Find the closing of x(m, n) and h(m, n). h(m, n)=[ 1 0 1 0 1 0 0

Find the closing of x(m, n) and h(m, n). h(m, n)=[ 1 ...

Find the closing of $x(m, n)$ and $h(m, n)$. $$ h(m, n)=\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right] $$ Verify the output using the equivalent expression in terms of the opening operation. (i) $$ x(m, n)=\left[\begin{array}{llll} 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right] $$ (ii) $$ x(m, n)=\left[\begin{array}{llll} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right] $$ (iii) $$ x(m, n)=\left[\begin{array}{llll} 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right] $$

Find the closing of $x(m, n)$ and $h(m, n)$.
$$
h(m, n)=\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 1 & 0 \\
0 & 1 & 1
\end{array}\right]
$$
Verify the output using the equivalent expression in terms of the opening operation.
(i)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 1 & 1 & 1 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 \\
0 & 1 & 1 & 0
\end{array}\right]
$$

Key Concepts

Structuring Element

A structuring element is a predefined binary matrix used to probe and interact with an image in morphological operations. It plays a crucial role in determining how the image is modified by highlighting spatial relationships between pixels, and its shape and size directly influence the outcome of operations like dilation, erosion, opening, and closing.

Morphological Dilation

Dilation is an operation that expands the boundaries of foreground regions in an image. It works by placing the structuring element over the image and taking the maximum value covered by the element. This operation is useful for connecting disjoint elements and filling small gaps within objects.

Morphological Erosion

Erosion is the process of shrinking the boundaries of foreground regions by taking the minimum value covered by the structuring element as it is moved across the image. This operation eliminates small objects and imperfections, thus helping to refine the shape of objects prior to subsequent operations.

Morphological Closing

Closing is a composite morphological operation that involves first dilating an image and then performing erosion with the same structuring element. This sequence effectively fills in small holes and gaps, connects nearby elements, and smooths contours while maintaining the overall shape and size of larger objects.

Morphological Opening

Opening is the dual operation to closing and consists of an erosion followed by a dilation. This operation is used to remove small noise or disjoint elements from an image, helping to emphasize the larger, more significant structures. In many cases, the results of opening can be used to verify the outcomes of closing through duality relations.

Transcript

00:01 Hi, i'm going to find out here for the coefficient of x2 power n and so on.

00:11 So we're given here the series here and this is equals to a polynomial with degree n.

00:18 Now when you compare this here, so see here it is x2 power n on the right hand side.

00:28 The highest power of x is n.

00:29 So find out here the highest power of x here.

00:32 So we have x multiplied x squared, multiplied x cube and so on with x2.

00:37 N so right this here on the left inside we have x2 part one i'm just checking though to find the highest power of x here on the left hand side so we have x2 times we have extra part two times x2 times x2 x2 and so on times x2 power n that will become x2 plus 2 plus 3 and so on plus n we compare this to left and right inside the highest power of xm should be equal to 1 plus 2 plus 3 and so on till n.

01:11 So from here we get n equals n plus 1 over 2.

01:19 Next if we look at here, i find the coefficient of x that is given as a1.

01:25 So this polynomial is equal to this series here product.

01:31 So here coefficient of x should be equal to coefficient of x here on the left hand side.

01:36 Now we've got to left inside here.

01:38 There's the first term that's going to be 1 times 1 times 1 and so on times 1.

01:42 That will become 1.

01:43 So, rightly we got a note that is coming out to be 1.

01:47 So we have a note is 1...