查看“︁闭运算”︁的源代码

[[File:Closing.png|right|thumb|如果選擇灰色圓盤當 B(kernel)，對兩個深藍色的正方形做closing，會多出淺藍色的部分.]]
在數學形態學中，集合B對集合''A''的Closing是先dilation再用erosion。

: <math>A\bullet B = (A\oplus B)\ominus B, \, </math>

<math>\oplus</math>和<math>\ominus</math>分别表示dilation和erosion。

在影像處理中，closing和[[开运算 (形态学)|opening]]是用來對影像除噪，這邊通常A就是我們輸入的影像，B則是kernel。

== 特性 ==

* Idempotent，即<math>(A\bullet B)\bullet B=A\bullet B</math> .
* Increasing，如果<math>A\subseteq C</math> ， 然後<math>A\bullet B \subseteq C\bullet B</math> .
* Extensive，即<math>A\subseteq A\bullet B</math> .
* Translation invariant.

== 實作 ==
這邊對binary image做closing的操作，kernel為5*5大小，4個角落被去掉。

closing不一定要先dilation一次再用erosion一次，也可以使用closing = dilation k times + erosion k times。

當想要closing的洞比較大時，可以使用更高的k次。<syntaxhighlight lang="python3">
def dilation(img):
  img1 = np.zeros((img.shape[0], img.shape[1]), dtype="int")
  for i in range(img.shape[0]):
    for j in range(img.shape[1]):
      if img[i][j]>0:
        row = i-2; col = j-2; # The octogonal 4-5-5-5-4 kernel
        for w in range(5):
          for h in range(5):
            if ( w==0 and h==0 ) or ( w==0 and h==4 ) or ( w==4 and h==0 ) or ( w==4 and h==4 ):
              continue
            else:
              i2 = row + w; j2 = col + h;
              if i2 >= 0 and j2 >= 0 and i2 < img.shape[0] and j2 < img.shape[1]:
                img1[i2][j2] = 255
  return img1

def erosion(img):
  img1 = np.zeros((img.shape[0], img.shape[1]), dtype="int")
  for i in range(img.shape[0]):
    for j in range(img.shape[1]):
        flag = 0
        row = i-2; col = j-2; # The octogonal 4-5-5-5-4 kernel
        for w in range(5):
          for h in range(5):
            if ( w==0 and h==0 ) or ( w==0 and h==4 ) or ( w==4 and h==0 ) or ( w==4 and h==4 ):
              continue
            else:
              i2 = row + w; j2 = col + h;
              if i2 > 0 and j2 > 0 and i2 < img.shape[0] and j2 < img.shape[1]:
                if img[i2][j2] == 0:
                  flag = 1
        if flag == 0:
          img1[i][j] = 255
  return img1
  
def closing(img):
  return erosion(dilation(img))
</syntaxhighlight>

== 參考 ==

* ''Image Analysis and Mathematical Morphology'' by Jean Serra, {{ISBN|0-12-637240-3}} (1982)
* ''Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances'' by Jean Serra, {{ISBN|0-12-637241-1}} (1988)
* ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, {{ISBN|0-8194-0845-X}} (1992)
[[Category:數位幾何學]]
[[Category:数学形态学]]