其中p=2*delta;
（2）将所有在此灰度区间里的像素值加起来；
（3）将总和除以总像素数，得到一个均值
（4）将均值赋给x'(i,j);

The two-sigma range is generally large enough to include 95.5% of the pixels from the same distribution in the window, yet in most cases it is small enough to exclude pixels representing high-contrast edges and subtle details. Linear features such as roads one or two pixels wide are retained, because only those pixels with intensity near that of the feature are included in the average. The main drawback is that sharp spot noise represented by clusters of one or two pixels not be smoothed. This could be very annoying especially for a fairly noisy image. To remedy this, we shall replace the two-sigma average with the center pixel's immediate neighbor average, if M, the number of pixels within the intensity range, is less than a prespecified value K, In other words, step(4) is replaced by
if(M>K) x'(i,j)=two-sigma average else x'(i,j)=immediate neighbour average
虽然2伽马区间囊括窗口中同种分布的大多数像素，但对于大多数情况下，还是太小不足以将高亮度边缘及细节排除。像道路类型的只有一两个像素的线性特征需要保留，因为在计算均值的时候只有与该特征相近的像素被包括。这样导致那样只有一两个像素组成的聚类不会得到平滑处理。这对那些被噪声污染比较严重的图像，处理效果会很差。为了修正这种缺陷，我们应该将中心像素的直接领域像素均值来代替2伽马均值，如果M，在灰度区间的像素总数，小于一个指定的值K，换句话说，上述的第4步可以用如下来代替：
if(M>K) x'(i,j)=2伽马均值 else x'(i,j)=直接邻域均值

The value of K should be carefully chosen to remove isolated spot noise without destroying thin features and subtile details. For a 7*7 window, K should be less than 4, and it should be less than 3 for a 5*5 window. It should be noted that subtle textures within the two-sigma range will be wiped out after a few iterations. If conservation of texture information is required, a small p range and one or two iterations should be used.
为了在不破坏细微的特征和细节的情况下，K值需慎重选取。需要注意在几次迭代之后，在伽马区间里细微的纹理特征可能会被丢弃。如果需要保留纹理特征，那么，要将p值选择小一些，并且只要用到一到2次迭代就可以了。

For images with unknown noise characteristics, the intensity range p can be determined either from a rough estimation of the noise standard deviation in a flat area, or from the desirability of retaining the gray level difference between the desirable features and its background. The sigma filter can be applied repeatedly with reduced p after each iteration. Two or three iterations are generally sufficient to reduce the noise level significantly.
在不清楚图像的噪声特点的情况下，灰度区间p可以通过在一个较为平坦区域的噪声标准偏差的粗略估计，或者从保留待选取特征及背景的灰度差别来选取。在每次迭代后，可以适当地减小p值，一般情况下，两到三次迭代，便能很明显的削减噪声了。

As an illustration, Fig.1(A) shows a medical image of cell structure. The results of applying the 7*7 sigma filter once, twice and three times are shown in Fig.1(B),(C),and (D), respectively. The result of applying the median filter twice is shown in (F). It should be noted that (E) is the result of applying a derivative version of the sigma filter, to be discussed in Section 4
作为一个例子，A图像显示了一个细胞的医学图像，使用7＊7的伽马滤波作为一次，两次及三次的结果分别显示在B，C，D图中。与此相对照的，使用均值滤波后图像结果显示于F图中。需要注意的是，E图是对图像使用了衍生后的伽马滤波后的结果，这将要在第4部分详细阐述。

3.A Comparison of locally smoothing algorithms
局部平滑算法性能比较

Numerous local image smoothing algorithms have been developed recently. It is impractial to compare all of them in detail. The straight local average method is known to blur edges and details. Lev et al. applied a template matching technique to detect edges and lines and then replaced the pixel by a weighted average corresponding to the particular pattern detected. Twelve 3*3 masks are created and relatively complicated weighting schemes are proposed. This algorithm is not computationally efficient, nor is it very effective in smoothing noise, since the window size is small. Lee using a local statistics method , produced good results for images corrupted by both additive and multiplicative noise. However, artifacts are observed in some cases, and the computation of the local variance makes this algorithm somewhat inefficient. These two filtering algorithms are excluded in the present comparision. The recently published gradient inverse method , the edge preserving smoothing scheme of Nagao and Matsuyama, and the well-known median filter are chosen instead.
近来出现了很多局部平滑算法，如果想去细细地比较它们的优劣实在是不可取的。直接的局部平滑会模糊边缘和细节。Lev et al采用一种模板匹配的技术来检测边缘和线，然后用针对相应的模式，采用相应的加权值来代替像素值。这种算法采用了12个3＊3的掩模，并且使用了相对复杂的权重模式。因为使用的窗口尺寸太小，这种算法计算起来不高效，而且也在平滑噪声上也不是很有效。Lee采用一种局部统计方法，对于累加噪声以及乘法噪声污染后的图像产生很好的结果，然后，在某些情况下，会出现虚假的信息，并且不是很高效。基于以上原因，在本部分的算法比较中不涉及这两种算法。我们只对最近发表的灰度反转算法，Nagao和Matsuyama的边缘保留平滑模式算法，还有众周所知的均值滤波算法进行比较。

For completeness, brief descriptions of these three algorithms are given in this section. The gradient inverse weighting scheme employs a 3*3 window and computes for each pixel its inverse gradient weighted average with its neighboring pixels. The idea is to weight less those pixels having greater absolute differences with their center pixel. The procedure for processing x(i,j) in a 3*3 window is given as follows:
(1)Compute the inverse gradients of the eight neiboring pixels:
for k=-1:1
for l=-1:1
if x(i+k,j+l)!=x(i,j) g(k,l)=1/abs(x(i+k,j+l)-x(i,j)); else g(k,l)=1/2
(2)Compute weights for the eight neighbors:
sum=0;
for k=-1:1
for l=-1:1
sum+=g(k,l);
w(k,l)=1/2*g(k,l)/sum, w(i,j)=1/2;
(3)
for k=-1:1
for l=-1:1
x'(i,j)+=w(k,l)*x(i+k,j+l);
在这一节里，我们将对这三种算法一一简短叙述。灰度反转权重模式使用3＊3的窗口，并且使用邻域像素求解每个像素的反转灰度权重。算法的主要思想是计算邻域像素与中心像素灰度值的差值的绝对值。对像素x(i,j)的计算过程如下：
(1)计算邻域8个像素的反转灰度值
for k=-1:1
for l=-1:1
if x(i+k,j+l)!=x(i,j) g(k,l)=1/abs(x(i+k,j+l)-x(i,j)); else g(k,l)=1/2
(2)计算8个像素的权重
sum=0;
for k=-1:1
for l=-1:1
sum+=g(k,l);
w(k,l)=1/2*g(k,l)/sum, w(i,j)=1/2;
(3)
for k=-1:1
for l=-1:1
x'(i,j)+=w(k,l)*x(i+k,j+l);

Nagao and Matsuyama proposed an algorithm which selects the most homogeneous neighborhood and replaces the pixel by its neighborhood average. They created nine overlapped subregions in a 5*5 window as shown in Fig.2. The means and variances of the nine subregions are computed, and the center pixel is replaced by the mean of the subregion having the minimum variance.
Nagao和Matsuyama提出一种算法，选取最同质的邻域像素，然后用这些邻域像素的均值来代替该像素。像在图2中所示，一共产生了9个重叠的子区域。计算每一个子区域的均值及差异度，取最小差异度的子区域，并用其均值把中心像素值替换。

The median filter is more flexible. It can be applied columnwise,rowwise, and areawise. In our study, a 3*3 window is used, and the median of the nine pixels in the window represents the smoothed pixel. The reason for not using a large window is that a large window will smear details and edges, not to mention the higher computational load.
中值滤波较为方便，它适用于以列方向，行方向，区域方向。在我们的实验中，我们采用3＊3的窗口，并且用九个像素值的中值来代表平滑像素。之所以不采用更大些的窗口是因为更大的窗口会抹去细节和边缘，更不用提更高的计算次数。

Two test images shown in Figs.3 and 4, of dimension 128*128 pixels, are used in our comparison. In Fig.3, a computer generated pattern of bars with increasing width(one pixel, three pixels....15 pixels) is created, and corrupted with noise to test the ability to preserve linear features, the ability to smooth noise along edges,and the effectiveness of noise reduction in general. The average intensity of the bar is 150 and of the background is 50. Figure 4 is a natural aerial scene artificially corrupted with noise. The intensity levels in all images in this paper are between 0 and 255. Each algorithm is applied to the noisy image repeatedly 3 times. The sigma filter is applied in a 7*7 window with the intensity intervals 2p, p, and p/2,and K=2.
为用于我们的比较，我们选用了两幅图像，128＊128的，图3和图4。在图3里，我们使用计算机产生了宽度分别为1，3，... 15个像素的长条，然后用噪声腐蚀，傅用这个图片来测试保留线性特征、在边缘处平滑噪声以及减弱噪声的能力。条形图的灰度平均值为150，背景灰度值是50。图4显示了自然场景中一个被噪声腐蚀的天线的图片。在这篇论文中所有图片的灰度值均在0到255间，每个作用于噪声图像的算法均迭代使用3次，伽马滤波使用7＊7的窗口，灰度间隔为2*p,p以及p/2, 且K＝2；

(1)Effectiveness in Noise Smoothing
The efficiency of smoothing noise can be measured by the reduction in noise standard deviation or variance. For the images of Fig.3 the standard deviations of each smoothed image are computed from a flat area in the lower left corner. The resluts are listed in Table 1.
对平滑噪声的有有效性我们可以采用噪声的标准偏差的减少量来度量。对于图3，可以通过对左下角的平坦区域，来计算每个平滑图像的标准偏差，结果如表1

The gradient iverse filter is apparently the least efficient smoothing algorithm due to its small mask and the nature of its weighting scheme. The sigma filter is significantly superior in smoothing noise with a reduction of standard deviation by approximately a factor of ten. The Nagao and median filters are comparable in their ability to reduce noise.
灰度反转滤波显然是效果最差的，因为它选取的掩码最小，还有它选取的权重模式的限制。伽马滤波显著地减少了标准偏差，平滑了噪声。Nagao滤波和中值滤波在减少噪声的效果是相似的。

Table 1
Comparison of Reduction in Noise Standard Deviation
噪声标准差的减少量对比
Noise Standard Deviation
Smoothing Algorithms Bar pattern(p=10) Bar pattern(p=30)
Sigma Filter 0.81 3.54
Gradient inverse 5.74 17.84
Nagao's filter 2.48 10.87
Median filter 2.55 8.11

(2)Presevation of Subtle Details and of Linear Feature
对微小细节及线性特征的保留
In some images it is important to retain highly distinguishable subtle details and line features,such as piers and roads. In other applications, such as image segmentation, it may be desirable to remove subtle details. The sigma filter is effective in preserving subtle details and line features as long as the intensity difference between them and their background is greater than the two-sigma intensity range. The background pixels will be excluded from the average when processing a pixel which represents the road or the subtle detail. In fact , it would preserve even a single outlying pixel, if we were not using the threshold K for spot noise reduction. The gradient inverse method theoretically will smear any feature of any size i applied a sufficient number of times, since it includes all pixels in the average and only weights them less if the difference is large. Similarly the Nagao filter will blur and eventually devour any feature with dimensions of three pixels or less in any direction. This can be easily seen in a noise free one-dimensional case in Fig.5, in which the Nagao filter is equivalent to replacing the center pixel with the average of itself and its two neighbours on either side, whichever has the minimum variance. The center pixel of the three-pixel-wide pulse will drop in value after one application. The deterioration will continue slowly in the one-dimensional case, but much faster in the two dimensional case. As seen in Figs.3(D) and (I), the bars of width one and three pixels are almost completely wiped out. The 3*3 median filter will wipe out single pixel lines in one application, since in a 3*3 mask, among the nine pixels, six of them will be background pixels. Thus the median will approach the background pixel value. A bar with two pixels wide is a critical case, It has five to six pixels depending on the orientation of the bar. The median filter will swallow slightly curving or broken two-pixel-wide bars. For a 5*5 median filter, a three-pixel-wide bar will be wiped out in one application. The images in Fig.4 further substantiate the characteristics of these algorithms. Figures 4(C),(D),(E),and (F)are the results of applying the smoothing algorithms three times. The gradient inverse scheme shown in Fig.4(D)did not do much about the noise and slightly reduced the contrast of the image. As shown in Fig.4(E),Nagao's filter smeared bridges and subtle detail and created artifacts. The 3*3 median filter smeared the bridges and generally blurred the image. The sigma filter performed fairly well except for the sharp spot noise problem.
在一些图像里，保留那些显著的细节特征和线性特征，比如码头和道路。在另外一些工程中，比如图像分割，可能需要去除细节特征。伽马滤波在保留微小细节以及保留线性特征方面是有效的，只要微小细节及线性特征与背景的灰度差值大于2伽马灰度区间。在处理道路或细节时，背景像素将会被计算均值的过程排除在外。实际上，它将会保留甚至是一个孤立的像素，如果我们不使用阀值K。灰度反转方法在理论上将会抹去任意尺寸的特征，因为它在计算均值时，把所有像素都包括进去了，即使在像素值差别过大时，它的权重也很小。同样地，Nagao滤波器将会模糊，并最终吞噬在任意方向的三个像素或更少的特征。这可以在图5中很容易地看出，在图5中，Nagao滤波就相当于将中心像素及两边的邻居像素的均值来代替中心像素，而不管邻居像素是否与中心像素的差值最小。在运行程序后，三个像素宽的中心像素值将会下降。在一维的情况下，效果持续恶化，但到了二维的情况下，就快了许多。就像在图3中D及I所示，只有一个和3个像素的条将几乎被完全清除。3＊3的中值滤波将会清除单个像素行，因为在3＊3的掩模下，9个像素中有6个是背景像素，因此中值更趋近于背景像素。只有两个像素的长条是关键，它有5或6个像素依赖长条的方向。中值滤波将会轻微的曲折或折断两像素宽的长条。对于5＊5的中值滤波，在一次执行后，三像素宽的长条将会被清除。图4中近一步表明了这些算法的特征，4图中C，D，E，F是使用了算法3次的结果。图4D采用了灰度反转模式，但对噪声并没有起多大作用，只是轻微地减弱了图像的对比度。而图4中E彩了Nagao滤波，它消除了“桥”以细节，并产生了假象。3＊3的中值滤波消除了“桥”并且模糊了图像，而伽马滤波性能良好，除了一些尖锐的噪声问题。

(3)Immunity from Shape Distortion
防图形变形

The gradient inverse method is not effective in smoothing noise, but it is relatively free from artifacts and shape distortion. Nagao's filter, on the other hand, as shown in Fig.4(E), does create significant distortion because of the directional subregion average. It will round off corners of less than 90 degrees. Median filter is known to create artifacts. The 3*3 median filter will round off corners and produce patterns of patches, the same as Nagao's filter. As shown in Fig.4(C),the sigma filter is practically free of shape distortion.
灰度反转方法对平滑噪声并不是很有效，但不会产生虚假信息，且不会使图形形状发生变形。Nagao滤波，就像在图4(E)中所示，由于方向性的子区域均值，确实会出现很明显的变形。它将会绕开角度小于90度的角落。中值滤波会产生虚假信息，3＊3的中值滤波会消除边角，并且会产生块状的区域，就像Nagao滤波那样，这在图4(c)中展示出来；伽马滤波可以免于图形变形

(4)Retention of Step Edges and Sharpening Ramp Edges
保留阶梯状边缘及锐化屋脊型边缘
The intensity variations in the direction perpendicular to a sharp edge in th image plane from a step edge. Retaining the sharpness of a step edge is highly desirable in both image smoothing and segmentation.The gradient iverse filter will blur the step edge, as it computes the average on all pixels. The meidan filter will maintain a noise free step edge, but it will smear a noise step edge. Figure 6 shows a 3*3 mask moving through an edge. Assuming the edge is contaminated by noise, the 3*3 median filter replaces the center pixel with the fifth least bright pixel of the six pixels on the left side of the edge, while as the window moves right by one pixel, the center pixel is replaced by the fifth brightest among the six pixels on the dark side of the edge. Consequently the sharpness of the edge is degraded. The sigma filter, however, retains its sharpness by replacing the center pixel by the average of the six pixels.
在垂直于图像面板上锐利边缘方向上剧烈的变化形成阶梯状的边缘。在图像平滑及分割领域很需要将阶梯边缘保留。灰度反转滤波会模糊阶梯边缘，因为它是对所有像素计算中值。而中值滤波会保留无噪声的阶梯边缘，但它会在涂抹一个阶梯边缘噪声。图6展示了一个沿边缘移动的3＊3的掩模。假定边缘已被噪声污染，3＊3的中值滤波使用边缘左边的6个像素中第5个最不亮的像素值来代替中心像素，当窗口向右移动一个像素时，就用边缘较暗的一边的6个像素中第5个最亮像素来代替中心像素，结果边缘的尖锐程度被减弱。然而，伽马滤波，可以通过采用取6个像素的均值来代替中心像素的方法来保留尖锐程度。

Sharpening a ramp edge is generally of interest in studies of image segmentation by gray level difference. In this application Nagao's filter is excellent due to its directional subregion average. The other three algorithms will not sharpen a ramp edge but all will maintain a ramp edge fairly well. A derivative of the sigma filter which will sharpen a ramp edge will be discussed in the next section.
在使用灰度级差别对图像进行分割时，一般会对屋脊型边缘的锐化感兴趣。在本实验里，Nagao滤波因为它彩了方向性的子区域均值，效果比较明显。而其它的三种算法并不会对屋脊型边缘进行锐化，但对屋脊型边缘保留的较好。在下一部分里我们将讨论锐化屋脊型边缘的伽马滤波的改进版本。

(5)Removing Spot Noise
去除斑点噪声
The median filter is well known for its effectiveness in removing sparsely positioned sharp spot noise, since the spot noise has intensity at either end of the intensity scale. Nagao's filter is also effective, but requires a few iterations. The gradient inverse filter weights the spot noise much higher than its surrounding pixels. Consequently, it is not effective. The sigma filter with large window size is highly susceptible to spot noise, since no other pixel but the spot noise itself is within the two sigma range. The modified version with threshold K(as shown in Fig.3(G)) discussed in the last section will remove most isolated spot noise. However, sport noise near edges remains because the 7*7 mask contains several edge pixels which will fall into the two-sigma range. Increasing the value of K will further reduce the spot noise, but at the expense of blurring edges and subtle details. The spot noise can be further reduced by applying a 3*3 sigma filter with K=1, or 2. Figure 7 shows the effect of spot noise reduction by applying it to Figs.3(G) and 4(C) for K=1 and K=2. Figures 3(G）and 4(C) are repeated in Figs.7 for comparison. As shown in Fig.7(C), the spot noise is almost completely removed; however, the one-pixel-wide bar is badly broken up. Figures 6(E) and (F) show the effect on the aerial image.
中值滤波在去除空间上的斑点噪声比较有效，因为斑点噪声的灰度值在灰度范围的两端。Nagao滤波也很有效，只不过需要多次迭代。灰度反转滤波对斑点噪声的权重要大于周边的噪声权重，因此，它并不是很有效。具有较大窗口的伽马滤波对斑点噪声是比较敏感的，因为没有其他像素，除了斑点噪声在2伽马区间里。增加k值会减少斑点噪声，但会模糊边缘和细节。斑点噪声可以进一步地进行削减，通过使用3＊3，K=1或K=2的伽马滤波. 图7显示了对图3中G及图4中C分别采用K=1,K=2的伽马滤波后的效果。图3(G)和图4(C)是重复的，为了在图7中作对照。就像在图7中(C)中所示，斑点噪声几乎被全部去除，然后，一个像素宽的长条被严重的折断了。图6(E)和(F)显示了对天线图像的作用效果。

(6)Computational Efficiency
计算效率
In our comparison, the algorithms were coded in FORTRAN and no special efforts were devoted to accelertate their executions. The computations were carried out on a Data General NOVA 800 with a Comtal 8000 image display. The ratio of computational time required for images of size 128＊128 for each filter(per iteration) is listed in increasing order as follows:
(1)The sigma filter(7*7），1 unit of time;
(2)The median filter(3*3),1.5 units of time;
(3)The gradient inverse filter, 4.0 units of time;
(4)Nagao's filter, 11.0 units of time;
The sigma filter is the fastest algorithm in this group even with a 7*7 window, In our simulation, it took no more time than computing a straight 7*7 average. Nagao's filter is extremely slow, since it requires the computation of variance for nine subregions.
在对比中，算法都用FORTRAN编写，并且没有用特殊的方法来加快它们的执行速度。计算样本是Data General Nova 800，一共有8000幅图像，对128＊128图像的处理，不同的算法，执行时间的比率如下：
（1）伽马滤波（7＊7），1个单位时间
（2）中值滤波（3＊3），1.5个单位时间
（3）灰度反转滤波，4.0个单位时间
（4）Nagao滤波，11。0个单位时间
使用7*7窗口的伽马滤波是最快的，在我们的仿真中，直接计算7*7的均值是最快的。Nagao滤波极慢，因为它需要对9个子区域进行计算。

4.The Extended Sigma Filter
改进后的伽马滤波

The sigma filter can be easily extended to perform image enhancement,segmentation, smoothing of signal-dependent noise, and even 3-D images, Here, only a few of these possibilities will be mentioned.
伽马滤波可以很容易地进行扩展，进行图像增强，图像分割，还有单一噪声的图像平滑，甚至用于3-D图像, 在这里，我们只提及几种。

(1)The Biased Sigma Filter
偏差伽马滤波

This extended sigma filter will sharpen a ramp edge and also enhance the contrast of subtle detail. Tha bias is introducted by separately averaging pixels in the upper intensity range of (x(i,j),x(i,j)+p)and in the lower intensity range of (x(i,j),x(i,j)-p). The absolute difference between the upper average and x(i,j), and also the absolute difference between the lower average and x(i,j),are computed. The center pixel is replaced by the average which has the smaller absolute difference. The function of the biased sigma filter can be easily explained in a one-dimensional case. Figure 8(A) shows the effect of a seven-pixel-wide biased sigma filter with p=3, With one application, the ramp edge becomes much sharper, and it will approach a step edge as the number of applications increase. It should be cautioned that the intensity range p should be chosen to be relatively large. As shown in Fig.8(B) a ramp edge may become a two step edge with p=1. This algorithm is useful in sharpening edges in preprocessing for image segmentation by gray level difference and also in bringing out subtle details in a smoothed image. Figure 1(E)shows the image of Fig.1(D) processed by the biased sigma filter.
扩展后的伽马滤波可以锐化屋脊型边缘，并且增强细节的对比度。前提是要将计算均值分为两步来计算，一步在区间(x(i,j),x(i,j)+p),一步在区间(x(i,j),x(i,j)-p）,分别在两个区间里求均值与x(i,j)的差值的绝对值。中心像素然后用较小的绝对值来代替，这种伽马滤波在一维的情况下很好解释，图8 A中显示了p=3，7个像素宽的滤波效果，执行一次，屋脊型边缘更尖锐，但当执行次数增多时，它更接近于阶梯边缘。需要注意灰度区间p应该被选得大一些，像图8 B中所展示的当p=1时，屋脊型边缘变成了两个阶梯形边缘。这种算法在为采用灰度级差别进行图像分割的预处理时进行边缘锐化，以及去除平滑图像的细节上，这种算法较为有用。图1 E显示了对图1 D进行偏差滤波后的结果。

(2)Signal-Dependent Noise
信号相关噪声

Signal-dependent noise or speckles occur in coherent optical images as well as in synthetic aperture radar images. To deal with this noise, a reasonably effective method based on local statistics was recently proposed by Lee. In our experiment the sigma filter modified for signal-dependent noise performs better in many cases and requires much less computational time. The intensity range will not only float up and down with x(i,j) but also shrink or grow with x(i,j), since theta is a function of x(i,j), A more detailed discussion will be give in a separate study.
相干光学图像及合成孔径雷达影像会出现信号相关噪声或者是散斑。为处理这种噪声，在局部统计的基础上，Lee最近提出了一种有效的方法。在我们的实验中，经过修改后的伽马滤波，在处理信号相关噪声上多数情况下表现较好，并且需要较少的计算时间。灰度区间不仅围绕在x(i,j)上下浮动，而且随着x(i,j)增减，因为theta是x(i,j)的函数，对此更详细的讨论在另外一份研究中给出。

(3)Extension to 3-D Images
扩展到3D图像

It is straightforward to extend the sigma filter and its derivatives to 3-D image smoothing. The two-dimensional window will be replaced by a three-dimensional cube. Pixels within the cube are processed by the same procedures established for the two-dimensional case.
将伽马滤波扩展到3D图像是很简单易懂的。二维的窗口将被三维的立方体代替，在立方体内部的像素的处理方法和二维情况下是相同的。

5.Remarks
评论

(a)Most local smoothing algorithms do not require prespecified parameters.Clearly,this is a distinct advantage if the algorithm is to be effective for all image categories. The sigma filter does require specification of the intensity range and the size of the window. However, these parameters permit us to fine-tune the filter to a specific image or class of images. Once the characteristics of the sigma filter with respect to the parameters are understood, It is fairly easy determine the appropriate values. In addition, the computational efficiency of this algorithm permits us to adjust the parameters interactively.
多数局部平滑算法并不需要指明参数。明显地，如果算法对所有图像类别都有效的话，这将是一个显著的优势。伽马滤波器需要指明灰度区间以及窗口的尺寸。然而，这些参数允许我们对滤波器进行调整，以使能适应指定的图像类别。一旦了解了具有相应参数的伽马滤波器的特性，那就很简单地可以确实合适的值了。另外，算法的计算有效性允许我们交互地调整参数。

(b)The basic principle of the sigma filter can be incorporated into other algorithms to modify the characteristics of these filters. For example, it could be included in Nagao's filter or Lee's local statistics algorithm to obtain the two-sigma average in the directional subregion after it has been chosen by the procedures of these algorithms.
伽马滤波器的原理可以融合到其他算法里，来改变滤波器的特性。举例来说，Nagao滤波器及Lee滤波器在选取了定向的子区域后，可以选用其子区域的2伽马均值。

6.Conclusion
结论

A simple,effective,and computationally efficient noise smoothing algorithm has been developed. Detailed comparisons with a few local smoothing algorithms are made to substantiate the basic characteristics of this filter.The procedure and strategy of utilizing this filter has been explored.Applications of this filter to image segmention and other problems are currently under investigation. It is hoped that the sigma filter will be accepted as a basic digital image processing technique because of its simplicity and effectiveness.
本文提出一个简单的、高效的、计算高效的噪声平滑算法。几中平滑算法的详细比较证明了该滤波器的基本特性。展示了实现该滤波器的程序及结构。将该滤波器运用于图像分割及其它问题上，目前正在研究。基于伽马滤波的简单及有效性，希望伽马滤波能成为一种基本的数字图像处理技术。

终于翻译完了，这篇具有15页的论文在翻译了两天后，终于翻译完了。终于体会到翻译的辛苦了，虽然文中的绝大部分单词我都认识，但对于有些句子还是很难翻译的。翻译完了，下一步就是实现伽马滤波了。