Texas A&M UniversityFall 2026
See the geometry. Build the computation. Use mathematics to understand how visual data is represented, transformed, projected, analyzed, and learned.
A = UΣVᵀ
x′ = Rx + t
λx = K[R|t]X
Course philosophy
The course develops mathematical intuition and computational skill through theory, hands-on exercises, and programming projects in visualization, graphics, computer vision, and computational media.
Understand what a mathematical object does before treating it as a formula to memorize.
Vectors become directions and measurements. Matrices become transformations. Subspaces become models. Cameras become projective maps. Fourier coefficients become visual frequencies. Neural layers become affine maps followed by nonlinearities.
What you will be able to do
Apply vectors, matrices, eigenanalysis, SVD, and projections to visual-computing problems.
Use complex numbers and quaternions to represent 2D and 3D orientation.
Use Fourier transforms to analyze and manipulate visual signals and images.
Understand neural networks as structured function approximators built from linear algebra.
Program advanced mathematical ideas and turn them into working visual demonstrations.
The learning arc
Each phase provides the language needed by the next: algebra supports geometry, geometry supports cameras, cameras create data, and data motivates signal and neural representations.
Weeks 1–3
Move from vectors and matrices to linear systems, subspaces, rank, basis, and dimension.
Enter FoundationsWeeks 4–6
Study rigid motion, invertibility, eigenstructure, and singular-value decomposition.
Enter GeometryWeeks 7–10
Connect least squares and PCA to image transformations, camera models, and calibration.
Enter Data & ProjectionWeeks 11–13
Use complex numbers, quaternions, and Fourier analysis to model motion and imagery.
Enter RepresentationsWeeks 14–15
Interpret multilayer perceptrons through linear algebra and synthesize the complete course pipeline.
Enter Learning & Synthesis
Weekly material
Every week combines the syllabus topics, a concise conceptual explanation, key equations, a visual-computing connection, and a mathematically generated concept video.
Showing all 15 weeks
Linear combination, vector projection, and the action of a matrix on a grid.
Week 1 · Intro to Linear Algebra for Visual Computing—I
Begin with the objects that carry visual information: vectors encode coordinates, directions, colors, and features, while matrices organize and transform them.
A linear combination builds a new vector from existing directions. The dot product measures alignment and produces projections. A matrix is best understood through the transformation it applies.
Visual computing connection. These ideas appear immediately in coordinate changes, feature similarity, surface normals, color transforms, and geometric modeling.
Open the Week 1 visualAn exact two-equation system, row elimination, and a verified LU factorization.
Week 2 · Intro to Linear Algebra for Visual Computing—I
Turn collections of constraints into solvable systems, then expose the triangular structure that makes repeated solution efficient.
Elimination replaces a system by an equivalent triangular one. LU decomposition records the elimination process as a product of lower- and upper-triangular matrices.
Visual computing connection. Linear systems arise in reconstruction, interpolation, image processing, geometry, and the estimation of model parameters.
Open the Week 2 visualA rank-two column space, a dependent column, and a null-space direction mapped to zero.
Week 3 · Vector Spaces and Subspaces
Move beyond individual vectors to the spaces they span, and learn how rank and null spaces reveal information, redundancy, and ambiguity.
A basis gives a minimal coordinate system for a space. Rank counts independent directions captured by a matrix; the null space contains directions that the matrix cannot observe.
Visual computing connection. Rank explains degeneracy in data, low-dimensional structure in images and shapes, and unobservable directions in inverse problems.
Open the Week 3 visualA shape undergoing rotation and translation while lengths, angles, and dot products remain unchanged.
Week 4 · Orthogonality and Rigid Body Motion
Use orthogonality to create stable coordinates and rotation matrices that preserve the metric structure of a visual scene.
Orthonormal bases preserve lengths and angles. A proper rotation satisfies RᵀR = I and det(R) = 1; translation then completes a rigid-body transformation.
Visual computing connection. Rigid transforms are the basic language of camera pose, object motion, registration, robotics, and 3D graphics.
Open the Week 4 visualArea scaling under a linear map and the approach to singularity as a parallelogram collapses.
Week 5 · Determinants and Inverses
Connect algebraic invertibility to a geometric question: does a transformation preserve information or collapse space?
The determinant measures signed area or volume scaling. A zero determinant means the transformation collapses at least one dimension and therefore has no inverse.
Visual computing connection. Invertibility matters in image warping, coordinate recovery, numerical stability, and the detection of degenerate configurations.
Open the Week 5 visualA unit circle mapped to an ellipse, orthogonal eigendirections, and the rotate–scale–rotate stages of SVD.
Week 6 · Eigen Values and Eigen Vectors
Find the directions a transformation treats specially, then decompose any linear map into orthogonal rotations and axis-aligned scaling.
An eigenvector keeps its direction under a matrix. For symmetric matrices, eigenvectors can be chosen orthonormal. SVD extends this geometric decomposition to any matrix.
Visual computing connection. Eigenanalysis and SVD support compression, denoising, conditioning, PCA, low-rank models, and many vision algorithms.
Open the Week 6 visualAn overdetermined line fit with vertical residuals, the normal-equation condition, and the pseudoinverse solution.
Week 7 · Least Squares and Pseudo Inverses
When noisy visual measurements do not satisfy a model exactly, estimate the parameters whose prediction is closest to the data.
Least squares projects the observation onto the model subspace. At the optimum, the residual is orthogonal to every column of the design matrix.
Visual computing connection. This is the backbone of line fitting, calibration, triangulation, motion estimation, and many inverse problems.
Open the Week 7 visualA correlated point cloud, its principal axes, and projection onto the dominant component.
Week 8 · Visual Computing Application—I
Use projection to retain the most informative directions in high-dimensional visual data while discarding redundancy.
PCA centers the data, finds covariance eigendirections, and projects onto the directions of largest variance. An orthogonal projector is symmetric and idempotent.
Visual computing connection. PCA provides compact models of appearance, shape, motion, and features, and clarifies the geometry of dimensionality reduction.
Open the Week 8 visualA true projective warp, a checkerboard homography, and the pinhole perspective relationship.
Week 9 · Visual Computing Application—II
Connect coordinates in images to projective geometry and understand how a camera converts a 3D scene into a 2D measurement.
A homography is a projective transformation of a plane. Perspective projection divides by depth, so farther points appear smaller in the image.
Visual computing connection. These models drive image rectification, mosaicing, planar tracking, augmented reality, and 3D-to-2D projection.
Open the Week 9 visualNormalized DLT for planar homographies followed by Zhang constraints that recover camera intrinsics.
Week 10 · Visual Computing Application—III
Estimate the parameters that connect rays in a camera to pixel coordinates, using correspondences and multiple views of a planar pattern.
DLT estimates a projective mapping from linear constraints. Zhang’s method combines planar homographies from different orientations to recover the intrinsic calibration matrix.
Visual computing connection. Calibration is required whenever metric geometry, accurate pose, measurement, or multi-view reconstruction is expected from images.
Open the Week 10 visualA point rotating on the complex plane with constant modulus and its equivalent 2D rotation matrix.
Week 11 · Understanding Rotation in 2D and 3D—I
Represent a 2D point as a complex number and turn rotation into multiplication by a unit-magnitude complex number.
Euler’s formula joins trigonometry and complex exponentials. Multiplication by eⁱᶿ rotates a point by θ without changing its magnitude.
Visual computing connection. Complex arithmetic gives a compact language for 2D rotations, periodic motion, phasors, and frequency-domain representations.
Open the Week 11 visualAn arbitrary 3D axis, Rodrigues rotation, and the equivalent unit-quaternion action on a vector.
Week 12 · Understanding Rotation in 2D and 3D—II
Represent 3D orientation using a rotation axis and angle, then encode the same rotation compactly with a unit quaternion.
Axis–angle expresses a rotation geometrically. The corresponding unit quaternion uses half the rotation angle and composes rotations through quaternion multiplication.
Visual computing connection. Quaternions are widely used for camera orientation, smooth interpolation, animation, and robot motion without coordinate singularities.
Open the Week 12 visualA square-wave Fourier series built from odd harmonics and a genuine 2D FFT low-pass image reconstruction.
Week 13 · Visual Computing Application—IV
Treat functions as combinations of basis functions and reveal how visual signals distribute their energy across frequencies.
Fourier analysis represents a signal as a weighted sum of sinusoids or complex exponentials. Frequency coefficients separate broad structure from fine detail.
Visual computing connection. The frequency domain explains filtering, denoising, sharpening, sampling, compression, and image reconstruction.
Open the Week 13 visualAn exact two-unit ReLU construction for XOR and the piecewise-linear regions created by affine maps plus activation.
Week 14 · Multi-layer Perceptron (MLP)
See a neural network as a sequence of affine maps and nonlinearities that bends input space into a useful representation.
Each layer applies a matrix transformation and bias, followed by a nonlinear activation. Without the nonlinearity, many layers collapse into a single affine map.
Visual computing connection. This viewpoint links classical linear algebra to learned visual representations, implicit functions, and nonlinear approximation.
Open the Week 14 visualA course-wide pipeline linking geometry, projection, low-rank models, Fourier representations, and neural approximation.
Week 15 · Course Revision; Final Project Presentation and Demo
Bring the semester together as a connected visual-computing pipeline rather than a collection of isolated formulas.
Geometry represents the scene; cameras project it; matrix models organize data; signal bases expose structure; neural functions approximate complex visual relationships.
Visual computing connection. The final project demonstrates how mathematical representation, computation, and visual reasoning work together in a complete application.
Open the Week 15 visualTry a topic such as “rotation,” “camera,” “Fourier,” or “SVD.”
Assessment
There are no written examinations. Evaluation centers on assignments, a final project, participation, and the ability to demonstrate authentic work.
Programming and concept work connected to material discussed in class.
Teams of two or three are strongly encouraged, with a final presentation and visual demonstration.
Attendance, in-class activity, timely work, final presentation participation, and a demo video clip.
Project-centered: the final exam period may be used for final project presentations.
Authentic work: students should retain records and be able to explain and demonstrate submitted work.
Schedule: weekly topics and timing may adjust in response to class interaction.
Resources & course information
Course access
The syllabus contains complete course policies, grading details, learning outcomes, recommended resources, and university statements.
Download syllabusComputing
Recommended references
Instructor
Office hours by appointment
suryanshkumar@tamu.edu