Merge pull request 'Melfely/working' (#1) from melfely/working into master

Reviewed-on: #1

presentation/main.typ

@@ -3,8 +3,8 @@
 #show: xwysyy-pre.with(
   theme: "midnight",
   config-info(
-    title: [Project 1],
+    title: [Project 1: Linear Camera Calibration],
-    subtitle: [Computer Vision],
+    subtitle: [ENGR 4350 - 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<i32, ng::Dyn, ng::Const<3>>,
+    image_points: ng::MatrixView<i32, ng::Dyn, ng::Const<2>>,
+) -> ng::OMatrix<i32, ng::Dyn, ng::Const<12>> {
+    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<f64>) 
+  -> (ng::Matrix3<f64>, ng::Matrix3<f64>, ng::Vector3<f64>) {
+    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_
+      ]
+    ]
+  ],
 )