Given a 4 ×4 4-bit image, find the histogram equalized...

Given a $4 \times 4$ 4-bit image, find the histogram equalized version of it. *(i) $$ \left[\begin{array}{llll} 4 & 4 & 3 & 3 \\ 4 & 4 & 4 & 3 \\ 5 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}\right] $$ (ii) $$ \left[\begin{array}{llll} 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 3 \\ 1 & 0 & 0 & 2 \\ 1 & 0 & 0 & 2 \end{array}\right] $$ (iii) $$ \left[\begin{array}{llll} 3 & 5 & 5 & 3 \\ 4 & 4 & 4 & 3 \\ 4 & 2 & 3 & 4 \\ 4 & 2 & 2 & 3 \end{array}\right] $$

Given a $4 \times 4$ 4-bit image, find the histogram equalized version of it.
*(i)
$$
\left[\begin{array}{llll}
4 & 4 & 3 & 3 \\
4 & 4 & 4 & 3 \\
5 & 4 & 4 & 4 \\
4 & 4 & 4 & 4
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
1 & 1 & 0 & 1 \\
1 & 1 & 0 & 3 \\
1 & 0 & 0 & 2 \\
1 & 0 & 0 & 2
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
3 & 5 & 5 & 3 \\
4 & 4 & 4 & 3 \\
4 & 2 & 3 & 4 \\
4 & 2 & 2 & 3
\end{array}\right]
$$

Key Concepts

Digital Image Representation

A digital image is composed of a grid of pixels, where each pixel represents an intensity value that is determined by the image's bit depth. For example, a 4-bit image can represent a limited range of intensity levels, meaning each pixel’s value is quantized into one of 16 possible levels. This discrete representation forms the foundation for various image processing techniques, including histogram equalization.

Image Histogram

The image histogram is a graphical representation of the distribution of pixel intensity values in an image. It shows the number of pixels for each intensity value, providing important statistical information about the image's overall brightness and contrast. This distribution is used to inform adjustments such as contrast enhancement in histogram equalization.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is derived from the image histogram and represents the cumulative frequency of pixel intensities. It accumulates the counts of all intensity levels up to a certain value, which is then normalized to form a mapping function used in histogram equalization. The CDF enables the redistribution of intensities so that the output image has a more uniform histogram.

Intensity Mapping

Intensity mapping is the process of reassigning the original image's pixel intensities to new values based on a transformation function. In histogram equalization, the mapping is derived from the normalized cumulative distribution function, ensuring that the output pixel values are spread across the available range. This helps to improve the contrast and visual detail in the image.

Histogram Equalization

Histogram equalization is an image enhancement technique aimed at improving the contrast of an image by redistributing its intensity values. By using the cumulative distribution function obtained from the image histogram, this method maps the original intensities to a new scale that tends to produce a uniform histogram. This results in an image where details in dark or bright areas become more visible, enhancing the overall visual quality.

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