基于粒子群算法（PSO）的灰度圖像閾值分割及多適應度函數實現

1. 算法框架設計

%% 核心流程
1. 圖像預處理 → 2. PSO參數初始化 → 3. 適應度函數計算 → 4. 粒子群迭代 → 5. 閾值輸出

2. PSO參數設置

% 粒子群參數
n_particles = 30;    % 粒子數量
max_iter = 100;      % 最大迭代次數
w = 0.7;             % 慣性權重
c1 = 1.5; c2 = 1.5;  % 學習因子
dim = 2;             % 搜索維度（二閾值時為2）
lb = [0, 0];         % 下界
ub = [255, 255];     % 上界

3. 多適應度函數實現

3.1 類間方差（Otsu準則）

function fitness = otsu_fitness(thresholds, hist)
    T1 = thresholds(1); T2 = thresholds(2);
    w0 = sum(hist(1:T1)); w1 = sum(hist(T1+1:T2)); w2 = sum(hist(T2+1:end));
    mu0 = sum((1:T1)' .* hist(1:T1)) / w0;
    mu1 = sum((T1+1:T2)' .* hist(T1+1:T2)) / w1;
    mu2 = sum((T2+1:end)' .* hist(T2+1:end)) / w2;
    fitness = w0*w1*(mu0-mu1)^2 + w1*w2*(mu1-mu2)^2;
end

3.2 最大熵準則

function fitness = entropy_fitness(thresholds, hist)
    T1 = thresholds(1); T2 = thresholds(2);
    H0 = -sum((hist(1:T1)./sum(hist(1:T1))).*log2(hist(1:T1)./sum(hist(1:T1))));
    H1 = -sum((hist(T1+1:T2)./sum(hist(T1+1:T2))).*log2(hist(T1+1:T2)./sum(hist(T1+1:T2))));
    H2 = -sum((hist(T2+1:end)./sum(hist(T2+1:end))).*log2(hist(T2+1:end)./sum(hist(T2+1:end))));
    fitness = H0 + H1 + H2;
end

3.3 區域均勻性

function fitness = region_uniformity(img, thresholds)
    T1 = thresholds(1); T2 = thresholds(2);
    mask1 = img < T1; mask2 = img >= T1 & img < T2; mask3 = img >= T2;
    uni1 = std2(img(mask1)); uni2 = std2(img(mask2)); uni3 = std2(img(mask3));
    fitness = 1 / (uni1 + uni2 + uni3 + eps);
end

3.4 梯度信息融合

function fitness = gradient_fitness(img, thresholds)
    [Gx, Gy] = imgradientxy(img);
    grad_mag = sqrt(Gx.^2 + Gy.^2);
    T1 = thresholds(1); T2 = thresholds(2);
    mask1 = grad_mag < T1; mask2 = grad_mag >= T1 & grad_mag < T2; 
    mask3 = grad_mag >= T2;
    fitness = sum(mask1(:)) + 0.5*sum(mask2(:)) + 2*sum(mask3(:));
end

4. PSO主循環實現

%% 初始化粒子群
particles = lb + (ub-lb) .* rand(n_particles, dim);
velocities = 0.1*(ub-lb) .* (2*rand(n_particles, dim) - 1);
pbest = particles;    % 個體最優
gbest = particles(1,:); % 全局最優

%% 適應度計算
fitness = arrayfun(@(i) otsu_fitness(particles(i,:), imhist(img)), 1:n_particles);

%% 迭代優化
for iter = 1:max_iter
    % 更新速度
    r1 = rand(n_particles, dim); r2 = rand(n_particles, dim);
    velocities = w*velocities + c1*r1.*(pbest - particles) + c2*r2.*(gbest - particles);
    velocities = min(max(velocities, -abs(ub-lb)), abs(ub-lb)); % 速度限制
    
    % 更新位置
    particles = particles + velocities;
    particles = min(max(particles, lb), ub); % 邊界處理
    
    % 計算新適應度
    new_fitness = arrayfun(@(i) otsu_fitness(particles(i,:), imhist(img)), 1:n_particles);
    
    % 更新個體最優
    update_idx = new_fitness < fitness;
    pbest(update_idx,:) = particles(update_idx,:);
    fitness(update_idx) = new_fitness(update_idx);
    
    % 更新全局最優
    [min_fit, min_idx] = min(fitness);
    if min_fit < otsu_fitness(gbest, imhist(img))
        gbest = particles(min_idx,:);
    end
    
    % 混沌擾動（防止早熟）
    if mod(iter,10) == 0
        particles = tent_map(particles);
    end
end

5. 多閾值擴展方法

5.1 多維PSO優化（三閾值示例）

dim = 3; % 三閾值
lb = [0, 0, 0]; ub = [255, 255, 255];
% 適應度函數改為多類間方差計算

5.2 協作學習策略

% 將高維問題分解為多個子問題
sub_swarm1 = particles(:,1:2); % 前兩個閾值
sub_swarm2 = particles(:,3:end); % 第三個閾值
% 各子群獨立優化后合并結果

參考代碼 將基本粒子群用于閾值灰度圖像分割，同時給出多種適應度函數 www.youwenfan.com/contentcnk/66080.html

6. 優化策略

1. 混沌初始化：使用Logistic映射生成初始粒子群，提升全局搜索能力

   function x = logistic_map(n, r=4)
       x = zeros(n,1);
       x(1) = rand();
       for i=2:n
           x(i) = r*x(i-1)*(1-x(i-1));
       end
   end
   

2. 動態參數調整：根據迭代次數自適應調整慣性權重

   w = 0.9 - 0.5*(iter/max_iter); % 線性遞減
   

3. GPU加速：利用CUDA并行計算適應度

   gpu_img = gpuArray(img);
   fitness = arrayfun(@(i) otsu_fitness(particles(i,:), imhist(gpu_img)), 1:n_particles);