AlgebraicVision.jl

AlgebraicVision.jl is a Julia package that generates an algebraic problem in geometric computer vision as a polynomial system depending on the setup of cameras that observe the scene and the scene itself.

Installation

Enter the Pkg REPL by pressing ] from the Julia REPL and then type

add https://github.com/MultivariatePolynomialSystems/AlgebraicVision.jl.git

To get back to the Julia REPL, press backspace.

Usage

Vision Problem Formulation

using AlgebraicVision

radial_4v13p = VisionProblem(
    cameras = CamerasSetup(
        ncameras = 4,
        camera_types = :radial,
        calibration = :calibrated
    ),
    scene = SceneSetup(
        npoints = 13,
        nlines = 0,
        incidences = []
    ),
    formulation = Formulation(
        pose = :rel,
        rotations = :cayley
    )
)

The result of the VisionProblem constructor is an object of type VisionProblem that is formulated by a polynomial system. The polynomial system can be viewed through

radial_4v13p.equations

System of length 104
 111 variables: c₁₋₁, c₂₋₁, c₃₋₁, c₁₋₂, c₂₋₂, c₃₋₂, c₁₋₃, c₂₋₃, c₃₋₃, c₁₋₄, c₂₋₄, c₃₋₄, t₁₋₁, t₂₋₁, t₁₋₂, t₂₋₂, t₁₋₃, t₂₋₃, t₁₋₄, t₂₋₄, α₁₋₁, α₂₋₁, α₃₋₁, α₄₋₁, α₅₋₁, α₆₋₁, α₇₋₁, α₈₋₁, α₉₋₁, α₁₀₋₁, α₁₁₋₁, α₁₂₋₁, α₁₃₋₁, α₁₋₂, α₂₋₂, α₃₋₂, α₄₋₂, α₅₋₂, α₆₋₂, α₇₋₂, α₈₋₂, α₉₋₂, α₁₀₋₂, α₁₁₋₂, α₁₂₋₂, α₁₃₋₂, α₁₋₃, α₂₋₃, α₃₋₃, α₄₋₃, α₅₋₃, α₆₋₃, α₇₋₃, α₈₋₃, α₉₋₃, α₁₀₋₃, α₁₁₋₃, α₁₂₋₃, α₁₃₋₃, α₁₋₄, α₂₋₄, α₃₋₄, α₄₋₄, α₅₋₄, α₆₋₄, α₇₋₄, α₈₋₄, α₉₋₄, α₁₀₋₄, α₁₁₋₄, α₁₂₋₄, α₁₃₋₄, X₁₋₁, X₂₋₁, X₃₋₁, X₁₋₂, X₂₋₂, X₃₋₂, X₁₋₃, X₂₋₃, X₃₋₃, X₁₋₄, X₂₋₄, X₃₋₄, X₁₋₅, X₂₋₅, X₃₋₅, X₁₋₆, X₂₋₆, X₃₋₆, X₁₋₇, X₂₋₇, X₃₋₇, X₁₋₈, X₂₋₈, X₃₋₈, X₁₋₉, X₂₋₉, X₃₋₉, X₁₋₁₀, X₂₋₁₀, X₃₋₁₀, X₁₋₁₁, X₂₋₁₁, X₃₋₁₁, X₁₋₁₂, X₂₋₁₂, X₃₋₁₂, X₁₋₁₃, X₂₋₁₃, X₃₋₁₃
 104 parameters: l₁₋₁₋₁, l₂₋₁₋₁, l₁₋₂₋₁, l₂₋₂₋₁, l₁₋₃₋₁, l₂₋₃₋₁, l₁₋₄₋₁, l₂₋₄₋₁, l₁₋₅₋₁, l₂₋₅₋₁, l₁₋₆₋₁, l₂₋₆₋₁, l₁₋₇₋₁, l₂₋₇₋₁, l₁₋₈₋₁, l₂₋₈₋₁, l₁₋₉₋₁, l₂₋₉₋₁, l₁₋₁₀₋₁, l₂₋₁₀₋₁, l₁₋₁₁₋₁, l₂₋₁₁₋₁, l₁₋₁₂₋₁, l₂₋₁₂₋₁, l₁₋₁₃₋₁, l₂₋₁₃₋₁, l₁₋₁₋₂, l₂₋₁₋₂, l₁₋₂₋₂, l₂₋₂₋₂, l₁₋₃₋₂, l₂₋₃₋₂, l₁₋₄₋₂, l₂₋₄₋₂, l₁₋₅₋₂, l₂₋₅₋₂, l₁₋₆₋₂, l₂₋₆₋₂, l₁₋₇₋₂, l₂₋₇₋₂, l₁₋₈₋₂, l₂₋₈₋₂, l₁₋₉₋₂, l₂₋₉₋₂, l₁₋₁₀₋₂, l₂₋₁₀₋₂, l₁₋₁₁₋₂, l₂₋₁₁₋₂, l₁₋₁₂₋₂, l₂₋₁₂₋₂, l₁₋₁₃₋₂, l₂₋₁₃₋₂, l₁₋₁₋₃, l₂₋₁₋₃, l₁₋₂₋₃, l₂₋₂₋₃, l₁₋₃₋₃, l₂₋₃₋₃, l₁₋₄₋₃, l₂₋₄₋₃, l₁₋₅₋₃, l₂₋₅₋₃, l₁₋₆₋₃, l₂₋₆₋₃, l₁₋₇₋₃, l₂₋₇₋₃, l₁₋₈₋₃, l₂₋₈₋₃, l₁₋₉₋₃, l₂₋₉₋₃, l₁₋₁₀₋₃, l₂₋₁₀₋₃, l₁₋₁₁₋₃, l₂₋₁₁₋₃, l₁₋₁₂₋₃, l₂₋₁₂₋₃, l₁₋₁₃₋₃, l₂₋₁₃₋₃, l₁₋₁₋₄, l₂₋₁₋₄, l₁₋₂₋₄, l₂₋₂₋₄, l₁₋₃₋₄, l₂₋₃₋₄, l₁₋₄₋₄, l₂₋₄₋₄, l₁₋₅₋₄, l₂₋₅₋₄, l₁₋₆₋₄, l₂₋₆₋₄, l₁₋₇₋₄, l₂₋₇₋₄, l₁₋₈₋₄, l₂₋₈₋₄, l₁₋₉₋₄, l₂₋₉₋₄, l₁₋₁₀₋₄, l₂₋₁₀₋₄, l₁₋₁₁₋₄, l₂₋₁₁₋₄, l₁₋₁₂₋₄, l₂₋₁₂₋₄, l₁₋₁₃₋₄, l₂₋₁₃₋₄

As we can see, it containts the rotation cayley coordinates cᵢ₋ⱼ, translation coordinates tᵢ₋ⱼ, depths αᵢ₋ⱼ of 3D points, coordinates Xᵢ₋ⱼ of 3D points, and finally the coordinates xᵢ₋ⱼ₋ₖ of the projections of 3D points into the cameras.

Fixing the world

TBA...

Elimination of variables

TBA...

Decomposition

TBA...