notation.md

Notation

Every Symbol in perception-guide.md, and the Ones That Collide

What this file is. A complete symbol table for the main document, and a standalone cheat sheet you can keep open in a second window. If you hit a symbol mid-chapter and cannot remember what it is, it is here, with the § where it is introduced.

Read §N.1 first, once. Reading conventions — what bold means, what a hat means, how frames are written — are worth five minutes and will save you from misreading equations for the rest of the document.

Then §N.2, the collisions. Perception borrows notation from computer vision, estimation theory, signal processing, and machine learning, and those four fields independently claimed the same letters. $\mathbf{K}$ is the camera intrinsics and the Kalman gain. $\mathbf{\Sigma}$ is a covariance and the SVD's singular values. Nobody is going to fix this. Knowing where the landmines are is the only defence, and §N.2 is the map.

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N.1 Reading conventions

Everything in the document follows these, and once they are automatic the equations get noticeably easier to read.

Convention	Meaning	Example
Bold lowercase	a vector	$\mathbf{p}$, $\mathbf{x}$, $\mathbf{r}$
Bold uppercase	a matrix	$\mathbf{R}$, $\mathbf{J}$, $\mathbf{\Sigma}$
Italic lowercase	a scalar	$f$, $\lambda$, $z$
$\mathcal{Calligraphic}$	a set, manifold, or distribution	$\mathcal{M}$, $\mathcal{N}$, $\mathcal{I}$
$\mathbb{Blackboard}$	a number space	$\mathbb{R}^3$, $\mathbb{E}[\cdot]$

Decorations — the accents that change meaning

These are the ones people skim past and then misread. Each one is doing real work.

Decoration	Reads as	Meaning	§
$\hat{\mathbf{x}}$	"x hat"	an estimate — inferred from data, not known	§2.7
$\bar{\mathbf{x}}$	"x bar"	a prior or mean value	§2.4
$\tilde{\mathbf{p}}$	"p tilde"	homogeneous coordinates (one extra element)	§1.6
$\dot{\mathbf{x}}$	"x dot"	time derivative	§32.3
$\mathbf{x}^-$	"x minus"	the predicted (pre-update) filter state	§32.2
$\boldsymbol{\phi}^\wedge$	"phi hat/wedge"	the skew-symmetric matrix built from a 3-vector	§4.4
$\mathbf{A}^\top$	"A transpose"	rows and columns swapped	§1.2
$\delta\mathbf{x}$	"delta x"	a small increment or error state	§3.3
$\mathbf{x}^*$	"x star"	the optimal value	§3.1

The one that bites. $(\cdot)^\wedge$ and $\hat{(\cdot)}$ are both spoken "hat" and mean entirely different things — the first turns a 3-vector into a $3\times3$ skew matrix, the second marks an estimate. This document writes the skew operator as a superscript wedge, $\boldsymbol{\phi}^\wedge$, to keep them apart. Much of the literature does not.

Frames — the leading superscript

$${}^{A}\mathbf{p} \quad\text{— the coordinates of point } p \text{ expressed in frame } A$$

$${}^{A}\mathbf{T}_{B} \quad\text{— the transform taking coordinates in } B \text{ to coordinates in } A$$

Subscripts chain by cancellation: ${}^{A}\mathbf{T}{B},{}^{B}\mathbf{T}{C} = {}^{A}\mathbf{T}_{C}$. That visual cancellation is the entire point of the convention — a composition error becomes something you can see on the page rather than something you debug at 2 a.m. Introduced in §1.1; the standard frame names (base_link, odom, map) are in §4.9.

Subscripts and superscripts you will meet constantly

Form	Meaning
$\mathbf{x}_k$	value at discrete time step $k$
$\mathbf{x}_{k	k-1}$
$f_x, f_y$	the $x$ and $y$ components of a quantity, not derivatives
$\mathbf{b}_g, \mathbf{b}_a$	gyroscope and accelerometer bias
$\sigma_g$, $\sigma_a$	noise density of gyroscope, accelerometer

Warning on $f_x$. In §7.2 this is the horizontal focal length in pixels. In most calculus texts a subscript denotes a partial derivative. It never does in this document.

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N.2 Collisions — read this section

Nine letters carry more than one meaning. In every case context resolves it, but only if you know there is something to resolve.

Symbol	Meaning A	Meaning B	Meaning C	How to tell
$\mathbf{K}$	camera intrinsic matrix, §7.2	Kalman gain, §32.2	—	A is $3\times3$ and constant; B changes every update and multiplies an innovation
$\mathbf{H}$	approximate Hessian $\mathbf{J}^\top\mathbf{W}\mathbf{J}$, §3.3	measurement Jacobian, §32.2	homography, §16.6	A is square and symmetric; B is (meas × state); C is $3\times3$ acting on pixels
$\mathbf{\Sigma}$	covariance, §2.2	singular values in the SVD, §1.4	—	B only ever appears between $\mathbf{U}$ and $\mathbf{V}^\top$
$\lambda$	eigenvalue, §1.3	LM damping, §3.4	wavelength (m), §9.2	B is always added to a diagonal; C appears in sensor physics
$\mathbf{R}$	rotation matrix, §1.1	measurement noise covariance, §32.2	—	A is orthonormal with $\det = 1$; B sits next to $\mathbf{Q}$
$\mathbf{Q}$	process noise covariance, §32.2	orthonormal factor in QR, §1.5	attention queries, §21.2	Context is filtering, linear algebra, or transformers
$\mathbf{P}$	state covariance, §2.4	$3\times4$ camera matrix $\mathbf{K}[\mathbf{R}	\mathbf{t}]$, §7.2	probability
$\mathbf{S}$	innovation covariance, §2.5	a generic symmetric matrix, §1.3	IMU scale-factor matrix, §10.2	A appears inside $\mathbf{S}^{-1}$ in a Mahalanobis distance
$\mathbf{\Lambda}$	information matrix $\mathbf{\Sigma}^{-1}$, §2.3	eigenvalue matrix, §1.3	—	B sits between $\mathbf{V}$ and $\mathbf{V}^\top$
$\boldsymbol{\eta}$	information vector $\mathbf{\Sigma}^{-1}\boldsymbol{\mu}$, §2.3	white noise, §10.2	—	A pairs with $\mathbf{\Lambda}$; B has a sensor subscript
$\sigma$	standard deviation, §2.2	singular value, §1.4	—	B is indexed and ordered, $\sigma_1 \ge \sigma_2 \ge \dots$
$\rho$	robust loss function, §3.1	surface reflectivity, §8.1	pseudorange (m), §11.1	A takes an argument; B and C are sensor quantities
$c$	principal point $(c_x,c_y)$ (px), §7.2	speed of light (m/s), §9.2	robust kernel scale, §3.5	Subscripted $x$/$y$ means A
$\mathbf{g}$	gradient vector, §3.6	gravity vector (m/s²), §10.1	—	B is $\approx 9.81$ and points down
$\mathbf{F}$	objective function, §3.1	fundamental matrix, §16.6	focal length in mm, §7.2	A is scalar-valued; B is $3\times3$ rank-2
$s$	skew in $\mathbf{K}$ (≈0), §7.2	RANSAC minimal sample size, §3.7	squared residual $|\mathbf{r}|^2$, §3.5	Context is optics, sampling, or robust loss
$w$	robust weight, §3.5	RANSAC inlier ratio, §3.7	quaternion scalar part, §4.3	—

The two worth memorising, because they appear on the same page as each other more than once: $\mathbf{K}$ (intrinsics vs Kalman gain) and $\mathbf{H}$ (Hessian vs measurement Jacobian vs homography). When §32.2's Kalman update writes $\mathbf{K} = \mathbf{P}^-\mathbf{H}^\top \mathbf{S}^{-1}$, none of those three letters means what Part IV means by them.

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N.3 Geometry and frames

Symbol	Meaning	Units	§
$\mathbf{p}$	a point, usually in $\mathbb{R}^3$	m	§1.1
${}^{A}\mathbf{p}$	point $p$ expressed in frame $A$	m	§1.1
${}^{A}\mathbf{T}_{B}$	transform from frame $B$ to frame $A$, an element of $SE(3)$	—	§1.1
$\mathbf{R}$	rotation matrix, an element of $SO(3)$	—	§1.1
$\mathbf{t}$	translation vector	m	§1.1
$\mathbf{T}$	$4\times4$ homogeneous rigid transform	—	§4.1
$\tilde{\mathbf{p}}$	homogeneous coordinates: $(x,y,z) \mapsto (x,y,z,1)$	—	§1.6
$\mathbf{q}$	unit quaternion ($w$-first in this document)	—	§4.3
$\boldsymbol{\phi}$	rotation vector / $SO(3)$ tangent vector, $\in\mathbb{R}^3$	rad	§4.4
$\boldsymbol{\xi}$	$SE(3)$ tangent vector, $\in\mathbb{R}^6$	rad, m	§4.5
$(\cdot)^\wedge$	hat / skew operator: $\mathbb{R}^3 \to$ skew-symmetric $3\times3$	—	§4.4
$\boxplus, \boxminus$	manifold-aware addition and subtraction	—	§4.7
$\mathrm{Ad}$	adjoint — moves a tangent vector between left and right conventions	—	§4.6
$\boldsymbol{\omega}$	angular velocity	rad/s	§4.4
$\mathbf{n}$	surface normal, unit length	—	§28.2
$\mathbf{e}_i$	$i$-th basis vector	—	§1.1
$\mathbf{I}$	identity matrix	—	§2.3
$\mathbf{0}$	zero vector or matrix	—	§1.6

Linear algebra

Symbol	Meaning	Units	§
$\mathbf{A}, \mathbf{B}, \mathbf{M}$	generic matrices	—	§1.1
$m, n$	number of rows, number of columns	—	§1.2
$r$	rank	—	§1.2
$\mathrm{col}(\mathbf{A})$	column space — everything reachable	—	§1.2
$\mathrm{null}(\mathbf{A})$	null space — everything annihilated; where lost information lives	—	§1.2
$\mathbf{V}, \mathbf{\Lambda}$	eigenvector matrix, eigenvalue matrix	—	§1.3
$\lambda_i$	$i$-th eigenvalue	—	§1.3
$\mathbf{v}_i$	$i$-th eigenvector	—	§1.3
$\mathbf{U}, \mathbf{\Sigma}, \mathbf{V}$	SVD factors: $\mathbf{A} = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^\top$	—	§1.4
$\sigma_i$	$i$-th singular value, ordered $\sigma_1 \ge \sigma_2 \ge \dots$	—	§1.4
$\kappa$	condition number $\sigma_1/\sigma_r$ — the error amplification factor	—	§1.4
$\mathbf{W}$	weight matrix in weighted least squares, usually $\mathbf{\Sigma}^{-1}$	—	§1.5
$|\cdot|$	Euclidean norm	—	§1.5
$O(\cdot)$	asymptotic cost — $O(n^2)$ storage, $O(1)$ lookup	—	§2.3

---

N.4 Probability and estimation

Symbol	Meaning	Units	§
$p(x)$	probability density	—	§2.2
$p(\mathbf{x}\mid\mathbf{z})$	density of $\mathbf{x}$ given $\mathbf{z}$	—	§2.4
$\mathbb{E}[\cdot]$	expectation	—	§2.2
$\boldsymbol{\mu}$	mean vector	varies	§2.2
$\sigma, \sigma^2$	standard deviation, variance	varies	§2.2
$\mathbf{\Sigma}$	covariance matrix — an ellipsoid of uncertainty, not a table	varies	§2.2
$\mathbf{P}$	state covariance (the filter's own uncertainty)	varies	§2.4
$\mathcal{N}(\mathbf{x};\boldsymbol{\mu},\mathbf{\Sigma})$	Gaussian density	—	§2.3
$\mathbf{\Lambda}$	information matrix, $=\mathbf{\Sigma}^{-1}$	—	§2.3
$\boldsymbol{\eta}$	information vector, $=\mathbf{\Sigma}^{-1}\boldsymbol{\mu}$	—	§2.3
$\mathbf{z}$	measurement	varies	§2.4
$h(\cdot)$	measurement function — predicts $\mathbf{z}$ from state	—	§2.4
$\mathbf{x}$	state	varies	§2.4
$\hat{\mathbf{x}}$	state estimate	varies	§2.7
$\mathbf{r}$	residual, or innovation $\mathbf{z} - h(\hat{\mathbf{x}})$	varies	§2.6
$\mathbf{S}$	innovation covariance $\mathbf{H}\mathbf{P}^-\mathbf{H}^\top + \mathbf{R}$	varies	§2.5
$d^2$	squared Mahalanobis distance	—	§2.5
$|\mathbf{e}|^2_{\mathbf{\Sigma}}$	$\mathbf{e}^\top\mathbf{\Sigma}^{-1}\mathbf{e}$, the squared Mahalanobis norm	—	§2.4
$\chi^2_k$	chi-squared distribution with $k$ degrees of freedom	—	§2.5
$\boldsymbol{\epsilon}$	measurement noise	varies	§2.7
$\mathcal{I}$	Fisher information $\mathbf{J}^\top\mathbf{\Sigma}^{-1}\mathbf{J}$	—	§2.7
$\mathbf{Q}, \mathbf{R}$	process noise, measurement noise covariance	varies	§32.2
$\mathbf{K}$	Kalman gain	—	§32.2
$\mathbf{H}$	measurement Jacobian $\partial h/\partial\mathbf{x}$	—	§32.2
$\mathbf{\Phi}$	state transition matrix	—	§32.5
$\ell$	log-odds $\log\frac{p}{1-p}$ — occupancy grids	—	§34.2
$\mathrm{NIS}$	normalized innovation squared; should average $\dim(\mathbf{z})$	—	§2.6
$\mathrm{NEES}$	normalized estimation error squared; should average $\dim(\mathbf{x})$	—	§2.6

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N.5 Optimization

Symbol	Meaning	Units	§
$F$	scalar objective / cost function	—	§3.1
$\mathbf{r}(\mathbf{x})$	residual vector — prediction minus measurement	varies	§3.1
$\mathbf{J}$	Jacobian $\partial\mathbf{r}/\partial\mathbf{x}$; rows are outputs, columns are inputs	—	§3.2
$\nabla F$	gradient of the cost	—	§3.2
$\nabla^2 F$	true Hessian	—	§3.2
$\mathbf{H}$	approximate Hessian $\mathbf{J}^\top\mathbf{W}\mathbf{J}$	—	§3.3
$\mathbf{g}$	gradient vector $\mathbf{J}^\top\mathbf{W}\mathbf{r}$	—	§3.6
$\delta\mathbf{x}$	the increment solved for at each iteration	varies	§3.3
$\lambda$	Levenberg–Marquardt damping	—	§3.4
$\mathbf{D}$	LM damping matrix, $\mathbf{I}$ or $\mathrm{diag}(\mathbf{H})$	—	§3.4
$\rho(\cdot)$	robust loss function	—	§3.1
$s$	squared weighted residual $|\mathbf{r}|^2_\Sigma$, the argument to $\rho$	—	§3.5
$\delta$	Huber threshold (≈1.345 for whitened residuals)	σ	§3.5
$c$	Cauchy / Geman–McClure kernel scale	σ	§3.5
$\alpha$	Barron adaptive-loss shape parameter	—	§3.5
$w(s)$	IRLS weight, $=\rho'(s)$	—	§3.5
MAD	median absolute deviation; $\hat\sigma = 1.4826,\mathrm{MAD}$	varies	§3.5
$k$	RANSAC iteration count	—	§3.7
$s$	RANSAC minimal sample size	—	§3.7
$w$	RANSAC inlier ratio	—	§3.7
$\tau$	RANSAC inlier threshold — set from the noise model, never twiddled	varies	§3.7
$p$	RANSAC target success probability	—	§3.7
$\mathcal{M}$	the manifold the state lives on	—	§3.1

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N.6 Sensors

Cameras — §7

Symbol	Meaning	Units	§
$\mathbf{K}$	intrinsic matrix	—	§7.2
$f_x, f_y$	focal length in pixels; $f_x = F/p_x$	px	§7.2
$F$	focal length as the lens is marked	mm	§7.2
$p_x$	pixel pitch	mm	§7.2
$c_x, c_y$	principal point — where the optical axis meets the sensor	px	§7.2
$s$	skew — effectively always 0 on modern sensors	px	§7.2
$(u,v)$	pixel coordinates: column, row	px	§7.2
$(X,Y,Z)$	3D point in the camera frame ($Z$ forward, $X$ right, $Y$ down)	m	§7.2
$\pi(\cdot)$	projection function, $\mathbb{R}^3 \to \mathbb{R}^2$	—	§7.2
$\mathbf{P}$	$3\times4$ camera matrix $\mathbf{K}[\mathbf{R}	\mathbf{t}]$	—
$W, H$	image width, height	px	§7.2
$k_1, k_2, k_3$	radial distortion coefficients (barrel if $k_1<0$)	—	§7.3
$p_1, p_2$	tangential distortion coefficients — usually tiny	—	§7.3
$(x,y)$	normalized image coordinates $(X/Z, Y/Z)$, before $\mathbf{K}$	—	§7.3
$r^2$	$x^2+y^2$ in normalized coordinates	—	§7.3
$t_l$	rolling-shutter line delay	s	§7.4
$B$	stereo baseline	m	§7.7
$d$	stereo disparity	px	§7.7

The f, c, k problem. §13.1 refers to "camera $f$, $c$, $k$" long before §7.2 defines them. They are focal lengths, principal point, and radial distortion coefficients — the intrinsics — and this table is where to look.

LiDAR — §8

Symbol	Meaning	Units	§
$r$	range	m	§8.1
$\rho$	surface reflectivity	—	§8.1
$\alpha$	atmospheric extinction coefficient	1/m	§8.1
$\theta_i$	incidence angle to the surface normal	rad	§8.1
$P_t, P_r$	transmitted, received optical power	W	§8.1
$\Delta\theta$	angular beam spacing	rad	§8.3
$t$	per-point timestamp — a scan is not a snapshot	s	§8.7

Radar — §9

Symbol	Meaning	Units	§
$\lambda$	wavelength (≈3.9 mm at 77 GHz)	m	§9.1
$D$	antenna aperture	m	§9.1
$B$	chirp bandwidth	Hz	§9.2
$T_c$	chirp duration	s	§9.2
$f_b$	beat frequency	Hz	§9.2
$f_d$	Doppler shift, $=2v_r/\lambda$	Hz	§9.2
$v_r$	radial velocity	m/s	§9.2
$c$	speed of light	m/s	§9.2
$N$	chirps per frame	—	§9.2
$\Delta r, \Delta v$	range resolution, velocity resolution	m, m/s	§9.2

IMU — §10

Symbol	Meaning	Units	§
$\boldsymbol{\omega}$	angular rate, body frame	rad/s	§10.1
$\mathbf{a}$	specific force — not acceleration; a stationary IMU reads $+g$ up	m/s²	§10.1
$\mathbf{g}$	gravity vector, $\approx 9.81$ m/s²	m/s²	§10.1
$\mathbf{b}_g, \mathbf{b}_a$	gyroscope bias, accelerometer bias — must be filter states	rad/s, m/s²	§10.2
$\mathbf{S}_g, \mathbf{S}_a$	scale-factor and axis-misalignment matrices	—	§10.2
$\boldsymbol{\eta}_g, \boldsymbol{\eta}_a$	white noise	—	§10.2
$\sigma_g$	gyro noise density (ARW)	rad/s/√Hz	§10.3
$\sigma_a$	accelerometer noise density (VRW)	m/s²/√Hz	§10.3

GNSS — §11

Symbol	Meaning	Units	§
$\rho_i$	pseudorange to satellite $i$	m	§11.1
$\delta t_{\text{rx}}$	receiver clock error — the 4th unknown	s	§11.1
$I_i, T_i$	ionospheric, tropospheric delay	m	§11.1
$\mathbf{G}$	design matrix; DOP comes from $(\mathbf{G}^\top\mathbf{G})^{-1}$	—	§11.1
$\epsilon_i$	receiver noise	m	§11.1
${}^{B}\mathbf{l}$	lever arm — antenna offset from the IMU in body frame	m	§11.4
$L$	dual-antenna baseline length	m	§11.4

---

N.7 Learning

Symbol	Meaning	Units	§
$\mathbf{w}$	network weights / parameters	—	§19.2
$L$	loss	—	§19.3
$\alpha$	learning rate	—	§19.5
$\beta_1, \beta_2$	Adam momentum coefficients	—	§19.5
$z_i$	logit — a raw class score, pre-softmax	—	§19.3
$p_i$	softmax probability	—	§19.3
$y_i$	ground-truth label (one-hot)	—	§19.3
$C$	channel count	—	§20.2
$H, W$	feature-map height, width	—	§20.2
$k$	kernel size	—	§20.2
$s$	stride	—	§20.2
$p$	padding	—	§20.2
$\mathbf{Q}, \mathbf{K}, \mathbf{V}$	attention queries, keys, values	—	§21.2
$d$	attention key dimension (the $\sqrt{d}$ scaling)	—	§21.2
$\mathrm{Attn}$	$\mathrm{softmax}(\mathbf{Q}\mathbf{K}^\top/\sqrt{d})\mathbf{V}$	—	§21.2
$\mathrm{SSIM}$	structural similarity, used in self-supervised depth	—	§24.4
$l$	layer index; also box length in $(l,w,h)$ for a 3D box	—	§19.4, §29.5

Note the density of collisions in this table alone. In §20.2 the letters $k$, $s$, $p$, $H$, $W$ mean kernel size, stride, padding, height, width. In §3.7 the same $k$, $s$, $w$, $p$ mean RANSAC iterations, sample size, inlier ratio, and success probability. In §21.2, $\mathbf{Q}$ and $\mathbf{K}$ are attention matrices, not process noise and Kalman gain. Part V borrows machine learning's conventions wholesale, and they do not agree with the estimation literature.

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N.8 Greek quick-lookup

When you know the shape but not the name.

	Name	Primary meaning	§
$\alpha$	alpha	LiDAR extinction / Barron shape / learning rate	§8.1, §3.5, §19.5
$\beta$	beta	Adam momentum coefficients	§19.5
$\gamma$	gamma	GNSS carrier-phase ambiguity term	§11.5
$\delta$	delta	small increment; Huber threshold; clock error	§3.3, §3.5, §11.1
$\Delta$	Delta	a resolution or difference ($\Delta r$, $\Delta v$, $\Delta\theta$)	§5.2
$\epsilon$	epsilon	noise	§2.7
$\boldsymbol{\eta}$	eta	information vector; white noise	§2.3, §10.2
$\theta$	theta	an angle	§3.9
$\kappa$	kappa	condition number	§1.4
$\lambda$	lambda	eigenvalue; LM damping; wavelength	§1.3, §3.4, §9.2
$\mathbf{\Lambda}$	Lambda	information matrix; eigenvalue matrix	§2.3, §1.3
$\boldsymbol{\mu}$	mu	mean	§2.2
$\boldsymbol{\xi}$	xi	$SE(3)$ tangent vector, $\in\mathbb{R}^6$	§4.5
$\pi(\cdot)$	pi	projection function (not 3.14159)	§7.2
$\rho$	rho	robust loss; reflectivity; pseudorange	§3.1, §8.1, §11.1
$\sigma$	sigma	standard deviation; singular value	§2.2, §1.4
$\mathbf{\Sigma}$	Sigma	covariance; SVD singular-value matrix	§2.2, §1.4
$\tau$	tau	RANSAC threshold; radar time delay	§3.7, §9.2
$\boldsymbol{\phi}$	phi	rotation vector, $\in\mathbb{R}^3$	§4.4
$\mathbf{\Phi}$	Phi	state transition matrix	§32.5
$\chi^2$	chi-squared	gating distribution	§2.5
$\psi$	psi	heading angle	§35.3
$\boldsymbol{\omega}$	omega	angular velocity	§4.4
$\Omega$	Omega	image domain	§15.1

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N.9 If a symbol is not here

Three possibilities, in order of likelihood.

1. It is a local index. $i$, $j$, $k$, $m$, $n$ used as loop or summation variables carry no meaning beyond the equation they appear in.
2. It is defined in place and used only within one section. That is fine and correct — house rule 2 requires a symbol be defined in place or carry a link to where it was defined.
3. It is missing, which is a bug in this file. The document is the authority; this table describes it. Report it.

For the mathematics behind these symbols rather than their meanings, see math-primer.md: §A.1 (vectors and frames), §A.3 (eigenvalues, and why a covariance is a shape), §A.5 (the Jacobian, at length), §A.6 (least squares), §A.7 (covariance), §A.8 (the Gaussian and Bayes).