Intuition

DIP is the bag of operations that act *on* the pixels — before any 'learning'. Three big paradigms: spatial (operate on pixel neighbourhoods directly), transform (move to a different basis like Fourier/DCT, operate there, transform back), and morphological (set-theoretic operations on binary images via a structuring element). Every modern CNN op (conv, pooling, padding, BN) has a DIP ancestor; understanding the ancestor explains the descendant.

Explanation

**A digital image is a function $f(x,y)\to z$.** Domain: discrete 2D grid of pixel locations. Range: intensity. Grayscale = 1 channel; RGB = 3. '8 bits/pixel' means each pixel uses 8 bits → $[0,255]$ values, $2^8=256$ levels. Image acquisition = analog (light hitting sensor) → digital, via sampling (which pixel positions exist) and quantisation (how many gray levels per pixel).

Compression matters. A 1920×1080 RGB image at 24 bits/pixel is $1920\cdot 1080\cdot 24=49,766,400$ bits = $\approx 6.22$ MB *uncompressed*. 30 s of 30 fps video → 900 frames × 6.22 MB $\approx$ 5.6 GB raw. Real videos are 100–1000× smaller because of (a) spatial redundancy — neighbouring pixels correlated — and (b) temporal redundancy — frame $t+1\approx$ frame $t$. Compression (JPEG, MPEG, H.264) exploits both via DCT + quantisation + entropy coding.

Three spatial-domain paradigms. *Point→Point* (operate on one pixel at a time — negatives, log, gamma, thresholding). *Neighbourhood→Point* (operate on a window — convolution, mean/Gaussian/Sobel/Laplacian, median, bilateral). *Global→Point* (operate on the whole image — histogram equalisation, normalisation, statistics).

Point operations — formulas to memorise. *Negative*: $s=(L-1)-r$, useful for inverting X-ray contrast. *Log*: $s=c\log (1+r)$ — compresses dynamic range, brightens dark. *Power-law (gamma)*: $s=c{r}^{\gamma }$. $\gamma <1$ → brighter; $\gamma >1$ → darker; $\gamma =1$ → identity. Used for gamma correction (CRT monitors have $\gamma \approx 2.2$, so cameras pre-compensate with $1/2.2$). Common bug: forgetting to normalise $r\in [0,1]$ before exponentiation.

Spatial filters — what each kernel is doing. *Mean*: $\frac{1}{9}{1}_{3\times 3}$ — sum to 1 (preserves brightness), blurs everything including edges → boxy artefacts. *Gaussian*: $G(x,y)=\frac{1}{2\pi {\sigma }^2}\exp (-\frac{x^2+y^2}{2{\sigma }^2})$ — weighted, smooth, separable (2D = 1D ⊗ 1D, $O(2K)$ not $O(K^2)$), isotropic, no ringing. $\sigma$ controls spread; kernel size ≈ $6\sigma +1$ captures ~99% mass. *Sobel-x*: $\left(\begin{array}{ccc}-1& 0& 1\\ -2& 0& 2\\ -1& 0& 1\end{array}\right)$ — first derivative, sums to 0, detects VERTICAL edges (horizontal change). *Sobel-y* = transpose, detects HORIZONTAL edges. *Laplacian*: ${\nabla }^{2}f={\partial }_x^2+{\partial }_y^2$; kernel $\left(\begin{array}{ccc}0& -1& 0\\ -1& 4& -1\\ 0& -1& 0\end{array}\right)$ — second derivative, isotropic, very noise-sensitive. *LoG (Laplacian of Gaussian)*: pre-smooth then Laplacian, 'Mexican hat' — robust edge detection (Marr-Hildreth).

Edge facts. Gradient $\nabla I=(G_x,G_y)$. Magnitude $|\nabla I|=\sqrt{G_x^2+G_y^2}$ (or $|G_x|+|G_y|$ for speed). Direction $\theta =atan2(G_y,G_x)$. Edge ⊥ gradient — gradient points across the edge (max intensity change), edge is the iso-intensity contour. Smoothing kernels sum to 1; edge/derivative kernels sum to 0. Coefficient sign rule: any kernel that detects change must sum to zero on a flat region (else flat regions would fire).

Sharpening — unsharp masking & highboost. Detail = original − Gaussian-blurred. $I^{\prime }=I+k(I-I_{\text{blur}})$ enhances edges. Highboost: $I^{\prime }=A\cdot I+(I-I_{\text{blur}})=(A-1)I+I_{\text{detail}}$. $A=1$ → standard unsharp; $A>1$ → also brightens. Used in printing and photography.

Median + bilateral — non-linear, edge-preserving. *Median*: sort the 9 values in a 3×3 window, take the middle. KILLS salt-and-pepper noise because outliers occupy the tails of the sorted list and never reach the median. Mean filter, being linear, smears outliers across the neighbourhood. Median's weakness: doesn't preserve fine 1-pixel lines (3 of 9 = minority → median picks background). *Bilateral*: two Gaussians — one spatial $G_{{\sigma }_s}$, one range/intensity $G_{{\sigma }_r}$. $I^{\prime }(p)=\frac{1}{W}{\sum }_{q}I(q)G_{{\sigma }_s}(\Vert p-q\Vert )G_{{\sigma }_r}(|I(p)-I(q)|)$. Across an edge, intensity-different neighbours get weight ≈ 0 → no blur across edges → edge-preserving smoothing.

Convolution vs cross-correlation. Cross-correlation $(I\star H)(x,y)=\sum H(i,j)I(x+i,y+j)$. Convolution flips the kernel: $(I⊛H)(x,y)=\sum H(i,j)I(x-i,y-j)$. For SYMMETRIC kernels (mean, Gaussian, Laplacian) the flip is invisible — same result. For ASYMMETRIC kernels (Sobel-x, Sobel-y) the two DIFFER. PyTorch's nn.Conv2d actually computes CROSS-CORRELATION internally despite the name — a quiz favourite. Convolution theorem: convolution in space ↔ multiplication in frequency.

Padding. A $K\times K$ kernel centred at a border pixel reaches outside the image. *Zero padding* — virtual pixels = 0, simple, common in CNNs, causes a dark halo at borders + fake Sobel edges along image boundary. *Replicate padding* — virtual pixels = nearest border pixel; no halo. Other: *reflect*, *wrap* (periodic). 'Same' padding = $P=(K-1)/2$ for odd $K$; 'valid' = no pad, output shrinks by $K-1$.

Histograms. $h(i)$ = number of pixels with intensity $i$. Tells you contrast & dynamic range — NOT spatial layout (a checkerboard and a half-black/half-white split have identical histograms). Low-contrast underexposed: mass in $[0,80]$. High-contrast: bimodal at extremes. Well-exposed: spread across $[0,255]$.

Histogram equalisation. Goal: redistribute intensities so output histogram is ≈ uniform → maximise contrast. Algorithm: (1) compute $h(i)$; (2) normalise to PDF $p(i)$; (3) compute CDF $C(i)={\sum }_{k\le i}p(k)$; (4) map $s=\mathrm{round}((L-1)C(r))$. The CDF is precisely the function that flattens any distribution to uniform. Caveat: enhances noise in flat regions; non-linear, can produce unnatural results.

Contrast stretching is the linear cousin: $s=(r-r_{\min })(L-1)/(r_{\max }-r_{\min })$. Preserves histogram shape, just stretches it horizontally. Gentle, predictable. Use stretching for gentle boost; equalisation for aggressive correction.

Why frequency domain? Three reasons: (1) Convolution theorem — spatial convolution = frequency multiplication. Large-kernel convolution via FFT: $O(N^2\log N)$ vs $O(N^2{K}^2)$ naive — wins for big $K$. (2) Compact representation — natural images concentrate energy in low frequencies; discard the tail and lose little perceptually (JPEG). (3) Periodic noise removal — scanlines/screen-tones appear as isolated spectrum spikes, trivially zeroed.

2D DFT. Forward $F(u,v)={\sum }_{x,y}f(x,y)e^{-2\pi j(ux/M+vy/N)}$. Inverse $f(x,y)=\frac{1}{MN}{\sum }_{u,v}F(u,v)e^{+2\pi j(ux/M+vy/N)}$. Complex-valued. FFT is the $O(N\log N)$ algorithm (vs $O(N^2)$ direct). Ideal LPF: $H(u,v)=1$ if $\sqrt{u^2+v^2}\le D_0$, else 0. Sharp cutoff → sinc spatial response → ringing artefacts. Gaussian LPF: $H(u,v)=\exp (-(u^2+v^2)/2D_0^2)$. Smooth, no ringing. High-pass = edges, low-pass = smoothing.

DCT vs DFT — why JPEG uses DCT. DCT is real-valued, cosines-only. DFT assumes periodic signal → boundary discontinuities → high-frequency energy (ringing). DCT mirror-extends → no boundary discontinuity → better energy compaction (most energy concentrates in fewer low-frequency coefficients) → fewer artefacts at 8×8 block boundaries in JPEG.

JPEG pipeline. RGB → YCbCr → chroma subsample (4:2:0; human eye less sensitive to colour resolution) → split into 8×8 blocks → 2D DCT per block → quantise (divide by quant table; low-freq small divisors, high-freq big divisors — kills the high-freq tail) → zigzag scan → run-length encode + Huffman/arithmetic. Lossy step = quantisation. Decompression is the inverse.

Morphology — set-theoretic operations on binary images. Structuring element (SE) = small binary pattern with origin and possibly 'don't-care' cells. Erosion $A\ominus B=\{z\mid B_z\subseteq A\}$ (SE must fit ENTIRELY inside A) → MIN filter → shrinks foreground. Dilation $A\oplus B=\{z\mid B_z\cap A\ne \varnothing \}$ (SE touches A) → MAX filter → grows foreground. Opening $A\circ B=(A\ominus B)\oplus B$ — erosion then dilation; kills small noise, preserves shape of large objects. Closing $A\bullet B=(A\oplus B)\ominus B$ — fills small holes. Both are idempotent. Boundary = $A-(A\ominus B)$ (or equivalently $A\oplus B-A$). Hit-or-miss locates isolated points, line endpoints, T-junctions.

Thresholding & Otsu. *Global* threshold: $T$ chosen once for the whole image (fails for uneven illumination). *Variable / adaptive* (Sauvola, Niblack): $T(x,y)$ depends on local statistics. Otsu (bimodal): pick $T$ that MAXIMISES between-class variance ${\sigma }_B^2={\omega }_0{\omega }_1({\mu }_0-{\mu }_1{)}^2$ (equivalently, MINIMISES within-class variance). Closed-form optimal threshold.

Geometric transformations. All expressed as $3\times 3$ matrices in homogeneous coordinates. *Translation* (2 DoF), *Rigid* (R+t, 3 DoF), *Similarity* (R+t+s, 4 DoF), *Affine* (6 DoF — preserves PARALLEL lines), *Projective/homography* (8 DoF — preserves only STRAIGHT lines, allows perspective). Forward vs inverse warping: forward iterates over source, pushes pixel to $T(x,y)$ → holes + overlaps (bad). Inverse iterates over destination, pulls from $T^{-1}(x^{\prime },y^{\prime })$ → every dest filled exactly once + need INTERPOLATION (nearest / bilinear / bicubic) at non-integer source coords. Always use inverse warping in practice.

Colour models. *RGB* (additive, screens, non-uniform perceptually). *CMYK* (subtractive, printing; K = black for deep darks instead of triple-overprint). *CIE Lab* (perceptually uniform — equal Δ in $(L,a,b)$ = equal perceived difference). *HSI/HSV* (intuitive: hue + saturation + intensity). *YCbCr* (luminance + 2 chrominance, used in JPEG with chroma subsampling).

Hough transform — line detection. Each image point votes for all lines passing through it. Parameterise lines in $(\rho ,\theta )$ polar form $x\cos \theta +y\sin \theta =\rho$ (not slope-intercept $y=mx+c$ — vertical lines have infinite slope). Build a 2D accumulator over $(\rho ,\theta )$; each edge pixel adds 1 to every $(\rho ,\theta )$ corresponding to a line through it; peaks = detected lines. Extensible to circles (3D accumulator) and arbitrary curves.

Template matching — three distance measures. *SSD* — sum of squared differences; sensitive to brightness. *NCC* (normalised cross-correlation) — invariant to brightness shifts, range $[-1,1]$. *Correlation coefficient* — invariant to brightness AND contrast. Limitation: not affine-invariant — fails under rotation/scale. Motivates affine-invariant features (SIFT, ORB) and learned features (CNN, ViT).

Definitions

• Sampling vs quantisation — Sampling = discretising spatial position (which pixels exist). Quantisation = discretising intensity (how many gray levels per pixel).
• Spatial domain — Operations on pixel values directly. Three flavours: Point→Point, Neighbourhood→Point, Global→Point.
• Transform domain — Operations after transforming to another basis (Fourier/DCT/wavelet); useful for convolution, compression, periodic-noise removal.
• Convolution theorem — Convolution in space ↔ multiplication in frequency. Justifies FFT-based filtering for large kernels.
• Separable filter — A 2D filter $H(x,y)=H_x(x)H_y(y)$ — can be applied as two 1D passes. Gaussian is separable; Laplacian is not.
• Bilateral filter — Edge-preserving smoothing. Weights = spatial Gaussian × range (intensity) Gaussian. Across an edge, intensity differs → range weight collapses → no blur across edge.
• Otsu's method — Automatic threshold for bimodal histograms. Picks T that maximises between-class variance (equivalently minimises within-class). $O(L)$ closed-form sweep over all intensity levels.
• Morphological erosion / dilation — Erosion $A\ominus B$: SE fits inside A → MIN filter, shrinks. Dilation $A\oplus B$: SE touches A → MAX filter, grows. Duals.
• Opening / Closing — Opening = erode then dilate (kills noise dots, preserves shape). Closing = dilate then erode (fills small holes). Both idempotent.
• Hit-or-Miss transform (HAM) — Match a foreground pattern AND its background simultaneously. Used to locate isolated points, line endpoints, T-junctions, corners.
• Affine vs projective — Affine (6 DoF) preserves parallel lines. Projective/homography (8 DoF) preserves only straight lines — allows perspective.
• Forward vs inverse warping — Forward: iterate over source pixels, push to T(x,y) — produces holes + overlaps. Inverse: iterate over destination, pull from T⁻¹(x', y') — every dest filled exactly once + needs interpolation.
• Hough transform — Voting in parameter space. Each image edge pixel votes for all lines through it; peaks in $(\rho ,\theta )$ accumulator = detected lines. Polar form avoids the vertical-line infinity problem.
• Template matching — Slide a template over the image; score similarity (SSD, NCC, correlation coefficient). Fast but NOT rotation/scale invariant — motivates feature descriptors.
• Histogram equalisation — Map intensity via CDF: $s=(L-1)\mathrm{CDF}(r)$. Output histogram is approximately uniform; contrast maximised; can amplify noise in flat regions.
• DCT vs DFT — DCT is real-valued (cosines only) and mirror-extends → no boundary discontinuity → better energy compaction. JPEG uses DCT; spectrum analysis uses DFT.

Formulas

• $$\text{Image as a function:}f(x,y)\to z,z\in [0,2^B-1]\text{for }B\text{ bits/pixel}$$
• $$\text{Negative:}s=(L-1)-r$$
• $$\text{Log:}s=c\log (1+r)$$
• $$\text{Gamma:}s=c{r}^{\gamma }(\gamma <1\text{ brighten},\gamma >1\text{ darken})$$
• $$\text{Convolution:}(I⊛H)(x,y)=\sum _{i,j}H(i,j)I(x-i,y-j)$$
• $$\text{Gaussian:}G(x,y)=\frac{1}{2\pi {\sigma }^2}\exp (-\frac{x^2+y^2}{2{\sigma }^2})$$
• $$\text{Gradient magnitude / direction:}|\nabla I|=\sqrt{G_x^2+G_y^2},\theta =atan2(G_y,G_x)$$
• $$\text{Sobel-x:}\left(\begin{array}{ccc}-1& 0& 1\\ -2& 0& 2\\ -1& 0& 1\end{array}\right)\text{Laplacian:}\left(\begin{array}{ccc}0& -1& 0\\ -1& 4& -1\\ 0& -1& 0\end{array}\right)$$
• $$\text{LoG:}\mathrm{LoG}(x,y)=-\frac{1}{\pi {\sigma }^4}\left[1-\frac{x^2+y^2}{2{\sigma }^2}\right]\exp (-\frac{x^2+y^2}{2{\sigma }^2})$$
• $$\text{Unsharp / Highboost:}{I}^{\prime }=AI+(I-I_{\text{blur}})$$
• $$\text{Bilateral:}{I}^{\prime }(p)=\frac{1}{W}\sum _{q}I(q)G_{{\sigma }_s}(\Vert p-q\Vert )G_{{\sigma }_r}(|I(p)-I(q)|)$$
• $$\text{Histogram equalisation:}s=(L-1)\mathrm{CDF}(r)$$
• $$\text{2D DFT:}F(u,v)=\sum _{x,y}f(x,y)e^{-2\pi j(ux/M+vy/N)}$$
• $$\text{Otsu:}{T}^{\star }=\arg \underset{T}{\max }{\omega }_0(T){\omega }_1(T)({\mu }_0(T)-{\mu }_1(T){)}^2$$
• $$\text{Boundary:}\partial A=A-(A\ominus B)$$
• $$\text{Homogeneous affine:}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right)=\left(\begin{array}{ccc}a& b& t_x\\ c& d& t_y\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ 1\end{array}\right)$$
• $$\text{Hough (polar line):}\rho =x\cos \theta +y\sin \theta$$

Derivations

**Why $\sqrt{w}$ — no wait, that's YOLO; here it's why Gaussian is separable.** $G_{2D}(x,y)=\frac{1}{2\pi {\sigma }^2}{e}^{-(x^2+y^2)/2{\sigma }^2}=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-x^2/2{\sigma }^2}\cdot \frac{1}{\sqrt{2\pi }\sigma }{e}^{-y^2/2{\sigma }^2}=G_x(x)G_y(y)$. Therefore $f⊛G_{2D}=(f⊛G_x)⊛G_y$ — two 1D passes instead of one 2D pass, $O(2K)$ vs $O(K^2)$.

Otsu's optimal threshold. Define within-class variance ${\sigma }_W^2={\omega }_0{\sigma }_0^2+{\omega }_1{\sigma }_1^2$ and between-class variance ${\sigma }_B^2={\omega }_0{\omega }_1({\mu }_0-{\mu }_1{)}^2$. Total variance ${\sigma }^2={\sigma }_W^2+{\sigma }_B^2$ is constant (independent of $T$). Therefore $\arg {\min }_T{\sigma }_W^2\equiv \arg {\max }_T{\sigma }_B^2$. Otsu sweeps all 256 thresholds and picks the max — $O(L)$ closed-form.

Convolution theorem (statement + sketch). $\mathcal{F}[f⊛h]=\mathcal{F}[f]\cdot \mathcal{F}[h]$. Proof sketch: substitute the convolution sum into the DFT definition; swap sums; the inner sum is the DFT of a shifted kernel, which factors. Practical consequence: large-kernel filtering by FFT-multiply-IFFT.

Why median kills salt-and-pepper but mean smears it. Salt-and-pepper replaces a small fraction of pixels with extreme values (0 or 255). In a 9-pixel window with 1 outlier (255) and 8 reasonable values, the sorted order is $[v_1,\dots ,v_8,255]$; the 5th element (median) is one of the reasonable values — outlier ignored. The mean is $(v_1+\dots +v_8+255)/9$, which is biased upward by $\approx 28$. Median is a rank statistic, mean is linear; outliers don't survive ranking but do survive averaging.

Bilateral preserves edges. At an edge, neighbours on the OTHER side have very different intensity. The range Gaussian $G_{{\sigma }_r}(|I(p)-I(q)|)$ collapses to $\approx 0$ for those neighbours. Effective kernel becomes one-sided → smoothing happens only WITHIN the side $p$ belongs to → edge preserved.

Examples

• 3×3 mean filter on center. Patch $\left(\begin{array}{ccc}10& 20& 30\\ 40& 50& 60\\ 70& 80& 90\end{array}\right)$. Output center = $(10+20+\dots +90)/9=450/9=50$.
• Sobel on a vertical edge. Patch with values $[0,0,255]$ in each row → $G_x=-1\cdot 0-2\cdot 0-1\cdot 0+1\cdot 255+2\cdot 255+1\cdot 255=1020$. $G_y=0$ (no horizontal change). Gradient points right; edge runs vertical. ✓
• Median on salt-and-pepper. Patch $\left(\begin{array}{ccc}20& 25& 255\\ 30& 0& 28\\ 22& 24& 26\end{array}\right)$. Sorted: $[0,20,22,24,25,26,28,30,255]$. Median = 25 — outliers 0 and 255 ignored. Mean would be $430/9\approx 47.8$ — distorted.
• Histogram equalisation on 16-pixel image. $h=[0,0,4,6,4,2,0,0]$ for $L=8$. PDF = $[0,0,.25,.375,.25,.125,0,0]$. CDF = $[0,0,.25,.625,.875,1,1,1]$. Map: $r=2\to s=7\cdot 0.25=1.75\to 2$. $r=3\to s=7\cdot 0.625=4.375\to 4$. $r=4\to s=7\cdot 0.875=6.125\to 6$. $r=5\to s=7$. Output uses $[2,7]$ instead of $[2,5]$ — contrast stretched.
• Otsu on a bimodal histogram. Two clean clusters at ${\mu }_0=50$ and ${\mu }_1=200$, with ${\omega }_0=0.4,{\omega }_1=0.6$. ${\sigma }_B^2=0.4\cdot 0.6\cdot (200-50{)}^2=0.24\cdot 22500=5400$. Maximised when classes are well separated — algorithm picks $T$ between the modes.
• JPEG quality knob. Quality 90 → quantisation table has small divisors → most DCT coefficients survive → file size large, artefacts invisible. Quality 30 → big divisors → only low-freq coefficients survive → small file, blocky artefacts visible at 8×8 grid boundaries.

Diagrams

• Cartesian map of filter taxonomy: linear/non-linear × smoothing/sharpening/edge. Place mean, Gaussian, median, bilateral, Sobel, Laplacian, LoG, unsharp.
• Sobel/Laplacian kernels side-by-side with a 3×3 input patch showing the convolution arithmetic.
• Histogram equalisation pipeline: histogram → PDF → CDF → mapped image, with arrows.
• Forward vs inverse warping: source-image grid mapped to dest with arrows; show 'holes' under forward and 'interpolation' under inverse.
• Erosion vs dilation: small binary shape with SE shown sliding; show shrunk vs grown output.
• JPEG block pipeline: image → YCbCr → 8×8 split → DCT → quant table → zigzag → entropy coded bitstream.
• Hough accumulator: image with 3 collinear points → 3 sinusoids in $(\rho ,\theta )$ space intersecting at one bin = the line.

Edge cases

• Mean filter at borders with zero padding produces a dark halo — the filter sees mostly zeros, output dims. Use replicate padding for image-processing tasks.
• Median on 1-pixel-wide lines removes them (only 3 of 9 pixels are line; median picks background).
• Ideal LPF rings. Sharp cutoff in frequency → sinc in space → oscillations. Always use a smooth filter (Gaussian, Butterworth).
• Otsu on unimodal histograms is meaningless — it will pick some arbitrary T. Use a different method when the histogram has no valley.
• Global threshold on uneven illumination fails — one half of the image may be entirely above $T$, the other below. Use local/adaptive (Sauvola, Niblack).
• Forward warping produces holes even on simple transforms because non-integer destinations leave gaps. Always invert.
• **Hough with too-coarse $(\rho ,\theta )$ bins** merges adjacent lines; too fine bins spread votes and miss the peak. Use about 1° in $\theta$ and 1 pixel in $\rho$.
• Template matching under rotation/scale fails entirely — pure pixel matching has no invariance.

Common mistakes

• **Confusing $\gamma <1$ and $\gamma >1$.** $\gamma <1$ BRIGHTENS dark regions; $\gamma >1$ DARKENS them. Forgetting to normalise $r\in [0,1]$ before exponentiation gives nonsense results.
• Saying 'edge IS the gradient direction'. No — edge is PERPENDICULAR to the gradient direction. Gradient points ACROSS the edge.
• Treating cross-correlation and convolution as identical. They differ for asymmetric kernels. PyTorch's nn.Conv2d is cross-correlation.
• Mean kernel sums to 1, edge kernel sums to 1. Wrong — smoothing sums to 1, derivative kernels (Sobel/Laplacian) sum to 0.
• JPEG uses DFT. Wrong — DCT. Better energy compaction, no boundary discontinuity.
• Otsu minimises between-class variance. Wrong — MAXIMISES between-class (equivalent to MINIMISING within-class).
• 'Histogram equalisation gives a uniform output.' Only approximately — discrete intensities + rounding mean it's 'as uniform as possible'.
• Erosion = MAX, dilation = MIN. Backwards. Erosion = MIN (shrinks), Dilation = MAX (grows).
• Forward warping is preferred. Wrong — INVERSE warping is preferred (avoids holes; needs interpolation).
• **Hough in $(m,c)$.** Slope $m$ goes to infinity for vertical lines. Use polar $(\rho ,\theta )$.

Shortcuts

• Conv output: $\lfloor (W-F+2P)/S\rfloor +1$. Same pad odd K: $P=(K-1)/2$.
• Smoothing kernels sum to 1; derivative kernels sum to 0.
• Gaussian is SEPARABLE → $O(2K)$, not $O(K^2)$.
• Edge ⊥ gradient. Memorise.
• Erosion = MIN (shrink), Dilation = MAX (grow). Opening kills noise; closing fills holes.
• Boundary = Dilation − Original (or Original − Erosion).
• Otsu maximises between-class variance.
• JPEG uses DCT (not DFT) — better energy compaction, no boundary discontinuity.
• Inverse warping ✓; forward warping has holes.
• **Hough polar form $\rho =x\cos \theta +y\sin \theta$ avoids vertical-line problem.**

Proofs / Algorithms

Separability of Gaussian. $G(x,y)=\frac{1}{2\pi {\sigma }^2}{e}^{-(x^2+y^2)/2{\sigma }^2}$. Factor: $e^{-(x^2+y^2)/2{\sigma }^2}=e^{-x^2/2{\sigma }^2}\cdot e^{-y^2/2{\sigma }^2}$. Therefore $G(x,y)=G_x(x)\cdot G_y(y)$ with $G_x,G_y$ 1D Gaussians (after normalising constants). Two 1D convolutions replace one 2D convolution.

Total variance = within + between. ${\sigma }^2={\sum }_i{p}_i(i-\mu {)}^2$. Expand using $\mu ={\omega }_0{\mu }_0+{\omega }_1{\mu }_1$. After algebra (Bishop §9): ${\sigma }^2={\omega }_0{\sigma }_0^2+{\omega }_1{\sigma }_1^2+{\omega }_0{\omega }_1({\mu }_0-{\mu }_1{)}^2={\sigma }_W^2+{\sigma }_B^2$. Since ${\sigma }^2$ is fixed, maximising ${\sigma }_B^2$ ≡ minimising ${\sigma }_W^2$.

Convolution theorem (statement). For $f,h$ with DFTs $F,H$: $\mathcal{F}[f⊛h](u)=F(u)H(u)$. Inverse: $f⊛h={\mathcal{F}}^{-1}[F\cdot H]$. Therefore an $N\times N$ convolution with a large kernel costs $O(N^2\log N)$ via FFT vs $O(N^2{K}^2)$ direct — wins for $K\gtrsim \log N$.