Image processing applications A

### Program Information

Name:  Image processing applications A
Domain: Graph and Image
Functionality: establish a method to generate additional test cases and to evaluate test results.
Input:  The input data are(two-dimensional) binary (i.e. black and white) images.   The intersection $A \cap B$ and the union $A\cup B$ of two images $A, B$ (of the same size) is defined is defined by the pixelwise application of the logical $\mathbf{AND}$ and $\mathbf{OR}$ operators, respectively.  The input $A$ is an arbitrary binary image of width $n_x$ and heigth $n_y$ generated according to an arbitrary strategy.
Output:  The Euclidean distance transform $D(\cdot)$  The distance transform $D(A)$ is an image operation that maps a binary image A onto a grayscale image with real values.   The counter-clockwise rotation $C(A)$ is defined in the usual (and intuitive) way;  The reflections $M_x(A)$ and $M_y(A)$ are given by the inversion of the direction of processing of the ordinate and the abscissa, respectively;  The transposition $T(A)$ is obtained by interchanging both coordinates;  The operations $max(A,B)$ and $min(A,B)$ are obtained through the pixelwise application of the respective operation;  The enlargement $E(A)$ constructs an image of roughly three times the size of the original image;

#### Reference

  On Random Testing of Image Processing Applications https://doi.org/10.1109/QSIC.2006.45

### MR Information

In the following, two additional properties of the Euclidean distance transform are identified, which are only intended to be used in combination with other properties, such as the metamorphic relations MR1–MR7. P1 Lower bound $D(A) \geq A$ P2 Local bound  Let $D(A)=(d_{i,j})$. Then $|d_{i,j}-d_{i+k,j+l}|\leq b_{k+1,l+1}$ for each $k,l \in \{-1,0,1\}$ and every inner pixel with $1\leq i < n_y-1$ and $1 \leq j < n_x-1$, where $$B=(b_{k,l})=\left( \begin{array}{ccc} \sqrt{2} & 1 & \sqrt{2} \\ 1 & 0 & 1 \\ \sqrt{2} & 1 & \sqrt{2} \end{array}\right)$$

#### MR1------ Counter-clockwise rotation by 90 degrees

Description:
Property:  $C(D(A))=D(C(A))$
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#### MR2------ Reflection at the ordinate

Description:
Property: $M_x(D(A))=D(M_x(A))$
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#### MR3------  Reflection at the abscissa

Description:
Property:  $M_y(D(A))=D(M_y(A))$
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#### MR4------  Transposition

Description:
Property:  $T(D(A))=D(T(A))$
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#### MR5------  Enlargement of the image

Description:
Property:  Let $D(A)=(d_{i,j})$ and $D(E(A))=(e_{i,j})$. Then $3d_{i,j}=e_{3i+2,3j+2}$ for each $0 \leq i < n_y$ and $0 \leq j < n_x$
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#### MR6------  Intersection of two images

Description:
Property:  $D(A\cap B)=min(D(A),D(B))$
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#### MR7------  Union of two images

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Property:  $D(A\cup B) \geq max(D(A),D(B))$
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