图像分割中限定4顶点的凸包求解需求及Matlab代码咨询

Absolutely, this is totally achievable! I’ve dealt with similar noisy contour issues in image segmentation before, so I know exactly what you’re up against. MATLAB’s standard convhull holds onto every convex vertex from your ragged quadrilateral edges, which is why you’re getting more than 4 points. What you need is to compute the tightest possible convex quadrilateral (4 vertices) that encloses all your original points—usually the one with the smallest area while keeping every point inside.

How to Approach This #

Let’s break it down into straightforward steps:

• Grab all boundary points from your irregular quadrilateral (the noisy ones from your image processing pipeline).
• Compute the initial convex hull to get the outermost convex boundary (this will likely have more than 4 vertices).
• Simplify that convex hull down to exactly 4 vertices, making sure the result stays convex and still encloses all your original points. Alternatively, you can directly calculate the smallest-area convex quadrilateral that fits your point set.

Sample Code Walkthrough #

I’ll use simulated noisy quadrilateral data (swap this out with your actual contour points):

Step 1: Simulate Your Irregular Quadrilateral #

% Create a core quadrilateral and add noisy edge points to mimic your data
theta = linspace(0, 2*pi, 4);
core_vertices = [cos(theta); sin(theta)]'; % Basic quadrilateral shape
noisy_points = [];

% Add noisy points along each edge
for i = 1:4
    next_idx = mod(i,4)+1;
    edge_points = linspace(core_vertices(i,:), core_vertices(next_idx,:), 20);
    noisy_edge = edge_points + 0.05*randn(size(edge_points)); % Add small noise
    noisy_points = [noisy_points; noisy_edge];
end

Step 2: Get the Initial Convex Hull #

hull_indices = convhull(noisy_points);
convex_hull_points = noisy_points(hull_indices(1:end-1), :); % Remove duplicate last point
fprintf('Initial convex hull has %d vertices\n', size(convex_hull_points,1));

Step 3: Simplify to 4 Vertices #

We’ll use MATLAB’s reducepoly function to trim the convex hull down to exactly 4 vertices. It works by keeping the most critical points that define the shape:

% Simplify the convex hull to 4 vertices
simplified_quad = reducepoly(convex_hull_points, 4);

% Double-check convexity (just to be safe)
is_convex = ispolycw(simplified_quad); % Returns true for convex clockwise polygons
if ~is_convex
    % If needed, reorder points to fix convexity (adjust indices as needed)
    simplified_quad = simplified_quad([1 3 2 4], :);
end

Step 4: Visualize the Results #

figure;
hold on;
scatter(noisy_points(:,1), noisy_points(:,2), 10, 'gray', 'filled', 'DisplayName', 'Noisy Edge Points');
plot(convex_hull_points(:,1), convex_hull_points(:,2), 'b-', 'LineWidth', 1.5, 'DisplayName', 'Original Convex Hull');
plot([simplified_quad(:,1); simplified_quad(1,1)], [simplified_quad(:,2); simplified_quad(1,2)], 'r-', 'LineWidth', 2, 'DisplayName', '4-Vertex Optimal Convex Hull');
axis equal;
legend;
title('4-Vertex Convex Hull for Irregular Quadrilateral');
hold off;

Alternative: Get the Smallest-Area Convex Quadrilateral #

If you want to guarantee the minimal possible area (instead of just simplifying the hull), use an optimization approach. Here’s a quick implementation:

% Objective function: minimize quadrilateral area, penalize if points are outside
function area = quad_area(vertices, all_points)
    quad = reshape(vertices, 4, 2);
    % Check if all points are inside the quadrilateral
    all_inside = inpolygon(all_points(:,1), all_points(:,2), quad(:,1), quad(:,2));
    if ~all(all_inside)
        area = 1e9; % Big penalty for invalid shapes
        return;
    end
    % Calculate area of the quadrilateral
    area = polyarea(quad(:,1), quad(:,2));
end

% Convexity constraint: ensure all interior angles are <= 180 degrees
function [c, ceq] = convex_constraint(vertices)
    quad = reshape(vertices, 4, 2);
    angles = [];
    for i = 1:4
        prev_idx = mod(i-2,4)+1;
        curr_idx = i;
        next_idx = mod(i,4)+1;
        vec1 = quad(curr_idx,:) - quad(prev_idx,:);
        vec2 = quad(next_idx,:) - quad(curr_idx,:);
        angle = atan2d(vec2(2), vec2(1)) - atan2d(vec1(2), vec1(1));
        angle = mod(angle, 360);
        if angle > 180
            angle = 360 - angle;
        end
        angles = [angles, angle];
    end
    c = 180 - angles; % All angles must be <= 180 (convex)
    ceq = [];
end

% Run the optimization
start_guess = convex_hull_points(randperm(size(convex_hull_points,1),4),:);
start_guess = start_guess(:); % Flatten for optimization

options = optimoptions('fmincon', 'Display', 'off');
optimal_vertices = fmincon(@(v) quad_area(v, noisy_points), start_guess, [], [], [], [], [], [], @convex_constraint);

% Reshape back to 4x2 points
optimal_quad = reshape(optimal_vertices, 4, 2);

Quick Tips #

• Tolerance Tweaks: If reducepoly gives a shape that’s too loose, you can adjust its tolerance parameter or write a custom simplifier that prioritizes keeping the most extreme points (like max/min x/y).
• Convexity Checks: Always verify your final quadrilateral is convex—ispolycw is a quick way to do this.
• Speed: The simplification method is faster for large point sets, while the optimization method gives the tightest possible shape. Pick whichever fits your workflow.

内容的提问来源于stack exchange，提问作者user84310