Yabewa (Tsarin Halitta)

**Zazzagewa** (yawanci ana wakilta ta **⊖** ) ɗaya ne daga cikin mahimman ayyuka guda biyu (ɗayan kasancewa dilation ) a cikin sarrafa hoto na ƙirar halitta wanda duk sauran ayyukan ƙirar halittar suka samo asali. An samo asali ne don hotunan binaryar, daga baya an mika shi zuwa hotuna masu launin toka, kuma daga baya don kammala lattices . Ayyukan zaizayarwa yawanci yana amfani da sifa don bincike da rage sifofin da ke cikin hoton shigarwar.

## Zazzagewar binary.
A cikin ilimin halittar jiki na binary, ana kallon hoto azaman yanki na sararin Euclidean \mathbb{R}^d ko grid ɗin lamba \mathbb{Z}^d, don wani girma *d* .

Mahimmin ra'ayi a cikin ilimin halittar jiki na binary shine bincika hoto tare da siffa mai sauƙi, wanda aka riga aka tsara, zana yanke shawara kan yadda wannan sifa ta dace ko rasa sifofi a cikin hoton. Wannan "bincike" mai sauƙi ana kiransa structuring element, kuma shi kansa hoton binary (watau yanki na sarari ko grid).

Bari *E* ya zama sararin Euclidean ko grid mai lamba, da *Hoton* binary a cikin *E.* **Rushewar** hoton binaryar *A* ta hanyar sifa *B* an bayyana shi ta:

:: A \ominus B = \{z\in E | B_{z} \subseteq A\} ,

inda *B* *z* shine fassarar *B* ta vector z, watau, B_z = \{b+z|b\in B\}, \forall z\in E .

Lokacin da structuring element *B* yana da cibiya (misali diski ko murabba'i), kuma wannan cibiyar tana kan asalin *E*, to ana iya fahimtar yazawar *A* ta *B* a matsayin wurin maki da tsakiyar *B* ya isa. lokacin da *B* ya motsa cikin *A.* Misali, lalacewar murabba'i na gefe 10, wanda ke a tsakiya a asalin, ta faifan radius 2, wanda kuma ya ke a tsakiya a asalin, murabba'in gefe 6 ne mai tsakiya a asalin.

Rushewar *A* ta *B* kuma an bayar da ita ta furcin: A \ominus B = \bigcap_{b\in B} A_{-b}, inda *A <sub>-b</sub>* ke nuna fassarar *A* ta *-b* .

Wannan kuma an fi sani da shi da bambancin Minkowski .

### Misali
Ace A shine matrix 13 x 13 kuma B shine matrix 3 x 3:
 1 1 1 1 1 1 1 1 1 1 1 1 1    
   1 1 1 1 1 1 0 1 1 1 1 1 1  
   1 1 1 1 1 1 1 1 1 1 1 1 1  
   1 1 1 1 1 1 1 1 1 1 1 1 1  
   1 1 1 1 1 1 1 1 1 1 1 1 1        
   1 1 1 1 1 1 1 1 1 1 1 1 1 1
   1 1 1 1 1 1 1 1 1 1 1 1 1 1
   1 1 1 1 1 1 1 1 1 1 1 1 1 1
   1 1 1 1 1 1 1 1 1 1 1 1 1    
   1 1 1 1 1 1 1 1 1 1 1 1 1  
   1 1 1 1 1 1 1 1 1 1 1 1 1  
   1 1 1 1 1 1 1 1 1 1 1 1 1  
   1 1 1 1 1 1 1 1 1 1 1 1 1
A ɗauka cewa asalin B yana tsakiyarsa, ga kowane pixel a cikin A **yana ɗaukaka** asalin B, idan B yana kuma ƙunshe da A gaba ɗaya pixel ɗin yana riƙe, in ba haka ba za a share shi.

Don haka **yazawar** A ta B ana ba da wannan matrix 13 x 13.
 0 0 0 0 0 0 0 0 0 0 0 0
   0 1 1 1 1 0 0 0 1 1 1 1 0
   0 1 1 1 1 0 0 0 1 1 1 1 0
   0 1 1 1 1 1 1 1 1 1 1 1 0
   0 1 1 1 1 1 1 1 1 1 1 1 0
   0 1 1 1 1 1 1 1 1 1 1 1 0 
   0 1 1 1 1 1 1 1 1 1 1 1 0
   0 1 1 1 1 1 1 1 1 1 1 1 0 
   0 1 1 1 1 1 1 1 1 1 1 1 0 
   0 1 1 1 1 1 1 1 1 1 1 1 0
   0 1 1 1 1 1 1 1 1 1 1 1 0
   0 1 1 1 1 1 1 1 1 1 1 1 0
   0 0 0 0 0 0 0 0 0 0 0 0
Wannan yana nufin cewa kawai lokacin da B ya **ƙunshi gabaɗaya** a cikin A za a kiyaye ƙimar pixels, in ba haka ba yana sharewa ko lalacewa.

### Kayayyaki

* Rushewar fassarar ba ta bambanta .
* Yana karuwa, wato, idan A\subseteq C, sannan A\ominus B \subseteq C\ominus B .
* Idan asalin *E* yana cikin tsarin tsarin *B*, to, yashwar ya zama *anti-extensive*, watau. A\ominus B\subseteq A .
* Zazzagewar ya gamsar (A\ominus B)\ominus C = A\ominus (B\oplus C), ku \oplus yana nuna fa'idar halittar jiki .
* Yazara yana rarraba kan mahadar da aka saita

## Yashwar launin toka.

A cikin ilimin halittar jiki mai launin toka, hotuna ayyuka ne da ke tsara sararin Euclidean ko grid *E* cikin \mathbb{R}\cup\{\infty,-\infty\}, ku \mathbb{R} shine saitin hakikanin gaskiya, \infty kashi ne mafi girma fiye da kowane lamba na gaske, kuma -\infty kashi ne karami fiye da kowane lamba na gaske.

Nuna hoto ta *f(x)* da sikelin sikeli mai launin toka ta *b(x)*, inda B shine sarari wanda b(x) aka siffanta, yashewar *f* by *b* yana bayarwa ta

:: (f\ominus b)(x)=\inf_{y\in B}[f(x+y)-b(y)] ,

inda "inf" ke nuna rashin lafiya .

Ma'ana lalacewar ma'ana ita ce mafi ƙarancin maki a cikin unguwarsu, tare da siffanta wannan unguwar ta hanyar sifa. Ta wannan hanyar yana kama da yawancin nau'ikan masu tace hoto kamar matatar tsaka-tsaki da tace gaussian .

## Karfewa a kan cikakken lattices.
Cikakkun lattices an yi odar jeri ne a jeri, inda kowane yanki yana da rashin ƙarfi da babba . Musamman, yana ƙunshe da ƙaramin sinadari da mafi girman sinadari (wanda kuma ake nuni da "duniya").

Bari (L,\leq) zama cikakken lattice, tare da maras lafiya kuma mafi girman alama ta \wedge kuma \vee, bi da bi. sararin samaniya da mafi ƙanƙanta abubuwan da *U* da ke nuna alamarta \emptyset, bi da bi. Bugu da ƙari, bari \{ X_{i} \} zama tarin abubuwa daga *L.*

Yazawa a ciki (L,\leq) kowane ma'aikaci ne \varepsilon: L\rightarrow L wanda ke rarraba kan marasa lafiya, kuma yana kiyaye duniya. Ie:

* \bigwedge_{i}\varepsilon(X_i)=\varepsilon\left(\bigwedge_{i} X_i\right) ,
* \varepsilon(U)=U .

## Dubi kuma

* Halin lissafi
* Rashin hankali
* Budewa
* Rufewa

## Manazarta.

* *Nazarin Hotuna da Nazarin Lissafi* na Jean Serra,   (1982)
* Hoton Bincike da Lissafin Lissafi, Volume 2: Ci gaban Ka'idoji na Jean Serra,   (1988)
* An *gabatar* da Morphological Image Processing by Edward R. Dougherty,   (1992)
* Morphological Image Analysis; Ka'idoji da Aikace-aikacen Pierre Soille,   (1999)
* R. C. Gonzalez da R. E. Woods, sarrafa hoto na dijital, 2nd ed. [Hasiya]