F #, morphology of binary images

Introduction

As applied to binary images, morphology is used to describe any properties of this image. And operations of mathematical morphology are used for transformations of sets. In this regard, they are conveniently used for processing binary images with subsequent analysis.

The article will discuss the basic operations of mathematical (binary) morphology in the context of F #. And also, at the end of the article I will give an example of the analysis of the result, which can be used in machine vision algorithms.

This article will use the material discussed in my previous article on the analysis of the labeling algorithm for related components. You can familiarize yourself with the article here: F #, an algorithm for marking the connected components of an image .

Basic operations of binary morphology

In order to conduct a binary morphology operation, we need

• Original binary image (B)
• Structural element (S)
  

  A structuring element is a description of an area of ​​some shape. This area can have any size and shape, which can be represented as a binary image
  

,

$$

$$Box(3, 5) = \begin{matrix}*&*&*&*&*\\*&*&*&*&*\\*&*&*&*&*\end{matrix}$$

$$

$$Disk(5, 5) = \begin{matrix}0&*&*&*&0\\*&*&*&*&*\\*&*&*&*&*\\*&*&*&*&*\\0&*&*&*&0\end{matrix}$$

$$

$$Ring(5, 5) = \begin{matrix}0&*&*&*&0\\*&0&0&0&*\\*&0&0&0&*\\*&0&0&0&*\\0&*&*&*&0\end{matrix}$$

, , .

. , . , .

.

B S B ⊕ S

$$

$$B \oplus S = \underset{b\in B}{\cup} S_b$$

. S . , , .

.

Figure 1: B

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0\\ *&*&*&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&*&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&*&*&0&0&0&0\\ 0&0&0&0&0&0&0&0 \end{matrix}$$

Figure 2: S

$$

$$\begin{matrix} *&*&*\\ *&*&*\\ *&*&* \end{matrix}$$

Figure 3: B ⊕ S

$$

$$\begin{matrix} *&*&*&*&*&*&*&*\\ *&*&*&*&*&*&*&*\\ *&*&*&*&*&*&*&*\\ 0&*&*&*&*&*&*&*\\ 0&*&*&*&*&*&*&*\\ 0&*&*&*&*&*&*&*\\ 0&*&*&*&*&*&*&*\\ 0&*&*&*&*&0&0&0\\ \end{matrix}$$

F#, .

let (>.) value checkValue = if value > checkValue then checkValue else value
let (<.) value checkValue = if value < checkValue then checkValue else value

let subArrayOfMaks array2d mask (centerX, centerY) (maskCenterX, maskCenterY) = 
        let (rows, cols) = Matrix.sizes {values = array2d}
        let (maskRows, maskCols) = Matrix.sizes {values = mask}

        let x1 = centerX - maskCenterX
        let y1 = centerY - maskCenterY
        let x2 = x1 + maskCols - 1
        let y2 = y1 + maskRows - 1

        (x1 <. 0, x2 >. (cols - 1), y1 <. 0, y2 >. (rows - 1))

let upbuilding array2d mask (maskCenterX, maskCenterY) =

        let sub source mask (x, y) (maskCenterX, maskCenterY) =
            let (x1, x2, y1, y2) = subArrayOfMaks source mask (x, y) (maskCenterX, maskCenterY)
            source.[y1..y2, x1..x2] <- mask

        let copy = (Matrix.cloneO {values = array2d}).values
        array2d |> Array2D.iteri (fun y x v -> if v = 1 then sub copy mask (x, y) (maskCenterX, maskCenterY))

        copy

upbuilding

let upbuilding array2d mask (maskCenterX, maskCenterY) =

, .

>. <.

let (>.) value checkValue = if value > checkValue then checkValue else value
let (<.) value checkValue = if value < checkValue then checkValue else value

value —
checkValue — ( )

value ( ) checkValue, checkValue, — value. nil Swift (??), , ( ).
,

2 <. 3 //  3

3 <. 2 //  3

, .

subArrayOfMaks

let subArrayOfMaks array2d mask (centerX, centerY) (maskCenterX, maskCenterY) = 

.

array2d — ( )
mask — ()
(centerX, centerY) — ,
(maskCenterX, maskCenterY) — . , . , .

upbuilding

array2d —
mask —
(maskCenterX, maskCenterY) —

let upbuilding array2d mask (maskCenterX, maskCenterY) =

        //      ,        
        let sub source mask (x, y) (maskCenterX, maskCenterY) =
            //    
            let (x1, x2, y1, y2) = subArrayOfMaks source mask (x, y) (maskCenterX, maskCenterY)
            //    
            source.[y1..y2, x1..x2] <- mask

        //    ,  
        let copy = (Matrix.cloneO {values = array2d}).values
        //      
        array2d |> Array2D.iteri (fun y x v -> if v = 1 then sub copy mask (x, y) (maskCenterX, maskCenterY))

        //    
        copy

, .

B S B ⊖ S

$$

$$B \ominus S = { b | b + s \in B ∀s \in S }$$

"" . , .

( , Figure 1).

Figure 4: B ⊖ S

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&*&*&0&0\\ 0&0&0&0&*&*&0&0\\ 0&0&0&0&*&*&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{matrix}$$

F#. , F#

//    ,     array2d   mask?
let contains array2d mask =
        if Matrix.isEquallySized (Matrix.ofArray2D array2d) (Matrix.ofArray2D mask) then
            not (array2d
                 |> Array2D.mapi (fun x y v -> if mask.[x, y ] = 0 then 1
                                               elif mask.[x, y] = v then 1
                                               else 0)
                 |> Seq.cast<int>
                 |> Seq.contains 0)
        else false

// ,   
let erosion array2d mask (maskCenterX, maskCenterY) =

        let sub source (dest: int [,]) mask (x, y) (maskCenterX, maskCenterY) =
            let (x1, x2, y1, y2) = subArrayOfMaks source mask (x, y) (maskCenterX, maskCenterY)
            let subArray2d = source.[y1..y2, x1..x2]

            if contains subArray2d mask then
                dest.[y, x] <- 1

        let copy = (Matrix.cloneO {values = array2d}).values
        array2d |> Array2D.iteri (fun y x v -> if v = 1 then sub array2d copy mask (x, y) (maskCenterX, maskCenterY))

        copy

, . , , .

.

B S B • S

$$

$$B • S = (B \oplus S) \ominus S$$

F#

let closure array2d mask (centerX, centerY) = 
        let step1 = upbuilding array2d mask (centerX, centerY)
        let step2 = erosion step1 mask (centerX, centerY)

        step2

Figure 5: B • S

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0\\ 0&*&*&*&*&*&*&0\\ 0&0&*&*&*&*&*&0\\ 0&0&*&*&*&*&*&0\\ 0&0&*&*&*&*&*&0\\ 0&0&*&*&*&*&*&0\\ 0&0&*&*&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{matrix}$$

B S B ◦ S

$$

$$B ◦ S= (B \ominus S) \oplus S$$

F#

let opening array2d mask (centerX, centerY) = 
        let step1 = erosion array2d mask (centerX, centerY)
        let step2 = upbuilding step1 mask (centerX, centerY)

        step2

Figure 6: B ◦ S

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0\\ 0&0&0&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&0&*&*&*&*&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{matrix}$$

, , , . , . , .

. , . , .

Figure 7:

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0&0&0\\ 0&0&*&0&0&0&0&0&0&0\\ 0&*&*&*&0&0&0&*&0&0\\ 0&0&0&0&0&0&0&0&*&0\\ 0&0&0&0&*&0&0&0&0&0\\ 0&0&0&0&*&*&0&0&*&0\\ 0&0&0&*&*&*&*&0&*&0\\ 0&0&0&*&0&0&0&0&*&0\\ 0&0&0&0&0&0&0&*&*&0\\ 0&0&0&0&0&0&0&0&0&0\\ \end{matrix}$$

Figure 8:

$$

$$\begin{matrix} *\\ *\\ *\\ \end{matrix}$$

, , ( , ). ,

Figure 9:

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&*&0&0&0&0&0\\ 0&0&0&0&*&*&0&0&*&0\\ 0&0&0&*&*&*&*&0&*&0\\ 0&0&0&*&0&0&0&0&*&0\\ 0&0&0&0&0&0&0&*&*&0\\ 0&0&0&0&0&0&0&0&0&0\\ \end{matrix}$$

F#

//    
let origin = array2D [[0;0;0;0;0;0;0;0;0;0]
                      [0;0;1;0;0;0;0;0;0;0]
                      [0;1;1;1;0;0;0;1;0;0]
                      [0;0;0;0;0;0;0;0;1;0]
                      [0;0;0;0;1;0;0;0;0;0]
                      [0;0;0;0;1;1;0;0;1;0]
                      [0;0;0;1;1;1;1;0;1;0]
                      [0;0;0;1;0;0;0;0;1;0]
                      [0;0;0;0;0;0;0;1;1;0]
                      [0;0;0;0;0;0;0;0;0;0]]

//  
let mask = array2D [[1]
                    [1]
                    [1]]

//      
let marked = Algorithms.markingOfConnectedComponents origin
//  
let erosion = Algorithms.erosion origin mask (0, 1)

printfn "origin = \n %A" origin
printfn "marked = \n %A" marked
printfn "erosion =\n %A" erosion

//        .       ()
let set = erosion |> Seq.cast<int> |> Seq.filter ((<>)0) |> Set.ofSeq

//     ,     
let copy = (Matrix.cloneO {values = marked}).values

//         
erosion
|> Array2D.iteri (fun row col v -> if v = 1 then
                            //     
                            let label = marked.[row, col]
                            if not (set.Contains label) then
                                //       ,       
                                marked |> Array2D.iteri (fun row1 col1 v1 -> if v1 = label then copy.[row1, col1] <- 1)
                         )

printfn "copy =\n %A" copy

origin = 
 [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 1; 0; 0; 0; 0; 0; 0; 0]
 [0; 1; 1; 1; 0; 0; 0; 1; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 1; 0]
 [0; 0; 0; 0; 1; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 1; 1; 0; 0; 1; 0]
 [0; 0; 0; 1; 1; 1; 1; 0; 1; 0]
 [0; 0; 0; 1; 0; 0; 0; 0; 1; 0]
 [0; 0; 0; 0; 0; 0; 0; 1; 1; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]
marked = 
 [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 1; 0; 0; 0; 0; 0; 0; 0]
 [0; 1; 1; 1; 0; 0; 0; 3; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 4; 0]
 [0; 0; 0; 0; 5; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 5; 5; 0; 0; 6; 0]
 [0; 0; 0; 5; 5; 5; 5; 0; 6; 0]
 [0; 0; 0; 5; 0; 0; 0; 0; 6; 0]
 [0; 0; 0; 0; 0; 0; 0; 6; 6; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]
erosion =
 [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 1; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 1; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 1; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]
copy =
 [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 1; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 1; 1; 0; 0; 1; 0]
 [0; 0; 0; 1; 1; 1; 1; 0; 1; 0]
 [0; 0; 0; 1; 0; 0; 0; 0; 1; 0]
 [0; 0; 0; 0; 0; 0; 0; 1; 1; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]

, , ,

F#

Figure 10:

$$

$$A= \begin{matrix} 0&*&0\\ 0&*&*\\ 0&*&0\\ \end{matrix} , B= \begin{matrix} 0&0&0\\ *&*&*\\ 0&0&0\\ \end{matrix}$$

A B

$$

$$A \cup B$$

Figure 11:

$$

$$\begin{matrix} 0&*&0\\ *&*&*\\ 0&*&0\\ \end{matrix}$$

A B

$$

$$A \cap B$$

Figure 12:

$$

$$\begin{matrix} 0&0&0\\ 0&*&*\\ 0&0&0\\ \end{matrix}$$

A

$$

$$A^c$$

Figure 13:

$$

$$\begin{matrix} *&0&*\\ *&0&0\\ *&0&*\\ \end{matrix}$$

A B

$$

$$A/B$$

Figure 13:

$$

$$\begin{matrix} 0&*&0\\ 0&0&0\\ 0&*&0\\ \end{matrix}$$

F#

// 
let union array2d1 array2d2 = 
        if Matrix.isEquallyArraySized array2d1 array2d2 then
            array2d1 |> Array2D.mapi (fun x y v -> v ||| array2d2.[x, y])
        else failwith "array2d1 is not equal to array2d2"

// 
let intersection array2d1 array2d2 = 
        if Matrix.isEquallyArraySized array2d1 array2d2 then
            array2d1 |> Array2D.mapi (fun x y v -> v &&& array2d2.[x, y])
        else failwith "array2d1 is not equal to array2d2"

//  (    )
let inverse array2d =
        array2d |> Array2D.mapi (fun _ _ v -> v ^^^ 1)

let complement array2d = 
        inverse array2d

// 
let difference array2d1 array2d2 = 
        if Matrix.isEquallyArraySized array2d1 array2d2 then
            array2d1 |> Array2D.mapi (fun x y v -> if v <> array2d2.[x, y] then v else 0)
        else failwith "array2d1 is not equal to array2d2"

, . , . .

Figure 14:

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&*&*&*&*&0&0\\ 0&*&*&*&*&*&0&0\\ 0&*&*&*&*&*&*&1\\ 0&*&*&*&*&*&*&*\\ 0&0&*&*&*&*&*&*\\ 0&0&0&0&0&0&0&0 \end{matrix}$$

Figure 15:

$$

$$\begin{matrix} 0&*&0\\ *&*&*\\ 0&*&0\\ \end{matrix}$$

F#

let borderAllocation array2d mask (maskCenterX, maskCenterY) = 
        let e = erosion array2d mask (maskCenterX, maskCenterY)
        let border = difference array2d e
        border

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&*&*&*&*&0&0\\ 0&*&0&0&0&*&0&0\\ 0&*&0&0&0&0&*&1\\ 0&*&0&0&0&0&0&*\\ 0&0&*&*&*&*&*&*\\ 0&0&0&0&0&0&0&0 \end{matrix}$$

. , . , . , , . , . , , .

, , : ,

$$

$$\begin{matrix} 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&*&*&*&*&0&0\\ *&*&*&*&0&0&0&0&0&*&*&*&*&*&*&0\\ *&*&*&*&0&0&0&0&*&*&*&*&*&*&*&*\\ *&*&*&*&0&0&0&0&*&*&*&*&*&*&*&*\\ *&*&*&*&0&0&0&0&*&*&*&*&*&*&*&*\\ *&*&*&*&0&0&0&0&0&*&*&*&*&*&*&0\\ *&*&*&*&0&0&0&0&0&0&*&*&*&*&*&0\\ *&*&*&*&0&0&0&0&0&0&0&0&0&0&0&0\\ *&*&*&*&0&0&0&0&0&0&0&0&0&0&0&0\\ *&*&*&*&0&0&*&*&*&0&0&0&0&0&0&0\\ *&*&*&*&0&0&*&*&*&0&0&0&0&0&0&0\\ *&*&*&*&0&0&*&*&*&0&0&0&0&0&0&0\\ *&*&*&*&0&0&0&0&0&0&0&0&0&0&0&0 \end{matrix}$$

F#

//     
    let multyBinImage = array2D [[0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0]
                                 [0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0]
                                 [0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0]
                                 [0;0;0;0;0;0;0;0;0;0;1;1;1;1;0;0]
                                 [1;1;1;1;0;0;0;0;0;1;1;1;1;1;1;0]
                                 [1;1;1;1;0;0;0;0;1;1;1;1;1;1;1;1]
                                 [1;1;1;1;0;0;0;0;1;1;1;1;1;1;1;1]
                                 [1;1;1;1;0;0;0;0;1;1;1;1;1;1;1;1]
                                 [1;1;1;1;0;0;0;0;0;1;1;1;1;1;1;0]
                                 [1;1;1;1;0;0;0;0;0;0;1;1;1;1;0;0]
                                 [1;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0]
                                 [1;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0]
                                 [1;1;1;1;0;0;1;1;1;0;0;0;0;0;0;0]
                                 [1;1;1;1;0;0;1;1;1;0;0;0;0;0;0;0]
                                 [1;1;1;1;0;0;1;1;1;0;0;0;0;0;0;0]
                                 [1;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0]]

,

//   
let markedMultyImages = Algorithms.markingOfConnectedComponents mulyBinImage
printfn "marked = \n %A" markedMultyImages

marked = 
 [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 1; 1; 1; 0; 0]
 [2; 2; 2; 2; 0; 0; 0; 0; 0; 1; 1; 1; 1; 1; 1; 0]
 [2; 2; 2; 2; 0; 0; 0; 0; 1; 1; 1; 1; 1; 1; 1; 1]
 [2; 2; 2; 2; 0; 0; 0; 0; 1; 1; 1; 1; 1; 1; 1; 1]
 [2; 2; 2; 2; 0; 0; 0; 0; 1; 1; 1; 1; 1; 1; 1; 1]
 [2; 2; 2; 2; 0; 0; 0; 0; 0; 1; 1; 1; 1; 1; 1; 0]
 [2; 2; 2; 2; 0; 0; 0; 0; 0; 0; 1; 1; 1; 1; 0; 0]
 [2; 2; 2; 2; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [2; 2; 2; 2; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
 [2; 2; 2; 2; 0; 0; 5; 5; 5; 0; 0; 0; 0; 0; 0; 0]
 [2; 2; 2; 2; 0; 0; 5; 5; 5; 0; 0; 0; 0; 0; 0; 0]
 [2; 2; 2; 2; 0; 0; 5; 5; 5; 0; 0; 0; 0; 0; 0; 0]
 [2; 2; 2; 2; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]

(1, 2, 5)

, — , , .

//   

// groups key - , value -    ,   
let groups = Dictionary<int, (int*int) list>()
markedMultyImages
|> Array2D.iteri (fun row col v -> if v <> 0 then
                                       if not (groups.ContainsKey(v)) then
                                            groups.Add(v, [row, col])
                                       else
                                            groups.[v] <- (row, col)::groups.[v])

printfn "groups = \n %A" groups

groups = 
 seq
  [[1, [(9, 13); (9, 12); (9, 11); ... ]];
   [2, [(15, 3); (15, 2); (15, 1); ... ]];
   [5, [(14, 8); (14, 7); (14, 6); ... ]]]

1. ( )
2. ( )
3. ( )

let mapAreas = groups
               |> Seq.map (|KeyValue|)
               |> Map.ofSeq

//   
let areas = mapAreas
            |> Map.map (fun k v -> v |> List.length)
printfn "areas =\n %A" areas

areas — , — , —

areas =
 map [(1, 44); (2, 48); (5, 9)]

, , 1 44 , 2 — 48 , 5 — 9 .

, , , 1 , . , . , — ,

//     
    let subarrays = mapAreas
                    |> Map.map (fun k v ->
                                    let minRow = v |> List.map (fst) |> List.min
                                    let maxRow = v |> List.map (fst) |> List.max
                                    let minCol = v |> List.map (snd) |> List.min
                                    let maxCol = v |> List.map (snd) |> List.max

                                    let dimRow = maxRow - minRow + 2 + 1 // add two zeros on left and right sides
                                    let dimCol = maxCol - minCol + 2 + 1

                                    let array2d = Array2D.create dimRow dimCol 0
                                    v |> List.iter (fun elem -> (array2d.[(fst elem) - minRow + 1, (snd elem) - minCol + 1] <- multyBinImage.[fst elem, snd elem]))

                                    array2d
                               )

    printfn "subarrays =\n %A" subarrays

subarrays =
 map
  [(1, [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
        [0; 0; 0; 1; 1; 1; 1; 0; 0; 0]
        [0; 0; 1; 1; 1; 1; 1; 1; 0; 0]
        [0; 1; 1; 1; 1; 1; 1; 1; 1; 0]
        [0; 1; 1; 1; 1; 1; 1; 1; 1; 0]
        [0; 1; 1; 1; 1; 1; 1; 1; 1; 0]
        [0; 0; 1; 1; 1; 1; 1; 1; 0; 0]
        [0; 0; 0; 1; 1; 1; 1; 0; 0; 0]
        [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]);

   (2, [[0; 0; 0; 0; 0; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 1; 1; 1; 1; 0]
         [0; 0; 0; 0; 0; 0]]);

   (5, [[0; 0; 0; 0; 0]
        [0; 1; 1; 1; 0]
        [0; 1; 1; 1; 0]
        [0; 1; 1; 1; 0]
        [0; 0; 0; 0; 0]])]

. — .

//       
    let erosionMaks = array2D [[0;1;0]
                               [1;1;1]
                               [0;1;0]]

    let borders = subarrays
                  |> Map.map (fun label array2d -> Algorithms.borderAllocation array2d erosionMaks (1, 1))
    let bordersLength = borders |> Map.map (fun label elem -> elem |> Seq.cast<int> |> Seq.filter ((<>)0) |> Seq.length)

    printfn "borders =\n %A" borders
    printfn "borders length =\n %A" bordersLength

borders =
 map
  [(1, [[0; 0; 0; 0; 0; 0; 0; 0; 0; 0]
        [0; 0; 0; 1; 1; 1; 1; 0; 0; 0]
        [0; 0; 1; 0; 0; 0; 0; 1; 0; 0]
        [0; 1; 0; 0; 0; 0; 0; 0; 1; 0]
        [0; 1; 0; 0; 0; 0; 0; 0; 1; 0]
        [0; 1; 0; 0; 0; 0; 0; 0; 1; 0]
        [0; 0; 1; 0; 0; 0; 0; 1; 0; 0]
        [0; 0; 0; 1; 1; 1; 1; 0; 0; 0]
        [0; 0; 0; 0; 0; 0; 0; 0; 0; 0]]);

(2, [[0; 0; 0; 0; 0; 0]
      [0; 1; 1; 1; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 0; 0; 1; 0]
      [0; 1; 1; 1; 1; 0]
      [0; 0; 0; 0; 0; 0]]);

   (5, [[0; 0; 0; 0; 0]
        [0; 1; 1; 1; 0]
        [0; 1; 0; 1; 0]
        [0; 1; 1; 1; 0]
        [0; 0; 0; 0; 0]])]

borders length =
 map [(1, 18); (2, 28); (5, 8)]

1 18 , 2 — 28 5 — 8 .

.

//    
    let centers = borders
                   |> Map.map (fun k v ->
                                 let centerRow = v |> Array2D.mapi (fun r _ _ -> float r) |> Seq.cast<float> |> Seq.average // list.average requires float
                                 let centerCol = v |> Array2D.mapi (fun _ c _ -> float c) |> Seq.cast<float> |> Seq.average
                                 (int centerCol, int centerRow)
                              )
    printfn "centers =\n %A" centers

centers =
 map [(1, (4, 4)); (2, (2, 6)); (5, (2, 2))]

.

$$

$$C=\frac{μR}{σR}$$

μR —
σR —

$$

$$μR = \frac{1}{K}\sum\limits_{k=1}^{K-1}||(r_k, c_k) - (\overline{r}, \overline{c})||$$

(r_k, c_k) —
(r, c) —

//    
    let diff point1 point2 = 
        let square x = x * x
        let s1 = square ((fst point1) - (fst point2))
        let s2 = square ((snd point1) - (snd point2))
        sqrt(double (s1 + s2))

μR .

//  ,  key - , value -   (double)
let mr = borders
         |> Map.map (fun label v ->
                        let aver = v
                                    |> Array2D.mapi (fun r c v -> diff (r, c) centers.[label])
                                    |> Seq.cast<double>
                                    |> Seq.sum
                        aver / (double v.Length)
                 )

σR

$$

$$σR = (\frac{1}{K}\sum\limits_{k=1}^{K-1}[||(r_k, c_k) - (\overline{r}, \overline{c})|| - μR]^2)^{1/2}$$

let qr = borders
     |> Map.map (fun label v ->
                    let aver = v
                            |> Array2D.mapi (fun r c v ->
                                                let t1 = diff (r, c) centers.[label]
                                                let t2 = (t1 - mr.[label]) * (t1 - mr.[label])
                                                t2
                                            )
                            |> Seq.cast<double>
                            |> Seq.sum
                            |> sqrt
                aver / (double v.Length)
         )

//  .   ,   
    let roundness = mr |> Map.map (fun label v -> v / qr.[label])
    printfn "roundness =\n %A" roundness

roundness =
 map [(1, 25.06054796); (2, 20.37134197); (5, 13.43281947)]

, 1 . .

() () F#. . , . , F# , . , , F# . , . .

The algorithms themselves discussed in the article can be found here. There, in the tests repository, there is an implementation of the last part of the article, where the most circular objects inside the image are searched.
Algorithms of binary morphology