GATE Geomatics Engineering Section II Syllabus

What does the spread and orientation of a scattergram indicate?

A tight, elongated diagonal cloud indicates high correlation (redundancy) between the two bands; a circular/diffuse spread indicates low correlation. Distinct clusters correspond to separable spectral classes.

Define the variance of a single image band with mean $\mu$ and $N$ pixels.

$$ \sigma^{2} = \frac{1}{N} \sum_{i=1}^{N} (x_{i} - \mu)^{2} $$ It measures the spread/contrast of grey values about the mean.

Write the covariance between two bands $X$ and $Y$.

$$ \operatorname{cov}(X,Y) = \frac{1}{N} \sum_{i=1}^{N} (x_{i} - \mu_{X})(y_{i} - \mu_{Y}) $$

What is the variance-covariance matrix for a multiband image and what are its diagonal/off-diagonal entries?

It is a symmetric $n \times n$ matrix $\Sigma$ where $n$ is the number of bands. Diagonal entries are band variances $\sigma_{ii}^{2}$; off-diagonal entries are covariances $\sigma_{ij} = \operatorname{cov}(i,j)$ between band pairs.

Why is the variance-covariance matrix significant in image processing?

It quantifies the variability within each band and the statistical dependence between bands. It is the foundation for PCA (its eigen-decomposition), for measuring band redundancy, and for many classification statistics.

Write the correlation coefficient between two bands in terms of covariance and standard deviations.

$$ \rho_{ij} = \frac{\operatorname{cov}(i,j)}{\sigma_{i}\,\sigma_{j}} $$ with $-1 \leq \rho_{ij} \leq 1$.

What is the correlation matrix and how does it differ from the covariance matrix?

The correlation matrix is a symmetric $n \times n$ matrix of correlation coefficients $\rho_{ij}$ with 1's on the diagonal. It is the normalized (scale-independent, dimensionless) version of the covariance matrix, with values bounded in $[-1, 1]$.

What is the significance of the correlation matrix for selecting bands?

High correlation between bands indicates redundant information; selecting weakly correlated bands maximizes the information content of a band combination. It also guides whether PCA will yield strong dimensionality reduction (high inter-band correlation favours PCA).

What is image registration?

Image registration is the process of geometrically aligning two or more images of the same scene (taken at different times, sensors, or viewpoints) into a common coordinate system, by establishing a geometric transformation between them.

What are Ground Control Points (GCPs) and their role in registration?

GCPs are clearly identifiable features whose coordinates are known in both the input image and the reference (map/image) system. They are used to compute the coefficients of the geometric transformation (e.g., polynomial warp) used for rectification/registration.

What is resampling and when is it required?

Resampling is the process of computing new pixel values for the output grid after a geometric transformation, because transformed coordinates rarely fall exactly on integer pixel positions. It is needed in rectification, registration, and reprojection.

Name three common resampling interpolation methods.

Nearest-neighbour, bilinear interpolation, and cubic convolution (bicubic).

Compare nearest-neighbour and bilinear resampling.

Nearest-neighbour assigns the value of the closest pixel: fast, preserves original DN values (good for thematic/classified data), but produces blocky/disjointed edges. Bilinear uses a weighted average of the 4 nearest pixels: smoother output, alters original DN values, mild smoothing/blurring.

What are the characteristics of cubic convolution resampling?

Cubic convolution uses the 16 nearest ($4\times4$) pixels with a weighted average. It produces the sharpest, smoothest result with the least blurring, but is the most computationally expensive and alters the original DN values.

What is contrast enhancement?

Contrast enhancement (contrast stretching) is the modification of the range/distribution of grey values to improve the visual distinction between features, expanding a narrow input range to use the full available display range.

Write the formula for a linear (min-max) contrast stretch.

$$ DN_{out} = \left( \frac{DN_{in} - DN_{min}}{DN_{max} - DN_{min}} \right) \times (L - 1) $$ where $L$ is the number of output levels (e.g., 256).

What is a percentage / saturation linear stretch and why is it used?

A percentage (saturation) linear stretch clips a small percentage of extreme pixels (e.g., 2% at each tail) and stretches the remaining values linearly. It prevents outliers from compressing the stretch, giving better contrast for the bulk of pixels.

Name three non-linear contrast enhancement methods.

Histogram equalization, logarithmic (and exponential) stretch, and Gaussian stretch. (Piecewise-linear is intermediate.)

What is histogram equalization and its effect?

Histogram equalization is a non-linear stretch that redistributes pixel values so the output histogram is approximately uniform, using the cumulative distribution function as the transformation. It assigns more display range to the most frequently occurring grey levels, maximizing global contrast.

In histogram equalization, what transformation function is applied to grey level $r$?

The CDF-based transformation: $$ s = T(r) = (L-1) \int_{0}^{r} p_{r}(w)\, dw $$ (continuous), or $s_{k} = (L-1)\sum_{j=0}^{k} \frac{n_{j}}{N}$ (discrete).

What is spatial (spatial-domain) enhancement and what image property does it operate on?

Spatial enhancement modifies pixel values based on the values of neighbouring pixels (using filters/convolution kernels). It operates on spatial frequency — the rate of change of brightness over distance — rather than on individual pixel statistics.

What is a low-pass (smoothing) spatial filter and its effect?

A low-pass filter averages or weights neighbourhood pixels (e.g., a mean filter) to attenuate high spatial frequencies. It smooths the image, reduces noise, but blurs edges and fine detail.

Write a $3\times3$ mean (box) smoothing kernel.

$$ \frac{1}{9} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} $$

What is a median filter and what type of noise is it best at removing?

A median filter is a non-linear filter that replaces each pixel with the median of its neighbourhood values. It is highly effective at removing salt-and-pepper (impulse) noise while preserving edges better than a mean filter.

What is a high-pass (sharpening) spatial filter and its effect?

A high-pass filter emphasizes high spatial frequencies (edges, fine detail) by subtracting a smoothed version or using a kernel with a positive centre and negative surround. It sharpens/enhances edges but also amplifies noise.

Name common types of image noise.

Gaussian (additive) noise, salt-and-pepper (impulse) noise, speckle (multiplicative, common in radar), and periodic noise.

What is Principal Component Analysis (PCA) in remote sensing?

PCA is a linear transformation that converts correlated multispectral bands into a new set of uncorrelated variables (principal components), ordered by decreasing variance. It concentrates information into fewer bands and is used for dimensionality reduction and decorrelation.

How are the principal components mathematically derived?

PCs are obtained from the eigenvalues and eigenvectors of the variance-covariance (or correlation) matrix $\Sigma$. The eigenvectors give the transformation axes (loadings) and the eigenvalues give the variance carried by each component; the PC with the largest eigenvalue holds the most variance.

What is the variance and correlation structure of the principal components produced by PCA?

The principal components are mutually uncorrelated (orthogonal), so the covariance matrix of the PCs is diagonal. The variance of the $i$-th PC equals the $i$-th eigenvalue $\lambda_{i}$, with $\lambda_{1} \geq \lambda_{2} \geq \dots$

What is Discriminant Analysis and how does its objective differ from PCA?

Discriminant analysis (e.g., Fisher's LDA) finds linear combinations of variables that best separate predefined classes. Unlike PCA (unsupervised, maximizes total variance), LDA is supervised and maximizes between-class separation relative to within-class scatter.

What criterion does Fisher's Linear Discriminant maximize?

It maximizes the ratio of between-class scatter to within-class scatter: $$ J(w) = \frac{w^{T} S_{B}\, w}{w^{T} S_{W}\, w} $$ where $S_{B}$ is the between-class and $S_{W}$ the within-class scatter matrix.

What is the RGB to IHS colour transformation used for?

It converts the Red-Green-Blue colour space into Intensity, Hue, and Saturation components. It is used for colour enhancement and especially for image (data) fusion — e.g., replacing intensity with a high-resolution panchromatic band, then transforming back (pan-sharpening).

Define the Intensity, Hue, and Saturation components in the IHS model.

Intensity = overall brightness (total energy), e.g., $I = \frac{R+G+B}{3}$. Hue = dominant wavelength / pure colour (the attribute we name a colour). Saturation = purity or amount of white light mixed with the hue (how vivid vs. pastel).

What does the CMYK colour model represent and how does it relate to RGB?

CMYK (Cyan, Magenta, Yellow, blacK) is a subtractive model used in colour printing; inks subtract light from white. Its primaries are the complements of the additive RGB primaries: cyan = white $-$ red, magenta = white $-$ green, yellow = white $-$ blue.

What is a band ratio and one benefit of using it?

A band ratio divides the DN of one band by another, pixel by pixel: $R = \frac{DN_{i}}{DN_{j}}$. It enhances spectral differences (slopes), suppresses topographic/illumination effects, and highlights specific materials.

Write the formula and value range for NDVI.

$$ NDVI = \frac{NIR - Red}{NIR + Red} $$ Range is $-1$ to $+1$; high positive values ($\approx 0.6$–$0.9$) indicate dense healthy vegetation.

Why is NDVI a normalized index rather than a simple ratio, and what does it exploit about vegetation?

Normalization to $[-1,1]$ reduces sensor/illumination/topographic effects and bounds the values. It exploits that healthy vegetation strongly reflects NIR (high) and strongly absorbs red light (low) due to chlorophyll, giving a large difference.

Write the formula for NDWI (McFeeters, water-detection form).

$$ NDWI = \frac{Green - NIR}{Green + NIR} $$ Positive values indicate water bodies (water reflects green, absorbs NIR); vegetation and soil give negative values.

In a typical NDVI image, what values do water, bare soil, and clouds/snow tend to show?

Water and clouds/snow give negative or near-zero NDVI (sometimes negative), bare soil gives small positive values (~0.1–0.2), and dense vegetation gives high positive values (~0.6–0.9).

What is the key difference between point operations and neighbourhood (spatial) operations in image enhancement?

Point operations (e.g., contrast stretch, histogram equalization) compute each output pixel from the single corresponding input pixel value only. Neighbourhood (spatial) operations (e.g., smoothing, sharpening filters) compute each output pixel from a window of surrounding input pixels.