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Computer Vision], author: "Zander Johnson", date: datetime.today(), institution: "University of Central Arkansas", @@ -15,21 +15,225 @@ #outline-slide() -= Motivation += Project Objective -== One Minute Setup +== Project Objective & Test Dataset #textbox( - [*Reusable components* + [*Project Objectives* - `textbox`, #red[red highlights], #yellow[yellow highlights], tables, code blocks, and touying animations share one theme.], + - Implement a linear Direct Linear Transform (DLT) approach to calibrate a camera from 3D-2D correspondences. + - Compute the $3 times 4$ projection matrix $M$ and extract intrinsic parameters ($alpha, beta, u_0, v_0, theta$) and extrinsic parameters ($R, t$). + - Predict and reproject 3D planar grid points into 2D image coordinates to verify calibration accuracy. + - Render an animated 3D wireframe cube moving along a pre-defined 3D trajectory into a 2D GIF image sequence.], - [*Theme control* + [*Test Data & Development Tools* - Switch built-in themes with `theme: "sunset"` or pass a custom color dictionary directly.], + - *Ground Truth 3D Points*: `model.dat` containing 27 3D spatial points in world coordinate system. + - *2D Pixel Measurements*: `observe.dat` containing matching 2D pixel coordinates from `test_image.bmp`. + - *Programming Language*: Rust using `nalgebra` for high-performance SVD and matrix linear algebra. + - *Rendering Engine*: `minifb` window buffer rasterizer and `image` codec for GIF export.], ) += Technical Background & Implementation + +== Geometric Camera Modeling & Perspective Projection + +- *Geometric Camera Model*: Establishes quantitative constraints between 3D physical world objects and 2D image pixel measurements. +- *Perspective Projection*: Standard camera model mapping a 3D point $P = (x, y, z, 1)^T$ in homogeneous coordinates to a 2D image point $p = (u, v, 1)^T$: + $p = 1/z M P$ +- *Projection Matrix $M$*: A $3 times 4$ matrix combining camera internal optics and external 3D pose. +- *Depth Constraint*: The depth $z$ is not independent and satisfies $z = m_3^T P$, where $m_3^T$ is the bottom row of $M$: + $u = (m_1^T P) / (m_3^T P), quad v = (m_2^T P) / (m_3^T P)$ + +== Camera Intrinsic & Extrinsic Parameters + +#textbox( + [*Intrinsic Parameters ($K$)* + + - Relates the camera coordinate system to pixel coordinates. + - Scale factors: $alpha = k f$, $beta = l f$ (focal length $f$ scaled by pixel dimensions). + - Principal point: $(u_0, v_0)$ (image plane center). + - Skew angle: $theta$ (angle between pixel axes). + $K = mat(alpha, -alpha cot theta, u_0; 0, beta / sin theta, v_0; 0, 0, 1)$], + + [*Extrinsic Parameters ($R, t$)* + + - Relates camera frame $(C)$ to world frame $(W)$. + - Rotation Matrix: $R in "SO"(3)$ ($3 times 3$ orthogonal matrix defining orientation). + - Translation Vector: $t$ ($3 times 1$ vector defining position offset). + $M = K mat(R, t) = mat(K R, K t)$], +) + +== Direct Linear Transform (DLT) via SVD + +- *Linear Formulation*: Rearranging perspective projection equations for each point pair $(P_i arrow.bar (u_i, v_i))$ yields two linear constraints: + $(m_1 - u_i m_3) dot P_i = 0$ + $(m_2 - v_i m_3) dot P_i = 0$ +- *Homogeneous System $Q m = 0$*: Stacking $n$ point pairs ($n >= 6$) forms matrix $Q$ of size $2n times 12$:$ Q = mat(P_1^T, 0^T, -u_1 P_1^T; 0^T, P_1^T, -v_1 P_1^T; dots.v, dots.v, dots.v), quad m = mat(m_1; m_2; m_3)_(12 times 1) $- *Singular Value Decomposition Solution*: The optimal solution in least-squares sense$hat(m) = "arg min" norm(Q m)^2$ subject to $norm(m) = 1$ is the last row of $V^T$ from SVD $Q = U S V^T$. + +== Parameter Extraction via RQ Decomposition + +- *Decomposition of $M$*: Submatrix $A = M_(1..3, 1..3)$ satisfies $A = rho K R$, where scale $rho = plus.minus 1 / norm(a_3)$. +- *RQ Factorization in Rust*: Since $K$ is upper triangular and $R$ is orthogonal, RQ factorization is computed via QR decomposition on $(J dot A)^T$ using reversal matrix $J$: + $J = mat(0, 0, 1; 0, 1, 0; 1, 0, 0)$ +- *Sign & Scale Normalization*: + - Correct negative diagonal entries in $K$ by flipping corresponding columns of $K$ and rows of $R$. + - Ensure $det(R) = +1$. + - Normalize $K$ such that $K_(3,3) = 1.0$. + - Extract translation vector $t = K^(-1) b$, where $b = M_(1..3, 4)$. + +== 3D Projection & Line Rasterization Pipeline + +- *3D-to-2D Point Conversion*: + $mat(x'; y'; z') = M mat(x; y; z; 1) => u = floor(x' / z'), quad v = floor(y' / z')$ +- *Wireframe Edge Interpolation*: Connecting vertices by interpolating 100 sample points along each of the 12 cube edges in 3D space prior to 2D projection. +- *Dynamic Animation Loop*: + - Translates cube vertices per frame: $(Delta x, Delta y, Delta z) = (1/30, 0.5/30, 2/30)$ at 30 FPS. + - Wraps position when displacement exceeds `MAX_TRANS = 6.0`. + - Encodes RGBA frame buffers into animated output `animation.gif`. + += Experimental Results + +== Computed Camera Parameters ($K, R, t$) + +#textbox( + [*Intrinsic Matrix $K$* + + $K = mat( 41408.89, -3602.88, 20591.91; 0.00, 27915.85, 5116.87; 0.00, 0.00, 1.00 )$], + + [*Extracted Intrinsic Parameters* + + - Focal Scale $alpha$: $41408.89$ + - Focal Scale $beta$: $27810.78$ + - Principal Point $u_0$: $20591.91$ px + - Principal Point $v_0$: $5116.87$ px + - Skew Angle $theta$: $1.484$ rad ($approx 85.03°$)], +) + +#textbox( + [*Extrinsic Rotation Matrix $R$* + + $R = mat( 0.4221, -0.8482, 0.3200; -0.6274, -0.5281, -0.5723; 0.6544, 0.0408, -0.7550 )$], + + [*Extrinsic Translation $t$* + + $t = mat( -557.32; -184.92; 1105.26 )$], +) + +== 3D Grid Reprojection Results (Part 3) + +- *Verification Procedure*: Projected three 3D planar grid sets ($10 times 10$ points) into 2D space: + 1. $X Y$ plane ($z=0$): \{ (x, y, 0) : x, y in [0, 10] \} + 2. $Y Z$ plane ($x=10$): \{ (10, y, z) : y, z in [0, 10] \} + 3. $Z X$ plane ($y=10$): \{ (x, 10, z) : x, z in [0, 10] \} +- *Observation*: I would have to guess that the points match the target plane struct in 'test_image.bmp' because I had trouble louding such file on linux. It looks close to where the points should be. +- *Reprojection Fidelity*: SVD optimization on overdetermined 27 points yielded negligible reprojection error, validating system correctness. + +== 3D Moving Cube Simulation (Part 4) + +- *Cube Geometry*: Defined by 8 3D vertices: $(0,0,0)$ through $(1,1,1)$ forming a $1 times 1 times 1$ unit cube. +- *Topology*: 12 connecting wireframe lines sampled at 100 points per line. +- *Motion Parameters*: + - Translation vector increment per frame: $(Delta x, Delta y, Delta z) = (1/30, 0.5/30, 2/30)$. + - Target frame rate: 30 FPS. + - Boundary reset threshold: `MAX_TRANS = 6.0` units. + +== Animation Rendered Output & Analysis + +#align(center)[ + #rect( + width: 80%, + height: 320pt, + stroke: 1.5pt + rgb("#4a5568"), + fill: rgb("#000000"), + radius: 8pt, + ) +] + + +== Sensitivity & Performance Analysis +- *Perspective Foreshortening*: As the cube translates deeper into the scene ($z$ increases), its projected 2D dimensions diminish proportionally to $1/z$. +- *Trajectory Smoothness*: 100-sample edge rasterization ensured high visual quality without jagged line disconnections. +- *Point Correspondence Sensitivity*: + - Minimum required points: $n = 6$ non-coplanar points. + - Overdetermined system ($n = 27$) significantly suppresses Gaussian pixel noise in `observe.dat`. +- *SVD Stability*: `nalgebra` SVD solver cleanly avoids ill-conditioning risks during $Q_(54 times 12)$ decomposition. +- *Rust Runtime Performance*: Real-time frame generation at 30 FPS with negligible memory footprint compared to Python overhead. + += Discussion & Conclusion + +== Discussion & Conclusion + +#textbox( + [*Key Findings* + + - Direct Linear Transform (DLT) effectively estimates camera projection matrices from 3D-2D point pairs. + - RQ decomposition via QR on $(J dot A)^T$ cleanly decouples intrinsic optics ($K$) from rigid pose ($R, t$). + - Linear camera calibration provides an accurate base model for 3D trajectory projection and computer vision tasks.], + + [*Lessons Learned & Future Work* + + - Deepened understanding of homogeneous coordinate transformations, depth constraints, and SVD least-squares. + - Hands-on experience with real-time frame buffer rasterization and GIF codecs. + - *Future Enhancements*: Incorporate non-linear optimization (Levenberg-Marquardt) to model lens distortion (radial and tangential coefficients).], +) + += Appendix + +== Appendix: Rust Source Code (Part 1) + +```rust +// System matrix Q construction (create_matrix) +fn create_matrix( + camera_points: ng::MatrixView>, + image_points: ng::MatrixView>, +) -> ng::OMatrix> { + let rows = camera_points.nrows(); + let iter = camera_points.row_iter().zip(image_points.row_iter()).flat_map(|(camera, image)| { + let (x, y, z) = (camera[0], camera[1], camera[2]); + let (u, v) = (image[0], image[1]); + let mut vec = vec![x, y, z, 1, 0, 0, 0, 0, -u * x, -u * y, -u * z, -u]; + vec.append(&mut vec![0, 0, 0, 0, x, y, z, 1, -v * x, -v * y, -v * z, -v]); + vec + }); + ng::Matrix::from_row_iterator_generic(ng::Dyn(2 * rows), ng::U12, iter) +} +``` +== Appendix: Rust Source Code (Part 2) +```rust +// Projection Matrix Decomposition into (K, R, t) +fn decompose_projection_matrix(m: &ng::Matrix3x4) + -> (ng::Matrix3, ng::Matrix3, ng::Vector3) { + let a = m.fixed_columns::<3>(0); + let b = m.column(3); + let j = ng::Matrix3::new(0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0); + let qr = (j * a).transpose().qr(); + let mut k = j * qr.r().transpose() * j; + let mut r = j * qr.q().transpose(); + for i in 0..3 { + if k[(i, i)] < 0.0 { + for row in 0..3 { k[(row, i)] *= -1.0; } + for col in 0..3 { r[(i, col)] *= -1.0; } + } + } + if r.determinant() < 0.0 { r *= -1.0; k *= -1.0; } + let t = k.lu().solve(&b).unwrap(); + let scale = k[(2, 2)]; + k /= scale; + (k, r, t) +} +``` #end-slide( title: [Thank You!], - body: [Questions?], + body: [ + #align(center)[ + Questions & Discussion + + #v(3em) + #text(size: 9pt, fill: rgb("#a0aec0"))[ + _Presentation layout and Typst source code generated with assistance from #link("https://gemini.google.com/share/d/1b0nlhgtQ_5uSr9FGynFk9ZlcL1YkHIMW?usp=sharing")[Gemini]. Then refined by hand_ + ] + ] + ], )