A rough outline of topics covered:

• A. coli
  • Review of calculus: functions, directional derivatives, gradients, vector fields, gradient ascent.
  • Vignette #1: Run-and-tumble behavior of A. coli.
  • The tangent plane is the best "linear" approximation to a surface near a point.
  • The gradient points in the direction of maximum increase of a function.
  • Gradient ascent (hill climbing) algorithm.
  • The tangent and normal (T, N) local frame attached to curves.
  • Tumble frees A. coli from local maxima.
• The Barnacle
  • Review of neuroscience.
  • Vignette #2: The barnacle.
  • Anatomy of Barnacle's nervous system.
  • From neural circuitry to elementary visual behavior.
  • Shadow reflex: if dI/dt > threshold, then withdraw.
  • Shadow reflex helps Barnacle avoid being food.
• Limulus linearis
  • Vignette #3: Limulus polyphemus.
  • The Limulus visual behavior and possible roles of the two visual systems.
  • Introduction of Linear Systems: (1) Additive; (2) Homogeneous.
  • The Principle of Superposition.
  • Unit step function, step response; Dirac Delta function, impulse response.
  • Representation of signals as a linear combination of step functions.
  • Convolution operator, the output of a linear system via convolution.
  • Check out Delta Function (Mathworld) and Representations of the Delta Function.
  • Demonstrations in Signals, Systems and Control by Wilson J. Rugh, et. al at Johns Hopkins University.
• Limulus linearis in Frequency
  • The Fourier transform - Frequency domain representation of continuous signals.
  • Sine and Cosine as eigenfunctions of linear systems.
  • Fourier basis are the eigenfunctions of linear systems.
  • Modulation Transfer Function (MTF) as a characterization of linear systems in the frequency domain.
  • Fourier transformation of convolution operation on functions.
  • Low pass filter, band pass filter, and high pass filter.
  • Linear systems being represented in frequency domain.
  • Sampling theory in a nutshell.
• Matching the Environmental Template
  • Nonlinearities in the human visual system.
  • The human visual system is band pass.
  • Vignette #4: Frog. "What Does the Frog's Eye Tell the Frog's Brain".
  • Image Blurring vs Image Sharpening.
  • The primate retina: rods, cones, fovea, the optical nerve, the blind spot.
• Introduction to Primate Visual Physiology
  • Retinal receptive fields : On-center and Off-center.
  • Visual Areas V1, six cortical layers.
  • Lateral Geniculate Nucleus (LGN, dLGN).
  • Receptive field types: simple cells, complex cells, hypercomplex cells.
  • Functional architecture of V1 ("Ice Cube Model"): Orientation columns, ocular dominance bands, hypercolumns, processes running within a column.
  • Models for composing simple features into more complex ones.
• First Paradigm Computer Vision
  • Hierarchies of feature detectors.
  • Histograms, differential operators in edge detection.
  • Composition of smoothing and differential operators (regularized edge detectors) followed by histogram peak selection.
  • Handout: Functional architecture of macaque monkey visual cortex" by D.H. Hubel and T.N. Wiesel.
• Second Paradigm Computer Vision
  • Physics Based Vision: Models for image formation, relation between image irradiance and scene radiance.
  • The BRDF (Bidirectional Reflectance Distribution Function).
  • Lambertian and specular surfaces.
  • Raise the Shape from Shading problem.
  • Handout from "Robot Vision", by Berthold K. P. Horn: Chapter 8 (Edges & Edge Finding).
  • The hough transform.
• Intro to Projective Geometry
  • Homogeneous coordinates, projective transformation.
  • Pinhole Camera Model: Image coordinate system, camera coordinate system and world coordinate system.
  • Projection matrix, intrinsic/internal parameters, extrinsic/external parameters.
• Shape from Shading
  • The Reflectance Map and the Image Irradiance Equation.
  • The foundation of Shape From Shading and its corresponding PDE.
  • Handout from "Robot Vision", by Berthold K. P. Horn: Chapter 11 (Reflectance Map: Shape from Shading).
• Inverse Problems
  • Other inverse problems: Structure from Motion, optical flow constraint equation, Stereoscopic Vision.
  • The Laplace equation and the heat equation, equilibrium distribution.
  • Image deblurring as solving the heat equation backwards in time.
• More Primate Physiology
  • Functions and connections of different areas: V1, V2, V4.
  • "What" pathway and "Where" pathway.
• The Local to Global Transition
  • Simple parametrized models: The hough transform.
  • Local to global transition for general shapes as an equilibrium problem.
  • Equilibrium configurations.
  • Solving Laplace's equation: The notions of harmonic functions, minimal surfaces, and the maximum principle.
  • Southwill relaxation, and its relation to lateral inhibition.
  • Handout: On the Foundations of Relaxation Labeling Processes, by Robert A. Hummel and Steven W. Zucker.
• Constraint Satisfaction
  • Discrete constraint satisfaction problem.
  • Discrete relaxation labeling.
  • Relaxation labeling: Nodes, labels, compatibilities, label discarding rule, average local consistency.
  • Handout: On the Foundations of Relaxation Labeling Processes, by Robert A. Hummel and Steven W. Zucker.
• Intro to Differential Geometry
  • The notions of curve, tangent, curvature and torsion.
  • The Frenet frame and the Frenet equations.
  • Handout from "Elementary Differential Geometry", by Barrett O'Neil: Chapter 1 (Calculus on Euclidean Space), Chapter 11 (Frame Fields).
• Edge Detection
  • Canny edge detector and its weakness.
  • Regular curves, piecewise regular curves.
  • Discontinuities in the image: Folds, cusps.
  • The principle of building reliable non-linear operators via logical combination of simple linear filters (logical/linear operators).
• Curvature
  • Endstopping.
  • The connection between V1, and the differential geometry of curves.
  • Introduction to cocircularity, and how to put it in a relaxation labeling network.
• Stereo
  • Brief discussion of stereo.
  • 3D point reconstruction.
  • Correspondence problem.
  • Intensity-based area correlation technique (fronto-parallel assumption in local neighborhood) and its weakness.
  • Expand cocircularity to 3D space, connecting the differential structures within images with 3D curves.
• Texture Flow
  • Tangential curvature and normal curvature, covariant derivatives, 1-forms
  • The minimal surfaces by Euler-Lagrange equation: plane, left helicoid, right helicoid.
  • Right helicoid has constant ratio of tangential curvature and normal curvature. Choose right helicoid as the compatibility structure.
• Conclusion
  • Wrapping up: Predictions for V1 and extensions to texture flow, shading and stereo.
• Additional Topics
  • Color
  • Cytochrome oxidize blobs, V4(color).
  • The comparison between the geometric aspect of early vision and color/contrast and other scalar measurements.
  • Experiments on Apparent Motion and Texture.
  • Bridging information processing, computational modeling, physiology, neurobiology.