Notation

What each symbol means, grouped by topic and in course order. The formulas themselves are on the Formula Sheet.

Read this first

Many letters are reused across the course: $\alpha$, $\sigma$, $T$, $t$, $f$, $W$, $h$ each mean at least two different things. In the exam, write down what your symbol means whenever it could be ambiguous. See Symbols with Several Meanings.

Coordinates and Cameras

Lectures 2.1, 3, 4

Symbol	Meaning	Unit / space
$(U,V,W)$, $P_W$	point in world coordinates	metric, 3D
$(X,Y,Z)$, $P_C$	point in camera coordinates; $Z$ = depth along the optical axis	metric, 3D
$(x,y)$	film (image) coordinates	metric, 2D
$(u,v)$	pixel coordinates	pixels, 2D
$\tilde{\cdot }$	homogeneous coordinates ($\tilde{u}=(u,v,1)$, defined up to scale)
$f$	focal length (pinhole: distance pinhole to sensor)	mm, or px after dividing by $s_x$
$s_o,s_i$	object distance and image distance of a lens
$s_x,s_y$	pixel size (width, height)	mm/px
$o_x,o_y$	principal point: where the optical axis hits the sensor	px
$K$, $M_{int}$	intrinsic matrix ($f/s_x$, $f/s_y$, $o_x$, $o_y$, skew)
$R$	rotation matrix, $R\in SO(3)$
$C$	camera center in world coordinates
$T$, $t$	translation of the extrinsics, $T=-RC$ (not the camera position)
$M_{ext}$, $[R\mid t]$	extrinsic matrix (world → camera)
$P$, $M$	projection matrix $K[R\mid t]$, $3\times 4$
$\pi (P,X)$	projection of the 3D point $X$ with camera $P$

Columns of $R$

In $P_C=R{P}_W$ the columns of $R$ are the world axes in camera coordinates. The columns of $R^T$ are the camera axes in world coordinates. Quick check: $R(1,0,0{)}^T$ is the first column. See Figuring out rotations.

Rotations

Lecture 2.2

Symbol	Meaning
$R_x,R_y,R_z$	rotation about one coordinate axis
$\alpha ,\beta ,\gamma$	Euler angles
$\omega$	rotation axis (unit vector)
$\theta$	rotation angle
$\hat{\omega }$, $[\omega {]}_{\times }$, ${\omega }_{\times }$	skew-symmetric matrix of $\omega$, the matrix form of the cross product
$\mathbf{q}=(q_w,\mathbf{v})$	quaternion: scalar part $q_w$, vector part $\mathbf{v}$
$\bar{\mathbf{q}}$	quaternion conjugate $(q_w,-\mathbf{v})$
$\xi =(v,\omega )$, $\hat{\xi }$	twist coordinates, twist matrix ($4\times 4$)
$G$	rigid transformation $\left[\begin{array}{cc}R& t\\ 0& 1\end{array}\right]$

Epipolar Geometry and SfM

Lectures 3, 4

Symbol	Meaning
$x_0,x_1$	normalized (calibrated) image points in camera 0 and camera 1
$u_0,u_1$	pixel coordinates of the same points
$E$	essential matrix $[t{]}_{\times }R$ (calibrated, rank 2, 5 DoF)
$F$	fundamental matrix $K_0^{-T}E{K}_1^{-1}$ (pixels, rank 2, 7 DoF)
${\tilde{e}}_0,{\tilde{e}}_1$	epipoles
${\tilde{l}}_0,{\tilde{l}}_1$	epipolar lines
$W$ (8-point)	the $N\times 9$ matrix, one row per correspondence
$W$ (pose from $E$)	the fixed $90°$ rotation matrix in $R=UW{V}^T$
$T_i$	normalization transform of the normalized 8-point algorithm
$u_{ij}$	observation of point $j$ in image $i$
$w_{ij}$	visibility indicator in bundle adjustment
$s$, $e$, $p$	RANSAC: sample size, outlier ratio, desired success probability
$b$, $d$, $z$	stereo: baseline, disparity, depth
$D_{max}$	number of disparity hypotheses in the cost volume

Two conventions for $R$ and $t$ in epipolar geometry

In the lecture, $R$ maps vectors from camera 1 to camera 0 and $t$ is the position of camera 1 in camera 0’s frame: $x_0^T[t{]}_{\times }R{x}_1=0$. Hartley and Zisserman use camera 1’s extrinsics $\hat{R}=R^T$, $\hat{t}=-R^{T}t$ and write $x_1^T[\hat{t}{]}_{\times }\hat{R}{x}_0=0$. The two are equivalent: $E={\hat{E}}^T$. Lecture 4 writes the same constraint as $p^{T}E{p}^{\prime }=0$ with $E=[T_{\times }]R$. See Assignment 2.

Surfaces, Fields and Point Clouds

Lectures 5, 6.1, 6.2

Symbol	Meaning
$\mathcal{S}$	the surface, $\{\mathbf{p}\mid f(\mathbf{p})=0\}$ for a distance field
$\mathcal{V}$	the volume (inside)
$f(\mathbf{p})$	a field: occupancy, UDF or SDF value at the point $\mathbf{p}$
$\nabla f$	gradient of the field = surface normal
$\chi$	indicator function (1 inside, 0 outside) in Poisson reconstruction
$V(x)$	vector field built from the oriented normals
$F(x,x_i)$	smoothing kernel (Poisson)
${\mathbf{p}}_i,{\mathbf{n}}_i$	Hermite data: edge intersection points and normals (dual contouring)
$\mathbf{X},\mathbf{Y}$	source and target point sets ($N\times 3$) in Procrustes / ICP
$\bar{\mathbf{X}},\bar{\mathbf{Y}}$	centered point sets; $\bar{\mathbf{x}},\bar{\mathbf{y}}$ centroids
$s,\mathbf{R},\mathbf{t}$	scale, rotation, translation of the similarity transform
$\mathbf{z}$	latent shape code (DeepSDF style)
${\mathbf{F}}_k(\mathbf{p})$	multi-scale deep features sampled at $\mathbf{p}$ (IF-Nets, NDF)
$h,g,f$	PointNet: per-point MLP, symmetric pooling (max), final MLP
$A$	T-Net feature transform
$\mathbf{q},\mathbf{k},\mathbf{v}$	query, key, value
$\delta =\theta ({\mathbf{p}}_i-{\mathbf{p}}_j)$	relative positional encoding (Point Transformer)

SDF sign

Lecture 6.1, slide 14 defines the SDF positive inside. DeepSDF and many papers use negative inside. State your convention. See Signed Distance Function.

Neural Rendering

Lectures 7, 8. These are the symbols of exam task 4.

Symbol	NeRF	3D Gaussian Splatting
$\mathbf{r}(t)=\mathbf{o}+t\mathbf{d}$	camera ray: origin $\mathbf{o}$, direction $\mathbf{d}$	(rays are not marched)
$t_n,t_f$	near and far bound of the ray
$i$	sample index along the ray	index of a Gaussian, sorted by depth
$\mathcal{N}$		the Gaussians overlapping the pixel
$\sigma$	volume density: $\sigma dt$ = probability that the ray stops in $[t,t+dt]$
${\delta }_i$	distance between samples, $t_{i+1}-t_i$
${\alpha }_i$	computed opacity of the segment, $1-\exp (-{\sigma }_i{\delta }_i)$	$o_i{G}_i^{2D}(\mathbf{x})$
$o_i$		learned opacity of Gaussian $i$
$T_i$	transmittance: probability that the ray reaches sample $i$ unoccluded, ${\prod }_{j<i}(1-{\alpha }_j)$	the same product, written out
${\mathbf{c}}_i$	color from the MLP, depends on position and view direction	color from spherical harmonics
$C$, $\hat{C}(\mathbf{r})$	rendered pixel color (hat = prediction)	rendered pixel color
$\mu$		center of the Gaussian
$\Sigma$		3D covariance $RS{S}^T{R}^T$
$R,S$		rotation (from a quaternion) and diagonal scale matrix
${\Sigma }^{\prime }$		projected 2D covariance $JW\Sigma W^T{J}^T$
$W$, $J$		viewing transform, Jacobian of the affine approximation of the projection
$G^{2D}(\mathbf{x})$		projected 2D Gaussian evaluated at pixel $\mathbf{x}$
$\gamma (p)$	positional encoding with $L$ frequencies	(not needed)
$\theta ,\varphi$	view direction angles in $F_{\theta }(x,y,z,\theta ,\varphi )$

Learning-Based Reconstruction

Lecture 9

Symbol	Meaning
$X^{v,1}$	point map of view $v$, expressed in the frame of camera 1
$\bar{X}$	ground-truth point map
$i$	pixel index; ${\mathcal{D}}^v$ the pixels with valid ground truth
$z,\bar{z}$	normalization factors (mean distance to the origin)
$C_i^v$	confidence of pixel $i$ in view $v$
$\alpha$	weight of the $-\log C$ regularizer
$\mathcal{E}$, $e$	edges (image pairs) of the pair graph
${\sigma }_e,P_e$	per-edge scale and rigid transform in global alignment
${\chi }^v$	global point map of view $v$

Diffusion and Flow Matching

Lectures 10, 11

Symbol	Meaning
${\mathbf{x}}_0$	clean data sample
${\mathbf{x}}_T$	pure Gaussian noise; $T$ = number of diffusion steps
${\mathbf{x}}_t$	noisy sample at step $t$
$q(\cdot )$	forward (noising) process, fixed
$p_{\theta }(\cdot )$	learned reverse (denoising) process
${\beta }_t$	noise schedule (variance added at step $t$)
${\alpha }_t=1-{\beta }_t$
${\bar{\alpha }}_t={\prod }_{s\le t}{\alpha }_s$	cumulative product; how much of ${\mathbf{x}}_0$ is left at step $t$
$\mathbf{ϵ}$	the true noise, $\mathcal{N}(0,\mathbf{I})$
${\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)$	the noise predicted by the network
${\mathbf{\mu }}_{\theta }$	mean of the learned reverse step
${\sigma }_t$	standard deviation of the noise re-injected during sampling
$\mathbf{z}$	fresh noise in DDPM sampling; latent in Stable Diffusion
$c(x,R,T)$	Zero-1-to-3 conditioning: input image and relative camera pose
$p$, $q$	flow matching: source (noise) and target (data) distribution
$X_0\sim p$, $X_1\sim q$	noise sample and data sample. Direction is reversed w.r.t. diffusion
$t\in [0,1]$	continuous time
${\psi }_t$	flow (where a point is at time $t$)
$u_t$, $u_t^{\theta }$	true and learned velocity field
$p_t$	probability path
$\theta$ (SDS)	parameters of the 3D representation; $\mathbf{x}=g(\theta )$ the render
$w(t)$	time weighting (SDS)

Time runs in opposite directions

Diffusion: ${\mathbf{x}}_0$ = data, ${\mathbf{x}}_T$ = noise. Flow matching (as in the lecture): $X_0$ = noise, $X_1$ = data. See Flow matching vs. diffusion.

Symbols with Several Meanings

Symbol	Meanings
$h$	object height in $y=fh/Z$ · sensor size in $\text{FOV}=2\arctan (h/2f)$ · per-point MLP in PointNet
$f$	focal length · field / implicit function $f(\mathbf{p})$ · vectorized $F$ in the 8-point algorithm · final MLP in PointNet
$F$	fundamental matrix · kernel in Poisson reconstruction · feature channels in a cost volume · radiance field $F_{\theta }$
$T$	translation of the extrinsics · baseline in rectified stereo · transmittance · number of diffusion steps · normalization $T_i$ in the 8-point algorithm
$t$	translation · ray parameter · diffusion / flow time
$\alpha$	Euler angle · NeRF / 3DGS opacity · $1-{\beta }_t$ in DDPM · cumulative product in DDIM · weight in screened PSR and in the confidence loss
$\sigma$	volume density · singular value · noise std in sampling · per-edge scale in global alignment
$W$	8-point data matrix · fixed matrix in $R=UW{V}^T$ · viewing transform in 3DGS · image width
$C$	camera center · rendered color · confidence (DUSt3R) · constant in the diffusion loss
$s$	pixel size $s_x,s_y$ · scale factor · RANSAC sample size · object distance $s_o$
$d$	disparity · ray direction $\mathbf{d}$ · $d_k$ key dimension in attention
$\theta$	rotation angle · network parameters · view direction angle
$\gamma$	Euler angle · positional encoding · attention MLP in the Point Transformer
$E$	essential matrix · energy (ICP, dual contouring) · encoder $\mathcal{E}$
$K$	intrinsic matrix · grid size of IF-Net features · number of points per group (PointNet++)

Focal length vs. focus distance

Focal length $f$ is a property of the lens. Focus distance is $s_o$, the distance to the plane in focus on the object side. In the pinhole model $f$ is the distance to the sensor; in the lens model the perspective is set by $s_i$. See Pinhole vs. lens model.

Related

• Formula Sheet
• Exam Structure · Study Plan · Assignments
• Course overview · Concepts