Find the autocorrelation of x(m, n). Assume zero-padding at...

Find the autocorrelation of $x(m, n)$. Assume zero-padding at the borders. (i) $$ x(m, n)=\left[\begin{array}{llll} 2 & 1 & 3 & 2 \\ 2 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 2 \end{array}\right] $$ (ii) $$ x(m, n)=\left[\begin{array}{llll} 3 & 1 & 1 & 2 \\ 0 & 3 & 1 & 2 \\ 1 & 2 & 1 & 0 \\ 1 & 2 & 0 & 2 \end{array}\right] $$ (iii) $$ x(m, n)=\left[\begin{array}{llll} 3 & 2 & 1 & 0 \\ 0 & 2 & 1 & 0 \\ 2 & 0 & 1 & 2 \\ 3 & 1 & 0 & 2 \end{array}\right] $$

Find the autocorrelation of $x(m, n)$. Assume zero-padding at the borders.
(i)
$$
x(m, n)=\left[\begin{array}{llll}
2 & 1 & 3 & 2 \\
2 & 1 & 1 & 0 \\
1 & 0 & 1 & 2 \\
1 & 1 & 0 & 2
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
3 & 1 & 1 & 2 \\
0 & 3 & 1 & 2 \\
1 & 2 & 1 & 0 \\
1 & 2 & 0 & 2
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
3 & 2 & 1 & 0 \\
0 & 2 & 1 & 0 \\
2 & 0 & 1 & 2 \\
3 & 1 & 0 & 2
\end{array}\right]
$$

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Key Concepts

Zero-Padding

Zero-padding is a technique used to extend the borders of a finite, discrete signal or image by adding zero values around the original data. This is particularly useful when performing operations like correlation or convolution since it allows the algorithm to process all parts of the signal uniformly, thereby managing edge effects and ensuring valid computation at the boundaries.

2D Signal Processing

2D signal processing involves the analysis and manipulation of two-dimensional data, such as images or matrices. It extends one-dimensional signal processing techniques—like filtering, convolution, and autocorrelation—to two dimensions, taking into account both horizontal and vertical spatial relationships. This field is essential for handling tasks in image analysis, computer vision, and pattern recognition.

Autocorrelation

Autocorrelation is a measure of the similarity between a signal and a shifted version of itself. In a two-dimensional context, this involves comparing a 2D array (such as an image) with its translated copies. The process helps in detecting patterns, periodicity, and identifying underlying structures in the data by assessing how well the patterns match at various displacements.

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