01 / Foundations
Geometry & Math
Linear algebra and vector calculus through geometric intuition, plus the statistical and information-theoretic tools for measuring relationships.
- Vectors & matrices
- Change of coordinates
- Gradient / divergence / curl
- Where quaternions came from
- From inner products to attention
Series
SeriesFields, Seen Through Local DerivativesGradient, Jacobian, divergence, curl, Hessian and Laplacian are facets of one idea. The derivative is a local linear map, and the other operators extract specific information from it. This series derives the field operators that appear most often in 3D vision and robotics along that single line.3 chaptersSeriesMeasuring Relationships: From Inner Products to AttentionCovariance, correlation, cross-correlation, convolution, attention and mutual information all answer one question, namely how strongly two things are related. This series derives them along a single line of reasoning and shows which parts of CNNs and Transformers that line explains, and which it does not.4 chapters
Published notes
- Inner products, covariance and correlationStarting from the inner product, centering gives covariance and normalizing gives correlation; the correlation coefficient is the cosine of the angle between the centered vectors. This chapter derives the chain in full and shows where these quantities appear in state estimation and point-cloud processing.
- The Jacobian and the gradient: the derivative as a local linear mapExtend the one-variable derivative to several variables and it turns from a number into a matrix, the Jacobian; the transpose of a scalar field's Jacobian is the gradient. This chapter derives why the gradient points uphill fastest and is normal to level sets, and works out the Jacobians of camera projection and pose optimization.
- Cross-correlation and convolution: the sliding inner productAdding a shift to the inner product gives cross-correlation; flipping the kernel gives convolution. This chapter derives how the two relate, why linear time-invariant systems are exactly convolutions, and why normalized cross-correlation is the correlation coefficient over a window, with uses in template matching and stereo.
- Divergence and curl: what they extract from the JacobianThe Jacobian fully describes what a vector field does near a point, and divergence and curl extract two parts of it. Divergence is the trace and reads off the local change of volume; curl comes from the antisymmetric part and equals twice the local angular velocity. This chapter derives both, their integral forms and the two zero identities, and uses optical-flow divergence to estimate time to contact.
- From convolution to attentionA convolutional layer takes inner products between shared learned templates and local patches; attention takes pairwise inner products between queries and keys generated from the input, then aggregates with softmax weights. This chapter derives both, the origin of the √d scaling and permutation equivariance, and asks how much "measuring relationships" really explains about why the two architectures succeed.
- Hessian, Laplacian and Poisson reconstructionTaking the Jacobian of the gradient gives the Hessian, and its trace gives the Laplacian. This chapter derives the second-order Taylor expansion, the link between the Laplacian and neighbourhood averages, and the second-order identities, then derives the Poisson surface reconstruction equation from a least-squares energy and shows where the Gauss–Newton approximation of the Hessian comes from.
- Mutual information: dependence beyond linearityZero correlation does not mean independence. Mutual information measures any form of statistical dependence through probability distributions. This chapter derives its three equivalent definitions and its non-negativity, shows that for Gaussians it is fully determined by the correlation coefficient, reads a confusion matrix as a joint distribution, and shows how it drives camera–LiDAR calibration.
Questions on the roadmap
- A matrix is a transform, not a tableWhy is matrix multiplication defined the way it is?
- Gradient, divergence, curlWhat does a field tell us locally?
- Where quaternions came fromWhy did Hamilton need a fourth number?
- Same thing, different frameChange of basis, similarity and invariants.