Open and Close

Morphological operations extract the image components used to represent and describe region shapes. Two common ones are opening and closing. First we fix how images are represented.

A binary image $A$ with $M$ rows and $N$ columns is the set of coordinates $(r, c)$ (where $1 \le r \le M$ and $1 \le c \le N$) at which the pixel equals $1$; the top-left corner is $(1, 1)$. A second binary image $B$, called the structuring element, has $2S+1$ rows and $2S+1$ columns and is represented the same way, except that its top-left pixel is at $(-S, -S)$ (so its center is at $(0, 0)$).

Two primitive operations are dilation and erosion. The dilation of $A$ by $B$ is

A (+) B = { a + b | a in A, b in B } ∩ Z

where coordinates are added componentwise and $Z$ is the set of $(i, j)$ with $1 \le i \le M$ and $1 \le j \le N$. The erosion of $A$ by $B$ is

A (-) B = { w | w + b in A for every b in B }

Using these, the opening of $A$ by $B$ is

A o B = (A (-) B) (+) B

and the closing of $A$ by $B$ is

A . B = (A (+) B) (-) B

Intuitively, opening removes small details while preserving the overall shape, and closing fills small gaps while preserving the overall shape.

The input consists of several test cases. Each case begins with a line containing three integers $M$, $N$, and $S$ separated by spaces ($10 \le M, N \le 256$, $1 \le S \le 4$). The next $M$ lines contain the rows of image $A$, each written as $N$ characters that are . (0) or * (1). The following $2S+1$ lines describe the structuring element $B$ in the same way. The input terminates with a line 0 0 0.

For each case, print the case number on its own line as Case x, followed by a blank line. Then print the opening $A \circ B$, followed by a blank line, followed by the closing $A \bullet B$. Each result image uses the same format as the input images. Separate the output of consecutive cases with a line of exactly 75 equals signs (=); do not print this separator after the last case.