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Computer Vision], + author: "Zander Johnson", + date: datetime.today(), + institution: "University of Central Arkansas", + ), +) + +#title-slide() + +#outline-slide() + += Project Objective + +== Project Objective & Test Dataset + +#textbox( + [*Project Objectives* + + - 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.], + + [*Test Data & Development Tools* + + - *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: [ + #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]. 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+[package] +name = "project" +version = "0.1.0" +edition = "2024" + +[dependencies] +anyhow = "1.0.104" +image = "0.25.10" +minifb = "0.28.0" +nalgebra = "0.35.0" diff --git a/project/src/main.rs b/project/src/main.rs new file mode 100644 index 0000000..d4b0b64 --- /dev/null +++ b/project/src/main.rs @@ -0,0 +1,482 @@ +use anyhow::Result as AnyResult; +use minifb as fb; +use std::ops::Mul; +use std::{fs::File, io::Read}; + +use nalgebra as ng; + +fn get_data(file_path: &str) -> AnyResult> { + let mut file = File::open(file_path)?; + + let mut buf = "".to_string(); + file.read_to_string(&mut buf)?; + + Ok(buf + .split_whitespace() + .filter_map(|f| f.parse::().ok()) + .collect()) +} + +fn create_matrix( + camera_points: ng::MatrixView>, + image_points: ng::MatrixView>, +) -> ng::OMatrix> { + //Verify row counts are equal + if camera_points.nrows() != image_points.nrows() { + panic!("Must have an equal number of rows between the image and camera points"); + } + + //Cache for later use + let rows = camera_points.nrows(); + + let iter = camera_points + .row_iter() //Iterate over each row + .zip(image_points.row_iter()) //Zip up the rows of image points + .flat_map(|(camera, image)| { + //Extract x,y,z from cords + let x = camera[0]; + let y = camera[1]; + let z = camera[2]; + + //Extract uv from image + let u = image[0]; + let v = image[1]; + + //Create a 12 long vector which is the first row + let mut vec = vec![x, y, z, 1, 0, 0, 0, 0, -u * x, -u * y, -u * z, -u]; + //Append another 12 elements to create another row + vec.append(&mut vec![ + 0, + 0, + 0, + 0, + x, + y, + z, + 1, + -v * x, + -v * y, + -v * z, + -v, + ]); + //return the vec which can be converted into an iterator + vec + }); + ng::Matrix::from_row_iterator_generic(ng::Dyn(2 * rows), ng::U12, iter) +} + +/// Returns (K,R,t) in that order +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); + + // Reversal matrix J + let j = ng::Matrix3::new(0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0); + + // RQ via QR on (J * A)^T + let ja_t = (j * a).transpose(); + let qr = ja_t.qr(); + let q = qr.q(); + let r_qr = qr.r(); + + let mut k = j * r_qr.transpose() * j; + let mut r = j * q.transpose(); + + // Fix negative diagonal elements in K + 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; + } + } + } + + // Ensure proper rotation determinant + if r.determinant() < 0.0 { + r *= -1.0; + k *= -1.0; + } + + // Extract translation vector t = K_raw^-1 * b + let t = k.lu().solve(&b).unwrap(); + + // Normalize K so K[2,2] == 1.0 + let scale = k[(2, 2)]; + k /= scale; + + (k, r, t) +} + +fn init() -> AnyResult> { + let observe_data = get_data("../observe.dat")?; + let observe_mat = ng::MatrixXx2::from_row_iterator_generic( + ng::Dyn(observe_data.len() / 2), + ng::Const::<2>, + observe_data, + ); + + println!("Image:\n{:}", observe_mat); + + let model_data = get_data("../model.dat")?; + let model_mat = ng::MatrixXx3::from_row_iterator_generic( + ng::Dyn(model_data.len() / 3), + ng::Const::<3>, + model_data, + ); + + println!("Camera:\n{:}", model_mat); + + let q = create_matrix(model_mat.as_view(), observe_mat.as_view()); + + let q = ng::OMatrix::from_iterator_generic( + ng::Dyn(q.nrows()), + ng::U12, + q.iter().map(|num| *num as f64), + ); + + println!("Q:\n{:}", q); + + let v_t = ng::SVD::new(q, false, true).v_t.expect("Expected"); + + println!("V:\n{:}", v_t); + + let m_flat = v_t.row(v_t.nrows() - 1); + + let m = ng::OMatrix::from_row_iterator_generic(ng::U3, ng::U4, m_flat.iter().copied()); + + println!("M:\n{:}", m); + + let model_mat = ng::OMatrix::from_iterator_generic( + ng::U4, + ng::Dyn(model_mat.nrows()), + model_mat + .row_iter() + .flat_map(|row| vec![row[0], row[1], row[2], 1]), + ); + + println!("P:\n{:}", model_mat); + + let predicted_mat = ng::OMatrix::from_iterator_generic( + ng::U2, + ng::Dyn(model_mat.ncols()), + model_mat.column_iter().flat_map(|column| { + let new_column = &m.mul(&column.map(|x| x as f64)); + let x = new_column[0]; + let y = new_column[1]; + let z = new_column[2]; + + vec![(x / z) as i32, (y / z) as i32] + }), + ); + + println!("Predict:\n{:}", predicted_mat); + + let (k, r, t) = decompose_projection_matrix(&m); + + println!("K:\n{:}\nR:\n{:}\nt:\n{:}", k, r, t); + + let alpha = k[0]; + let theta = 1.0_f64.atan2(k[3] / -alpha); + let beta = k[4] * theta.sin(); + let u_0 = k[6]; + let v_0 = k[7]; + + println!( + "alpha: {:}, theta: {:}, beta: {:}, u_0: {:}, v_0: {:}", + alpha, theta, beta, u_0, v_0 + ); + + Ok(m) +} + +//Returns (u,v) +fn convert_3d_to_2d(m: &ng::Matrix3x4, point: &Point3d) -> Point2d { + let dim_3 = ng::Matrix4x1::new(point.x, point.y, point.z, 1.0); + + let dim_2 = m * dim_3; + + Point2d::new((dim_2[0] / dim_2[2]) as i32, (dim_2[1] / dim_2[2]) as i32) +} + +const WINDOW_HEIGHT: usize = 480; +const WINDOW_WIDTH: usize = 640; + +fn main() -> AnyResult<()> { + //Handles all of the startup init code! Instead of keeping it in the way of the actual image drawing. + let m = init()?; + let mut window = fb::Window::new( + "Display Buffer", + WINDOW_WIDTH, + WINDOW_HEIGHT, + fb::WindowOptions::default(), + )?; + + let mut points_3d = Vec::new(); + + for i in 0..(10 * 10) { + points_3d.push(Point3d::new((i / 10) as f64, (i % 10) as f64, 0.0)); + } + + for i in 0..(10 * 10) { + points_3d.push(Point3d::new((i % 10) as f64, 10.0, (i / 10) as f64)); + } + + for i in 0..(10 * 10) { + points_3d.push(Point3d::new(10.0, (i % 10) as f64, (i / 10) as f64)); + } + + let points = points_3d + .into_iter() + .map(|point| convert_3d_to_2d(&m, &point)) + .collect::>(); + + let mut buffer = vec![0 as u32; WINDOW_WIDTH * WINDOW_HEIGHT]; + + update_buffer_with_pixels(&mut buffer, points); + + //Clone this, so we can clone this back into buffer later, for each frame. So we can keep the BG pixels. + let background_buffer = buffer.clone(); + + window.update_with_buffer(&buffer, WINDOW_WIDTH, WINDOW_HEIGHT)?; + window.set_target_fps(30); + + let trb = Point3d::new(1.0, 1.0, 1.0); + let trf = Point3d::new(0.0, 1.0, 1.0); + let tlb = Point3d::new(1.0, 1.0, 0.0); + let tlf = Point3d::new(0.0, 1.0, 0.0); + let blf = Point3d::new(0.0, 0.0, 0.0); + let blb = Point3d::new(1.0, 0.0, 0.0); + let brf = Point3d::new(0.0, 0.0, 1.0); + let brb = Point3d::new(1.0, 0.0, 1.0); + + let mut cube = Cube::new(trb, trf, tlb, tlf, blf, blb, brf, brb); + + let mut gif_buffer: Vec> = Vec::new(); + + let mut x_trans = 0.0; + let mut y_trans = 0.0; + let mut z_trans = 0.0; + + const X_TRANS_RATE: f64 = 1.0 / 30.0; + const Y_TRANS_RATE: f64 = 0.5 / 30.0; + const Z_TRANS_RATE: f64 = 2.0 / 30.0; + + const MAX_TRANS: f64 = 6.0; + + while window.is_open() { + //Reload the background for each frame + let mut buffer = background_buffer.clone(); + + cube.translate(Point3d::new(X_TRANS_RATE, Y_TRANS_RATE, Z_TRANS_RATE)); + + update_buffer_with_pixels(&mut buffer, cube.render(&m)); + + x_trans += X_TRANS_RATE; + y_trans += Y_TRANS_RATE; + z_trans += Z_TRANS_RATE; + + if x_trans > MAX_TRANS { + cube.translate(Point3d::new(-x_trans, 0.0, 0.0)); + x_trans = 0.0; + } + if y_trans > MAX_TRANS { + cube.translate(Point3d::new(0.0, -y_trans, 0.0)); + y_trans = 0.0; + } + if z_trans > MAX_TRANS { + cube.translate(Point3d::new(0.0, 0.0, -z_trans)); + z_trans = 0.0; + } + + //Required to close, and update frame; + if window.is_key_down(fb::Key::Q) { + break; + } + window.update_with_buffer(&buffer, WINDOW_WIDTH, WINDOW_HEIGHT)?; + gif_buffer.push( + buffer + .iter() + .flat_map(|pixel| pixel.to_be_bytes()) + .collect(), + ); + } + + export_gif(WINDOW_WIDTH, WINDOW_HEIGHT, 30, gif_buffer)?; + + Ok(()) +} + +///A Cube defined by 8 3d points. +#[derive(Debug)] +struct Cube { + pub trb: Point3d, + pub trf: Point3d, + pub tlb: Point3d, + pub tlf: Point3d, + + pub blf: Point3d, + pub blb: Point3d, + pub brf: Point3d, + pub brb: Point3d, +} + +impl Cube { + pub fn new( + trb: Point3d, + trf: Point3d, + tlb: Point3d, + tlf: Point3d, + blf: Point3d, + blb: Point3d, + brf: Point3d, + brb: Point3d, + ) -> Self { + Cube { + trb, + trf, + tlb, + tlf, + blf, + blb, + brf, + brb, + } + } + + pub fn render(&self, m: &ng::Matrix3x4) -> Vec { + let mut points = Vec::new(); + + let trb = self.trb.project(m); + let trf = self.trf.project(m); + let tlb = self.tlb.project(m); + let tlf = self.tlf.project(m); + + let blf = self.blf.project(m); + let blb = self.blb.project(m); + let brf = self.brf.project(m); + let brb = self.brb.project(m); + + points.append(&mut generate_line_between_verticies(&trf, &trb)); // Line 1 + points.append(&mut generate_line_between_verticies(&trb, &tlb)); // Line 2 + points.append(&mut generate_line_between_verticies(&tlb, &tlf)); // Line 3 + points.append(&mut generate_line_between_verticies(&tlf, &trf)); // Line 4 + points.append(&mut generate_line_between_verticies(&blf, &brf)); // Line 5 + points.append(&mut generate_line_between_verticies(&brf, &brb)); // Line 6 + points.append(&mut generate_line_between_verticies(&brb, &blb)); // Line 7 + points.append(&mut generate_line_between_verticies(&blb, &blf)); // Line 8 + points.append(&mut generate_line_between_verticies(&blf, &tlf)); // Line 9 + points.append(&mut generate_line_between_verticies(&blb, &tlb)); // Line 10 + points.append(&mut generate_line_between_verticies(&brb, &trb)); // Line 11 + points.append(&mut generate_line_between_verticies(&brf, &trf)); // Line 12 + + points + } + + ///translates the cube by Point amount + pub fn translate(&mut self, translate_by: Point3d) { + self.trb.translate(&translate_by); + self.trf.translate(&translate_by); + self.tlb.translate(&translate_by); + self.tlf.translate(&translate_by); + + self.blf.translate(&translate_by); + self.blb.translate(&translate_by); + self.brf.translate(&translate_by); + self.brb.translate(&translate_by); + } +} + +#[derive(Debug)] +struct Point3d { + pub x: f64, + pub y: f64, + pub z: f64, +} + +impl Point3d { + pub fn new(x: f64, y: f64, z: f64) -> Self { + Point3d { x, y, z } + } + + pub fn project(&self, m: &ng::Matrix3x4) -> Point2d { + convert_3d_to_2d(m, self) + } + + pub fn translate(&mut self, translate_by: &Point3d) { + self.x += translate_by.x; + self.y += translate_by.y; + self.z += translate_by.z; + } +} + +#[derive(Debug)] +struct Point2d { + pub x: i32, + pub y: i32, +} + +impl Point2d { + pub fn new(x: i32, y: i32) -> Self { + Self { x, y } + } +} + +fn update_buffer_with_pixels(buf: &mut Vec, points: Vec) { + points.into_iter().for_each(|point| { + let index = (point.y as usize) * WINDOW_WIDTH + (point.x as usize); + buf[index] = u32::MAX + }); +} + +fn generate_line_between_verticies(point_1: &Point2d, point_2: &Point2d) -> Vec { + let mut points = Vec::new(); + + let x_diff = point_2.x - point_1.x; + let y_diff = point_2.y - point_1.y; + + let x_change = x_diff as f64 / 100.0; + let y_change = y_diff as f64 / 100.0; + + for i in 0..100 { + points.push(Point2d::new( + point_1.x + (i as f64 * x_change) as i32, + point_1.y + (i as f64 * y_change) as i32, + )); + } + + points +} + +use image::codecs::gif::{GifEncoder, Repeat}; +use image::{Delay, Frame, RgbaImage}; + +fn export_gif( + width: usize, + height: usize, + fps: u32, + frames: Vec>, // Each inner Vec is width * height * 4 bytes (RGBA) +) -> AnyResult<()> { + let file = File::create("animation.gif")?; + let mut encoder = GifEncoder::new(file); + + // Loop forever + encoder.set_repeat(Repeat::Infinite)?; + + for raw_buffer in frames { + // Construct the image from your raw pixel slice + let img = RgbaImage::from_raw(width as u32, height as u32, raw_buffer).unwrap(); + + // Calculate frame timing (e.g., 1000 / 30 = 33.3ms) + let delay = Delay::from_numer_denom_ms(1000, fps); + let frame = Frame::from_parts(img, 0, 0, delay); + + encoder.encode_frame(frame)?; + } + + Ok(()) +} diff --git a/project1.md b/project1.md new file mode 100644 index 0000000..a99b1ea --- /dev/null +++ b/project1.md @@ -0,0 +1,100 @@ +**ENGR4350 Computer Vision** + +**Project 1. Linear Approach to Camera Calibration** + +**Due Data: September 20, 2026** + +**Objective:** To use a linear approach to calibrate a camera and to use +the camera parameters to predict the 2D image-domain location of a set +of 3D points or a 3D moving object. + +Procedure: + +1. Study the lecture note by showing the use of projection matrix that + maps objects from 3-D to 2-D, the definition of intrinsic and + extrinsic parameters of a camera, and the linear approach to + geometric camera calibration. +2. Given an image *test_image.bmp* captured by a camera which is posed + toward a 3D chess board, we manually selected 27 points whose 2-D + image coordinates are saved in the file *observe.dat*, and whose 3-D + coordinates in a world coordinate are stored in the file + *model.dat*. Please use the linear approach to calibrate the camera + system by computing the projection matrix (M) from which you are + required to compute the intrinsic and extrinsic parameters, + including ï±, u0, v0, α, β, and the rotation matrix (R), and the + shift vector (t). +3. In order to verify the correctness of the camera calibration, please + use the projection matrix to map the three set of 3D points, whose + 3-D coordinates are {x,y,0\|x,y=0,...,10}, {x,10,z\|x,z=0,...,10}, + {10,y,z\|y,z=0,...,10}, respectively, into the 2-D image space. + Discuss your results. +4. Create a video file that shows a 3D object moving in the 3D scene + along a specific pre-defined path. For example, a 3D cube (1x1x1) + can be displayed by nine lines connecting seven vertexes (see the + example below). Each line can be drawn with around 100 samples. (You + can see a video example from Slide 13 of the Lecture 8 handout in + the slideshow mode.) + +**Report Requirements:** + +1. Briefly discuss the basics of geometric camera modeling. +2. Briefly discuss the linear approach to camera calibration. +3. Show the simulation results by figures (Part 3). +4. Include some sample frames of the video (Part 4) in the report. +5. The Python source code should be included as the appendix of the + report with detailed comments. A separate Python file should also be + provided for testing. +6. Zip all files (DOC, AVI, M-file) into one package and upload it to + Blackboard by the due date. + +**Useful Python functions:** + +You need to import the required libraries: + +> import torch\ +> import numpy as np\ +> import matplotlib.pyplot as plt\ +> from PIL import Image\ +> import time + +> import cv2\ +> import os + +- Inner product: torch.dot(t1,t2), Cross product : torch.cross(t1,t2). + +> v0 = (s \*\* 2) \* torch.dot(a2, a3)\ +> \ +> cross_a1_a3 = torch.cross(a1, a3) + +> + +- Singular value decomposition (SVD) : + +> \# \-\-- Solve for the projection matrix M \-\--\ +> \# The solution to Qm=0 is the last column of V from the SVD of Q\ +> \# In PyTorch, torch.linalg.svd returns U, S, Vh (V transpose)\ +> U, S, Vh = torch.linalg.svd(Q)\ +> \ +> \# The solution is the last row of Vh, which corresponds to the +> smallest singular value\ +> m = Vh\[-1, :\]\ +> \ +> \# Reshape the 12x1 vector m into the 3x4 projection matrix M\ +> M = m.reshape(3, 4) + +- Norm of a vector: torch.linalg.norm(vector). + +> torch.linalg.norm(cross_a1_a3) + +> + +- Inverse of a matrix: torch.linalg.inv(M) + +> torch.linalg.inv(M) + +> + +- Reshape of a vector to a matrix: + +> \# Reshape the 12x1 vector m into the 3x4 projection matrix M\ +> M = m.reshape(3, 4) diff --git a/project1.odt b/project1.odt new file mode 100644 index 0000000..23c3cf5 --- /dev/null +++ b/project1.odt @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9ef93fb84eb41134fc4ee8bfbacfa20fd6a94847c35ca5b8b5018df1fb5eb299 +size 41461