JNTU B.Tech III DIGITAL IMAGE PROCESSING Previous Question Paper (January/February - 2021 - R16)

Digital image processing requires the conversion of a continuous-space, continuous-amplitude image into a discrete-space, discrete-amplitude format. This process involves two fundamental steps: sampling and quantization.

Sampling is the process of digitizing the coordinate values of a continuous image. It involves partitioning the image into a grid of discrete locations, where the intensity value at each point is captured. If a continuous image f(x, y) is sampled into a grid of size M x N, the resulting image is represented as a matrix where each element (pixel) corresponds to a specific spatial coordinate. The sampling rate determines the spatial resolution; if the sampling rate is too low, it leads to spatial aliasing, where high-frequency details are lost or distorted.

Quantization is the process of digitizing the amplitude values of the sampled image. Each sampled pixel value is assigned a discrete integer value from a finite set of levels. If the gray levels are represented by L levels, where L = 2^k, then each pixel requires k bits of storage. The number of quantization levels determines the intensity resolution (or bit depth) of the image. A low number of quantization levels results in false contouring, where smooth gradients appear as distinct bands of color or gray levels.

The relationship between the image size and storage requirements can be expressed as:

where b is the total number of bits required to store the image, M and N are the number of rows and columns, and k is the number of bits per pixel. Increasing the sampling rate (M, N) improves spatial detail, while increasing the quantization levels (k) improves the accuracy of intensity representation. Both processes are essential for representing real-world visual data in a format suitable for computer processing and storage.

The 2D Discrete Fourier Transform (DFT) is a fundamental tool in image processing for frequency domain analysis. Its key properties include:

1. Separability: The 2D DFT can be computed by performing a 1D DFT along the rows followed by a 1D DFT along the columns (or vice versa). This reduces computational complexity from O(N^4) to O(N^2 log N).
2. Translation: Shifting an image in the spatial domain by (x0, y0) results in a phase shift in the frequency domain, but the magnitude spectrum remains unchanged: F(u, v) * exp(-j2π(ux0/M + vy0/N)).
3. Rotation: If an image is rotated by an angle θ in the spatial domain, its Fourier transform is also rotated by the same angle θ in the frequency domain.
4. Scaling: Scaling the spatial coordinates by a factor 'a' results in an inverse scaling in the frequency domain by 1/a, accompanied by a change in amplitude.
5. Periodicity: The 2D DFT is periodic with period N in both the u and v directions, such that F(u, v) = F(u+N, v) = F(u, v+N).
6. Linearity: The transform is linear, meaning the DFT of a weighted sum of two images is equal to the weighted sum of their individual DFTs: DFT[a*f1(x,y) + b*f2(x,y)] = a*F1(u,v) + b*F2(u,v).
7. Convolution Theorem: Convolution in the spatial domain is equivalent to point-wise multiplication in the frequency domain: F[f(x,y) * h(x,y)] = F(u,v) * H(u,v), where * denotes convolution.
8. Correlation Theorem: Correlation in the spatial domain corresponds to the multiplication of one transform by the complex conjugate of the other in the frequency domain.