二维中值滤波的快速算法

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"A Fast Algorithm for Two-Dimensional Median Filtering - A method for efficient 2D median filtering in image processing, utilizing previous window partition results to enhance speed and comparing its performance with Huang's histogram algorithm." 在图像处理领域，中值滤波是一种常用的去噪技术，尤其适用于消除椒盐噪声和斑点噪声。中值滤波器通过将像素点替换为其邻域内的中值来工作，这个邻域通常是一个窗口。传统的中值滤波算法会逐个移动窗口并计算每个位置的中值，这在处理大型图像时效率较低。标题提及的“一种快速的二维中值滤波算法”是由M. Omairahmad和Duraisamy Sundararajan提出的。他们在1987年的《IEEE Transactions on Circuits and Systems》上发表了这一研究成果。这篇论文的核心是设计了一种新的方法，可以在计算窗口中值时利用前一个窗口的分区结果，从而显著提高算法的速度。中值是将一组数值分为两个具有相等元素数量的子集的数，该数在其中一个子集中大于或等于所有元素，并在另一个子集中小于或等于所有元素。在二维滤波中，这意味着需要对窗口内的所有像素值进行排序，然后找到位于中间的值。该算法的独特之处在于，它利用了历史数据的连续性，减少了重复计算，特别是排序步骤的开销。这使得算法的执行时间大大缩短，并且与数据值所用位数的多少无关，因此对于各种数据类型和精度都能保持高效。为了验证其性能，研究人员在VAX 11/780计算机上运行了该算法，并将其与黄光裕（Huang）提出的基于直方图的中值滤波算法进行了对比。结果显示，新提出的算法在执行速度上更快，而且速度不受数据表示位数的影响，这表明它在处理大数据量时具有更好的可扩展性和效率。这篇论文提出的快速二维中值滤波算法是对传统中值滤波方法的重要改进，它降低了计算复杂度，提升了实时处理图像的能力，尤其适用于需要实时处理或处理大量数据的应用场景，如视频处理、遥感图像分析和医学成像等领域。

1366

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL.

CAS-34,

NO. 11, NOVEMBER

1987

24 31 28

11 23

21 24 24 21 23 14

23 25 14 23 8

22 24 6 15 16

23 25 5

24 6 15 16 7

Subset1

Subset 1

5 '2 8 3

13 11 7

4 15 16 , 8

24 31 28 23 23

Subset2

31 28 23 23 23

Subset 2

(b)

Fig. 2. (a) Pixels in a section of the image of Exam le 1 with the

window corresponding to the pixel f(3,2). (b) The win B ow of (a) after

Fig. 3. (a) The window corresponding to the pixel f(3,3) of Example 1.

partitioning.

(b) The partitioned window indicating the position of the border at

various stages.

from one subset to the other is usually found to be less

than

.S + 1)/2 by first pushing all the elements that are

less than the median of the past first window into Subset 1.

Example I:

Consider a section of an image shown in Fig. 2(a).

Assuming a window size of 5

5, the elements in the

window corresponding to the pixel

f(3,2)

is shown en-

closed in a box. The elements in each column of the

window are individually sorted in ascending order and

then the window is partitioned into two subsets, as shown

in Fig. 2(b). Subset 1 has 13 smallest elements of the

window and the 13th largest element of this set (element

22 for our example) is the median corresponding to the

pixel f(3,2). The median corresponding to the pixel

f (3,3)

of Fig. 2(a) with its window shown in Fig. 3(a), is found

using the following four steps.

1) Discard the five elements (21, 22, 23, 23, and 24) of

the leftmost column of the window corresponding to f(3,2)

(Fig. 2(b)). Due to this operation, the number of elements

in Subset 1 reduces to 11 elements (i.e., elements 21 and 22

are removed) and that in Subset 2 decreases to 9 elements

(i.e., elements 23, 23, and 24 are removed).

2) The ordered rightmost column of the window cor-

responding to f(3,3) is obtained from the ordered right-

most column corresponding to the pixel

f(2,3)

(i.e., the

elements 7, 7, 8, 14, and 23) by deleting the pixel 7 and

inserting the pixel 3 at an appropriate position. Thus the

elements of the rightmost column corresponding to f(3,3)

are 3, 7, 8, 14, and 23. This process results in the window

shown in Fig. 3(b).

3) The Subsets 1 and 2 of step 1) are updated by adding

the elements of the rightmost column obtained in step 2.

24 24 21 23 14

25 14 23 8 8

The elements whose values are less than or equal to the

past median (22) corresponding to the pixel f(3,2) is

pushed into Subset 1 while the remaining elements go into

Subset 2. After this operation, Subset 1 has 15 elements

and Subset 2 has 10 elements as shown by the thin-line

partition of Fig. 3(b).

4) Since the number of elements in Subset 1 is 15, the

two largest elements of Subset 1 along the border is

pushed into Subset 2, thus moving the border between the

two subsets upward. The first time, element 21, the largest

element along the border in Subset 1, is pushed into Subset

2 with the position of the border indicated by the dotted

line of Fig. 3(b). Next, the element 17, the largest element

along the new border is pushed into Subset 2 resulting in

the final partitioning shown by the solid thick line of Fig.

3(b). At this point, Subset 1 has 13 elements which are less

than or equal to the smallest number in Subset 2. The

largest element (16) in Subset 1 which is the largest ele-

ment along the final border is the median corresponding to

the pixel f(3,3).

From the above discussion, it is clear that there are three

major steps in the implementation of the algorithm: (i)

deleting and inserting of an element in forming a new

column vector, (ii) finding the position of partitioning of

the new column vector, and (iii) moving the elements from

one subset to the other. These operations are repeated for

each window. A direct implementation of these operations

results in a large execution time. The algorithm coded in C

language is presented in the Appendix. As seen from the

Appendix and from the explanation in the next section, a

careful implementation of the algorithm is essential to

achieve a fast execution time.

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