JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV DIGITAL IMAGE PROCESSING Previous Question Paper (July - 2021 - R15)

Sampling and quantization are the two fundamental processes required to convert a continuous analog image into a digital format suitable for computer processing. Sampling refers to the digitization of the spatial coordinates of the image, where a continuous image is divided into a grid of discrete pixels. The sampling rate determines the spatial resolution; if the rate is too low, aliasing artifacts occur. Quantization refers to the digitization of the amplitude (intensity) values of each sampled pixel. It maps the continuous gray-level values to a finite set of discrete values, typically represented by a specific number of bits (e.g., 8-bit quantization provides 256 gray levels).

For example, consider a continuous grayscale image represented by a function f(x, y). Sampling discretizes x and y into a grid of size M x N. If we sample an image of a smooth gradient, a low sampling rate results in a visible grid pattern (pixelation). Quantization then assigns each pixel an integer value. If we use only 2 bits for quantization, the image will show only 4 distinct gray levels, leading to false contouring, whereas 8-bit quantization provides a smooth transition indistinguishable to the human eye.

The Discrete Cosine Transform (DCT) is a critical mathematical tool in digital image processing, primarily used for image compression. It transforms an image from the spatial domain into the frequency domain, representing the image as a sum of cosine functions oscillating at different frequencies.

The primary role of DCT includes:

• Energy Compaction: DCT effectively concentrates the majority of the image signal's energy into a small number of low-frequency coefficients located in the top-left corner of the transformed block.
• Decorrelation: It reduces the statistical redundancy between neighboring pixels, making the data more independent and easier to compress.
• Compression Efficiency: Because most information is packed into low-frequency components, high-frequency components (which represent fine details and noise) can be discarded or heavily quantized with minimal perceptual loss. This is the core principle behind the JPEG compression standard.

The 2D-DCT for an N x N image block is defined as:

where α(u) and α(v) are normalization factors. By discarding the high-frequency coefficients (quantization), the storage requirements are significantly reduced while maintaining high visual quality upon reconstruction via the Inverse DCT.