Digital Image Processing (AD-603 (B)) - Unit 3 Short Notes

How unit 3 is examined

This unit covers spatial and frequency domain enhancement, histogram processing, image subtraction and averaging, smoothing (low pass and median) and sharpening (high pass); no topic was asked in the supplied papers, so each is kept short.

Image enhancement, Filters in spatial and frequency domains

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Definition. <mark>Image enhancement is the process of processing an image so that the result is more suitable than the original for a specific application, such as better contrast or clearer edges.</mark>

Key points.

1. Spatial domain methods work directly on pixels: $g(x,y)=T[f(x,y)]$, where $T$ is a point operation or a neighbourhood (mask) operation.
2. Frequency domain methods take the Fourier transform, multiply by a filter, and invert: $G(u,v)=H(u,v)F(u,v)$, then $g=\mathcal{F}^{-1}[G]$.
3. Enhancement is subjective and application dependent, unlike restoration, which models the degradation.
4. A spatial mask convolution is equivalent to multiplication by $H(u,v)$ in the frequency domain.

Histogram based processing

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Definition. <mark>A histogram plots the number of pixels $n_k$ at each gray level $r_k$, and histogram equalization spreads the gray levels to give a flat histogram and higher contrast.</mark>

Key points.

1. The normalised histogram is $p(r_k)=n_k/n$, where $n$ is the total number of pixels.
2. Equalization uses the transform $s_k=T(r_k)=\sum_{j=0}^{k}p(r_j)$ scaled by $(L-1)$ and rounded.
3. It is automatic and needs no parameters, and it stretches a dark or low contrast image over the full gray range.
4. Histogram specification (matching) instead forces the histogram to a chosen shape.

Image subtraction, Averaging

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Definition. ==Image subtraction computes $g(x,y)=f(x,y)-h(x,y)$ to show the difference between two images, and image averaging adds many noisy images of one scene and divides by their number to reduce noise.==

Key points.

1. Subtraction highlights change, for example motion between frames or blood vessels in mask-mode radiography.
2. Averaging uses $\bar g(x,y)=\frac{1}{K}\sum_{i=1}^{K}g_i(x,y)$, where $g_i=f+\eta_i$.
3. For zero-mean uncorrelated noise, the noise variance falls to $\sigma^2_{\bar g}=\sigma^2_\eta/K$, so the noise standard deviation falls by $\sqrt{K}$.
4. Averaging needs the images to be registered, since the scene must not move.

Image smoothing, Nedion (median) filtering, Low pass filtering

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Definition. <mark>Smoothing blurs an image to remove noise and small details; a low pass filter passes low frequencies and attenuates high frequencies.</mark>

Key points.

1. The averaging (box) mask is $\frac{1}{9}\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}$, and it blurs edges as well as noise.
2. Ideal, Butterworth and Gaussian low pass filters are the frequency domain forms; the ideal one causes ringing.
3. The median filter replaces each pixel with the median of its neighbourhood, a nonlinear operation.
4. The median filter removes salt-and-pepper noise well while preserving edges better than averaging.

Image sharpening by High pass filtering

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Definition. <mark>Sharpening highlights edges and fine detail by a high pass filter, which attenuates low frequencies and passes high frequencies.</mark>

Key points.

1. A high pass mask has a positive centre and negative neighbours, for example $\begin{bmatrix}-1&-1&-1\\-1&8&-1\\-1&-1&-1\end{bmatrix}$, and its coefficients sum to zero.
2. In the frequency domain, $H_{hp}(u,v)=1-H_{lp}(u,v)$.
3. High-boost filtering gives $f_{hb}=Af-f_{lp}$, which keeps some low frequency content.
4. Sharpening also amplifies noise, so it is usually applied after smoothing.

Last-minute revision

• Enhancement is subjective; restoration models the degradation.
• Spatial: $g=T[f]$; frequency: $G=HF$.
• Histogram: $p(r_k)=n_k/n$; equalization uses the cumulative sum $s_k$.
• Averaging $K$ images cuts noise variance by $K$.
• Subtraction shows change between two images.
• The box mask divides by 9 for 3x3.
• The median filter suits salt-and-pepper noise.
• High pass equals $1-$ low pass; mask coefficients sum to zero.
• High-boost: $Af-f_{lp}$.

Memory hooks

• Equalize means flatten the histogram.
• Median means middle value, so spikes are dropped.
• Low pass blurs, high pass sharpens.
• Average $K$ images, noise shrinks by $\sqrt{K}$.

Coverage checklist

• Image enhancement, Filters in spatial and frequency domains: no past questions.
• Histogram based processing: no past questions.
• Image subtraction, Averaging: no past questions.
• Image smoothing, Nedion filtering, Low pass filtering: no past questions.
• Image sharpening by High pass filtering: no past questions.