{"id":22092,"date":"2026-02-23T20:42:42","date_gmt":"2026-02-23T12:42:42","guid":{"rendered":"https:\/\/www.orczhou.com\/?p=22092"},"modified":"2026-02-24T09:41:16","modified_gmt":"2026-02-24T01:41:16","slug":"forward-diffusion-process","status":"publish","type":"post","link":"https:\/\/www.orczhou.com\/index.php\/2026\/02\/forward-diffusion-process\/","title":{"rendered":"\u6269\u6563\uff08diffusion\uff09\u6a21\u578b\uff1aForward Diffusion Process"},"content":{"rendered":"\n<p>\u6700\u8fd1 <a href=\"https:\/\/www.midjourney.com\/\">midjourney<\/a> \u7528\u7684\u6bd4\u8f83\u591a\uff0c\u5f88\u597d\u5947\uff0c\u8fd9\u80cc\u540e\u7684\u6280\u672f\u5927\u6982\u662f\u600e\u6837\u7684\u3002\u597d\u4e86\uff0c\u90a3\u5c31\u53bb\u505a\u4e00\u4e9b\u4e86\u89e3\u5427\uff08\u8865\u5145\u6ce8\uff1a\u540e\u6765\u6211\u624d\u53d1\u73b0\u8fd9\u4e2a\u5751\u6709\u591a\u5927&#8230;\uff09\u3002<\/p>\n\n\n\n\n\n\n<p>\u9664\u4e86\u5927\u8bed\u8a00\u6a21\u578b\u4e4b\u5916\uff0c\u53e6\u4e00\u4e2a\u975e\u5e38\u6d3b\u8dc3\u7684\u9886\u57df\u5c31\u662f\u56fe\u7247\u5904\u7406\u6280\u672f\uff0c\u4f8b\u5982\u6587\u751f\u56fe\uff08Text-to-Image\uff09\u3001\u591a\u6a21\u6001\u6a21\u578b\u7b49\u3002\u9664\u4e86\u5728\u539f\u6709\u7684 CNN \u6280\u672f\u67b6\u6784\u4e0a\uff0c\u4e5f\u51fa\u73b0\u4e86\u5f88\u591a\u65b0\u7684\u7a81\u7834\u3002\u867d\u7136\u540c\u6837\u4f7f\u7528\u7684\u662f\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\uff0c\u4f46\u56fe\u50cf\u751f\u6210\u6280\u672f\u4e0e\u5927\u8bed\u8a00\u6a21\u578b\u6280\u672f\uff08LLM\uff09\u662f\u975e\u5e38\u4e0d\u540c\u7684\u3002\u5176\u6a21\u578b\u67b6\u6784\u4e0d\u540c \uff0c\u5e95\u5c42\u7684\u6570\u5b66\u3001\u7269\u7406\u539f\u7406\u4e5f\u975e\u5e38\u4e0d\u4e00\u6837\u3002\u5728\u5bf9\u5927\u8bed\u8a00\u6a21\u578b\u6709\u4e00\u4e2a\u6846\u67b6\u6027\u7684\u4e86\u89e3 \u4e4b\u540e\uff0c\u73b0\u5728\u6253\u7b97\u5f00\u4e00\u4e2a\u65b0\u7684\u201c\u5751\u201d\uff0c\u5f53\u7136\uff0c\u4e5f\u5c31\u6ca1\u6709\u6253\u7b97\u722c\u51fa\u6765\uff0c\u53ef\u80fd\u5c31\u201c\u6d45\u5c1d\u8f84\u6b62\u201d\u505a\u4e2a\u6700\u4e3a\u57fa\u7840\u7684\u4e86\u89e3\u3002\u4e00\u822c\u5b66\u4e1c\u897f\uff0c\u90fd\u662f\u4ece \u201cWhy\/How\/What\u201d \u7684\u987a\u5e8f\uff0c\u4f46\u5728\u5c1d\u8bd5\u4e86\u4e00\u4e0b\uff0c\u65e0\u6cd5\u7406\u89e3\u201cWhy\u201d\u540e\uff0c\u4e8e\u662f\u5c31\u6253\u7b97\u7ed5\u9053\uff0c\u770b\u770b How \u548c What \u4e86\u3002\u90a3\u4e48\uff0c\u201cWhat\u201d\u6253\u7b97\u5c31\u4ece\u201cFoward Diffusion Process\u201d\u5f00\u59cb\u5427\u3002<\/p>\n\n\n\n<p>\u6574\u4e2a\u6587\u751f\u56fe\uff08Text-to-Image\uff09\u7684\u67b6\u6784\u662f\u6bd4\u8f83\u590d\u6742\u7684\uff0c\u5176\u4e2d\u8f83\u4e3a\u201c\u7b80\u5355\u201d\u3001\u57fa\u7840 \u7684\u4e00\u6b65\u5373\u4f4d\u201cForward Diffusion Process\u201d\uff0c\u672c\u6587\u4ece\u201c<a href=\"https:\/\/arxiv.org\/pdf\/2006.11239\">Denoising Diffusion Probabilistic Models<\/a>\u201d\u8bba\u6587\u4e2d\u4e3a\u4f8b\uff0c\u8bf4\u660e\u8fd9\u4e2a\u8fc7\u7a0b\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-1\"><\/span>1. \u6982\u8ff0\uff1a\u201cFDP\u201d \u662f\u4e00\u4e2a\u52a0\u968f\u673a\u566a\u58f0\u7684\u8fc7\u7a0b<\/h3>\n\n\n\n<p>\u5f88\u591a\u5730\u65b9\u90fd\u4f1a\u8bf4\uff1a\u201cForward Diffusion Process\u201d \u5c31\u662f\u4e00\u4e2a\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u7684\u8fc7\u7a0b\u3002\u8fd9\u8bdd\u5bf9\u3001\u4e5f\u4e0d\u5bf9\u3002\u9996\u5148\uff0c\u201cFDP\u201d \u786e\u5b9e\u662f\u4e00\u4e2a\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u7684\u8fc7\u7a0b\uff0c\u4f46\u662f\u8fd9\u4e2a\u968f\u673a\u566a\u58f0\u6dfb\u52a0\u5f97\u975e\u5e38\u6709\u201c\u8bb2\u7a76\u201d\u3002\u6211\u4eec\u5c31\u4ece\u201c\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u201d\u548c\u201c\u8bb2\u7a76\u201d\u4e24\u4e2a\u89d2\u5ea6\u53bb\u4ecb\u7ecd\u3002<\/p>\n\n\n\n<p>\u201cForward Diffusion Process\u201d \u8fc7\u7a0b\u7684\u6570\u636e\u662f\u7528\u4e8e\u8bad\u7ec3\u795e\u7ecf\u7f51\u7edc\uff08\u901a\u5e38\u662f\u4e00\u4e2aU-Net\u67b6\u6784\uff09\u7684\u53c2\u6570\u7684\uff0c\u4ece\u800c\u6700\u7ec8\u5b9e\u73b0\u5176\u9006\u8fc7\u7a0b\uff08\u5373\u201cReverse Diffusion Process\u201d\uff09\uff0c\u800c\u751f\u6210\u56fe\u7247\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-2\"><\/span>1.1 \u6dfb\u52a0\u968f\u673a\u566a\u58f0<\/h4>\n\n\n\n<p>\u201cFDP\u201d \u505a\u5982\u4e0b\u7684\u4e8b\u60c5\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5148\u8bfb\u53d6\u4e00\u5f20\u56fe\u7247\uff08\u8fd9\u91cc\u4f7f\u7528\u7684\u4e00\u5f20\u535a\u5ba2\u56fe\u7247\uff09<\/li>\n\n\n\n<li>\u5c06\u56fe\u7247\u7684\u6bcf\u4e2a\u50cf\u7d20\u70b9\u53d6\u503c\uff0c\u968f\u673a\u201c\u8fed\u4ee3\u52a0\u4e0a\u201d\u4e00\u4e2a\u968f\u673a\u503c \\(N(0,1) \\)<\/li>\n<\/ul>\n\n\n\n<p>\u4f8b\u5982\uff0c\u6211\u4eec\u4f7f\u7528\u8bba\u6587 DDPM \u4e2d\u7684\u8bbe\u5b9a\u5bf9\u5982\u4e0b\u56fe\u7247\u6dfb\u52a0\u566a\u58f0\uff0c\u5c31\u53ef\u4ee5\u89c2\u5bdf\u5230\u5982\u4e0b\u8fc7\u7a0b\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"180\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-1024x180.png\" alt=\"\" class=\"wp-image-22090\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-1024x180.png 1024w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-300x53.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-768x135.png 768w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11.png 1182w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\u66f4\u4e3a\u8be6\u7ec6\u7684\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list is-style-no-disc\">\n<li>1. \u5148\u8bfb\u53d6\u4e00\u5f20\u56fe\u7247\uff0c\u5e76\u5c06\u5176 RGB \u901a\u9053\u7684\u6570\u636e\u8bfb\u53d6\u51fa\u6765<\/li>\n\n\n\n<li>2. \u5c06\u50cf\u7d20\u503c\u4ece [0, 255] \u7684\u6574\u6570\u7f29\u653e\u5230 [0.0, 1.0] \u7684\u6d6e\u70b9\u6570<\/li>\n\n\n\n<li>3. \u6bcf\u4e2a\u901a\u9053\u6bcf\u4e2a\u70b9\u968f\u673a\u201c\u8fed\u4ee3\u52a0\u4e0a\u201d\u4e00\u4e2a\u968f\u673a\u503c\uff0c\u6309\u7167 \\(N(0,1) \\) \u5206\u5e03\u751f\u6210\u8be5\u968f\u673a\u503c<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u201c\u7ebf\u6027\u201d\u6dfb\u52a0\u566a\u58f0<\/h3>\n\n\n\n<p>\u5728 <a href=\"https:\/\/arxiv.org\/pdf\/2006.11239\">Denoising Diffusion Probabilistic Models<\/a> \u8bba\u6587\u4e2d\u4f7f\u7528\u4e86\u201c\u7ebf\u6027\u8c03\u5ea6\u201d\u7684\u65b9\u5f0f\u6dfb\u52a0\u566a\u58f0\u3002\u5373\u6dfb\u52a0\u566a\u58f0\u7684\u5f3a\u5ea6\u201c\u7ebf\u6027\u201d\u7684\u9010\u6e10\u589e\u5f3a\uff0c\u8fd9\u91cc\u7684\u201c\u7ebf\u6027\u201d\u662f\u6307\u589e\u52a0\u7684\u566a\u58f0\u7684\u201c\u65b9\u5dee\u201d\u7ebf\u6027\u589e\u52a0\u3002<\/p>\n\n\n\n<p>\u5148\u7528\u66f4\u52a0\u5f62\u5f0f\u5316\u7684\u6570\u5b66\u8bed\u8a00\u63cf\u8ff0\u4e0a\u8ff0\u7684\u566a\u58f0\u6dfb\u52a0\uff0c\u5373\uff1a<\/p>\n\n\n\n<p>$$ x_t = A x_{t-1} + B \\epsilon \\quad \\text{where} \\quad \\epsilon \\sim \\mathcal{N}(0,1) $$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2.1 \u8c03\u6574\u6743\u91cd\u7cfb\u6570<\/h4>\n\n\n\n<p>\u76f4\u89c9\u4e0a\uff0c\u53ef\u4ee5\u8fd9\u6837\u7406\u89e3\uff0c\u5728\u6dfb\u52a0\u566a\u58f0\u7684\u8fc7\u7a0b\u4e2d\uff0c\u521a\u5f00\u59cb\u662f\u6e05\u6670\u56fe\u7247\uff0c\u6240\u4ee5\u566a\u58f0\u6dfb\u52a0\u7684\u8f83\u5c11\uff0c\u800c\u540e\uff0c\u968f\u7740\u56fe\u7247\u53d8\u5f97\u6a21\u7cca\uff0c\u4e5f\u9010\u6b65\u589e\u52a0\u4e86\u6dfb\u52a0\u566a\u58f0\u7684\u5f3a\u5ea6\uff08\u65b9\u5dee\uff09\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4e2d \u201c\u7ebf\u6027\u8c03\u5ea6\u201d \u6a21\u5f0f\u505a\u4e86\u5982\u4e0b\u8bbe\u8ba1\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_t = \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I}) \\tag{1}<br \/>\n$$<\/p>\n\n\n\n<p>\u8fd9\u91cc\u7684 \\(\\beta_t \\) \u662f\u4e00\u4e2a\u968f\u7740\u65f6\u95f4\u5e8f\u5217\u63a8\u8fdb\u9010\u6e10\u589e\u5927\u7684\u503c\uff0c\u4ece\u800c\u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\uff08\u6216\u8005\u8bf4\u8fd9\u4e2a\u9a6c\u5c14\u79d1\u592b\u94fe\u4e2d\uff09\uff0c\u9010\u6b65\u589e\u52a0\u566a\u58f0\u5728\u56fe\u7247\u4e2d\u7684\u5f71\u54cd\u3002\u8bba\u6587\u4e2d\uff0c\\(\\beta_t \\) \u662f\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u7684\u5e8f\u5217\uff0c\u4ece \\(10^{-4} \\)\uff0c\u901a\u8fc71000\u6b65\u8fed\u4ee3\uff0c\u589e\u52a0\u5230\\(0.02 \\)\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2.2 \u8ba1\u7b97\u7684\u201c\u7b80\u5316\u201d<\/h4>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u201c\u516c\u5f0f(1)\u201d\u662f\u4e00\u4e2a\u8fed\u4ee3\u8ba1\u7b97\u7684\u201c\u6570\u5217\u201d\uff08\u6216\u8005\u8bf4\u662f\u201c\u9a6c\u5c14\u79d1\u592b\u94fe\u201d\uff09\uff0c\u5728\u5b9e\u9645\u7684\u8ba1\u7b97\u4e2d\uff0c\u4f1a\u7ecf\u5e38\u4f7f\u7528\u5982\u4e0b\u7684\u201c\u901a\u9879\u516c\u5f0f\u201d\u8ba1\u7b97\u4e0a\u8ff0\u7684\u8fed\u4ee3\u201c\u6570\u5217\u201d\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_t = \\sqrt{\\bar{\\alpha}_t} x_0 + \\sqrt{1-\\bar{\\alpha}_t}\\epsilon   \\quad \\epsilon \\sim \\mathcal{N}(0,1) \\\\<br \/>\n\\text{Where} \\quad \\alpha_t := 1-\\beta_t ,\\, \\bar{\\alpha}_t = \\prod_{s=1}^{t}\\alpha_s \\tag{2}<br \/>\n$$<\/p>\n\n\n\n<p>\u5173\u4e8e\u8be6\u7ec6\u7684\u5982\u4f55\u4ece\u201c\u516c\u5f0f(1)\u201d\u4e25\u683c\u7684\u63a8\u5bfc\u5230\u4e0a\u8ff0\u8868\u8fbe\u5f0f(2)\uff0c\u53c2\u8003\u672c\u6587\u5c0f\u7ed3\u201c4.5 \u201c\u8c03\u5ea6\u516c\u5f0f\u201d\u7684\u63a8\u5bfc\u201d\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u8868\u8fbe\u5f0f\uff0c\u5728\u8bba\u6587\u4e2d\u51fa\u73b0\u7684\u5f62\u5f0f\u5219\u662f\uff1a<\/p>\n\n\n<p>$$<br \/>\nq(x_t|x_0) = \\mathcal{N}(x_t; \\sqrt{\\bar{\\alpha}}x_0,(1-\\bar{\\alpha_t}\\mathbf{I}) ) \\tag{3}<br \/>\n$$<\/p>\n\n\n\n<p>\u8fd9\u91cc\u7684\u201c\u516c\u5f0f(2)\u3001(3)\u201d\u6240\u8868\u8fbe\u7684\u610f\u601d\u662f\u7b49\u4ef7\u7684\u3002\u7b80\u5355\u7684\u8bf4\u660e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list is-style-no-disc\">\n<li>(a) \\(\\mathcal{N}(x; \\mu \\, , \\sigma^2) \\) \u8868\u793a\u6b63\u6001\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf \\(x \\)\uff0c\u5747\u503c\u4e3a\\(\\mu \\) \u65b9\u5dee\u4e3a \\(\\sigma^2 \\)<\/li>\n\n\n\n<li>(b) \u4e0a\u8ff0\u7684\u8868\u8fbe\u5f0f\u4e2d\u7684 \\(I \\) \u8868\u793a\u5355\u4f4d\u77e9\u9635\u3002\u8fd9\u662f\u56e0\u4e3a\u516c\u5f0f\u4e2d\u7684 \\(x_t \\) \u662f\u4e00\u4e2a\u8868\u793a\u6240\u6709\u50cf\u7d20\u503c\u7684\u5411\u91cf\uff08\u4f8b\u5982\uff0c128&#215;128\u7684\u5411\u91cf\uff0c\u5373\u53ef\u80fd\u670916384\u4e2a\u968f\u673a\u53d8\u91cf\uff09\uff0c\\(I \\) \u8868\u793a\u534f\u65b9\u5dee\u77e9\u9635\u662f\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff0c\u5373\u6240\u6709\u968f\u673a\u53d8\u91cf\u90fd\u662f\u5b8c\u5168\u72ec\u7acb\u7684\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-2\"><\/span>3. \u4ee3\u7801\u5b9e\u73b0<\/h3>\n\n\n\n<p>\u5b8c\u6574\u7684\u4ee3\u7801\u53c2\u8003\uff1a<a href=\"https:\/\/colab.research.google.com\/drive\/1kzydAUGoeHPLSg55Y163jP6mKNJaE_vF?usp=sharing\">Forward-Diffusion-Process.ipynb.ipynb<\/a><\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-4\"><\/span>3.1 \u8bfb\u53d6\u5e76\u9884\u5904\u7406\u56fe\u7247<\/h4>\n\n\n\n<p>\u8be5\u51fd\u6570\u8f93\u5165\u539f\u59cb\u56fe\u7247\u3001\u8fed\u4ee3\u6b21\u6570\u3001\u521d\u59cb\u566a\u58f0\uff0c\u5373 (x_0, t,noise) \u3002\u539f\u59cb\u56fe\u7247\u5728\u8bfb\u53d6\u540e\uff0c\u9700\u8981\u505a\u51e0\u4e2a\u5904\u7406\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7edf\u4e00 Resize \u5230 128 x 128 \u50cf\u7d20\u6765\u51fa\u6765<\/li>\n\n\n\n<li>\u6309 RGB \u4e09\u901a\u9053\u8f6c\u6210\u4e00\u4e2a 1 x 3 x 128 x 128 \u7684\u5f20\u91cf\/\u6570\u7ec4<\/li>\n\n\n\n<li>\u50cf\u7d20\u503c\u4ece [0, 255] \u7684\u6574\u6570\u7f29\u653e\u5230 [0.0, 1.0] \u7684\u6d6e\u70b9\u6570<\/li>\n<\/ul>\n\n\n\n<p>\u5bf9\u5e94\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\"># 3. \u52a0\u8f7d\u6d4b\u8bd5\u56fe\u7247\nurl = \"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/12\/IMG_1710-scaled.jpg\"\nimg = Image.open(requests.get(url, stream=True).raw).convert(\"RGB\")\ntransform = transforms.Compose([transforms.Resize((128, 128)), transforms.ToTensor()])\nx_0 = transform(img).unsqueeze(0) # \u53d8\u4e3a (1, 3, 128, 128)<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-5\"><\/span>3.2 \u751f\u6210\u968f\u673a\u566a\u58f0<\/h4>\n\n\n\n<p>\u751f\u6210\u4e00\u4e2a\u968f\u673a\u6309 \\(N(0,1) \\) \u5206\u5e03\u7684\u5bf9\u8c61\uff0c\u4e0e\u4e0a\u8ff0\u56fe\u7247\u76f8\u540c\uff0c\u5373\uff1a1 x 3 x 128 x 128<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">noise = torch.randn_like(x_0) # \u91c7\u6837\u7eaf\u566a\u58f0 epsilon<\/code><\/pre>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-6\"><\/span>3.3 \u539f\u59cb\u56fe\u7247\u53e0\u52a0\u566a\u58f0<\/h4>\n\n\n\n<p>\u566a\u58f0\u7684\u53e0\u52a0\u5e76\u4e0d\u662f\u7b80\u5355\u7684\u76f4\u63a5\u76f8\u52a0\uff08 \\(x_0 +  \\text{noise} \\)  \uff09\uff0c\u800c\u662f\u4e00\u4e2a\u8fed\u4ee3\u5f0f\u7684\uff0c\u5e76\u8003\u8651\u539f\u59cb\u56fe\u7247\u5f71\u54cd\u7684\u65b9\u5f0f\uff08\u7ebf\u6027\u91c7\u6837\u8003\u8651\uff09\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nx_t &#038;= \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I}) \\\\[0.5em]<br \/>\nx_t &#038;= \\sqrt{\\bar{\\alpha}} x_0 + \\sqrt{1-\\bar{\\alpha}}\\epsilon<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. \u516c\u5f0f(1)\u7684\u8bf4\u660e<\/h3>\n\n\n\n<p>\u6211\u4eec\u518d\u6765\u770b\u770b \u201c\u516c\u5f0f(1)\u201d \u7684\u8bbe\u8ba1\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_t = \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I}) \\tag{1}<br \/>\n$$<\/p>\n\n\n\n<p>\u4e3a\u4ec0\u4e48\u4e0d\u7528\u6700\u4e3a\u76f4\u89c2\u3001\u81ea\u7136\u7684 \\(1-\\beta_t \\) \u4e0e \\(\\beta_t \\) \u4f5c\u4e3a\u4e0a\u8ff0\u8868\u8fbe\u5f0f\u4e2d\u7684\u7cfb\u6570\uff0c\u800c\u662f\u4f7f\u7528\u4e86\u4ed6\u4eec\u7684\u5e73\u65b9\u6839\uff1f<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u201c\u8bbe\u8ba1\u201d\u601d\u8def\u7684\u6839\u6e90\u662f\u56e0\u4e3a\uff1a\u9ad8\u65af\u5206\u5e03\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\u540e\uff0c\u5176\u65b9\u5dee\u5219\u4e3a\u8be5\u5e38\u6570\u7684\u5e73\u65b9\u518d\u4e58\u4ee5\u539f\u6765\u7684\u65b9\u5dee\u3002\u5373\uff1a \\(X \\sim \\mathcal{N}(\\mu,\\sigma^2) \\)\uff0c\u90a3\u4e48 \\(aX \\sim \\mathcal{N}(a\\mu,a^2\\sigma^2) \\)\u3002<\/p>\n\n\n\n<p>\u6574\u4f53\u4e0a\uff0c\u8003\u8651\u662f\u5e0c\u671b\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\uff0c\u65b9\u5dee\u4e0d\u8981\u504f\u79bb\u592a\u5927\u3002\u5e76\u4e14\u968f\u7740\u65f6\u95f4\u7684\u63a8\u8fdb\uff0c\u6700\u7ec8\u7684\u6570\u503c\u662f\u4e00\u4e2a\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\uff0c\u5982\u679c\u6765\u770b\u63a8\u8fdf\u5230\u51fa\u6765\u7684\u201c\u516c\u5f0f3\u201d\uff1a<\/p>\n\n\n<p>$$<br \/>\nq(x_t|x_0) = \\mathcal{N}(x_t; \\sqrt{\\bar{\\alpha}}x_0,(1-\\bar{\\alpha_t})\\mathbf{I}) \\tag{3}<br \/>\n$$<\/p>\n\n\n\n<p>\u53ef\u4ee5\u770b\u5230\u5728\u8fd9\u6837\u7684\u201c\u8bbe\u8ba1\u201d\uff08\u5373\u4f7f\u7528\u201c\u6839\u53f7\u201d\uff09\u4e0b\uff0c\u6700\u7ec8\u8fed\u4ee3\u7684\\(x_t \\) \u7684\u5747\u503c\u662f \\(\\sqrt{\\bar{\\alpha}}x_0 \\)\uff0c\u65b9\u5dee\u4e3a \\(1-\\bar{\\alpha_t} \\)\uff0c\u968f\u7740\\(t \\)\u7684\u589e\u52a0\uff0c\u5c31\u9010\u6b65\u8d8b\u5411\u4e8e \\(\\mathcal{N}(0,1) \\)\u4e86\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">5. \u4e00\u4e9b\u6570\u5b66\u516c\u5f0f\u4e0e\u63a8\u5bfc<\/h3>\n\n\n\n<figure class=\"wp-block-image alignright size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"938\" height=\"560\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-3.png\" alt=\"\" class=\"wp-image-22189\" style=\"aspect-ratio:1.6750238998365559;width:394px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-3.png 938w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-3-300x179.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-3-768x459.png 768w\" sizes=\"auto, (max-width: 938px) 100vw, 938px\" \/><\/figure>\n\n\n\n<p>Diffusion \u76f8\u5173\u7684\u6570\u5b66\u57fa\u7840\u8fd8\u662f\u975e\u5e38\u3001\u975e\u5e38\u590d\u6742\u7684\uff0c\u800c\u8fd9\u91cc\u7684\u516c\u5f0f\u63a8\u5bfc\u770b\u8d77\u6765\u867d\u7136\u6709\u70b9\u590d\u6742\uff0c\u4f46\u53ef\u80fd\u662f\u6574\u4e2aDiffusion\u6a21\u578b\u7684\u6570\u5b66\u57fa\u7840\u4e2d\u6700\u4e3a\u7b80\u5355\u7684\u90e8\u5206\u4e86\u3002\u8fd9\u91cc\uff0c\u52c9\u5f3a\u796d\u51fa\u53f3\u8fb9\u7684\u56fe\u7247\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u8fd9\u91cc\u8fd8\u662f\u6765\u505a\u4e00\u4e9b\u5c1d\u8bd5\u5427\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-8\"><\/span>5.1 Forward Diffusion Process \u516c\u5f0f<\/h4>\n\n\n\n<p>$$x_t = \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I}) \\tag{1} $$<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u516c\u5f0f\u672c\u8eab\u5df2\u7ecf\u6709\u4e00\u5b9a\u7684\u590d\u6742\u5ea6\u4e86\uff0c\u8981\u641e\u6e05\u695a\u5927\u6982\u9700\u8981\u5173\u6ce8\u5982\u4e0b\u70b9\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u8fd9\u91cc\u7684 \\(\\epsilon  \\) \u662f\u4ec0\u4e48\u610f\u601d<\/li>\n\n\n\n<li>\\(\\beta_t \\) \u7684\u8ba1\u7b97\u8bbe\u8ba1<\/li>\n\n\n\n<li>\u4ece\u201c\u8fed\u4ee3\u516c\u5f0f(1)\u201d\u5230\u201c\u901a\u9879\u516c\u5f0f(2)\u201d<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-9\"><\/span>5.2 \u552c\u4eba\u7684 \\(\\epsilon \\)<\/h4>\n\n\n\n<p>\u591a\u7ef4\u53d8\u91cf\u7684\u6982\u7387\u5df2\u7ecf\u5fd8\u5f97\u5dee\u4e0d\u591a\u4e86&#8230;\uff0c\u597d\u5728\u8fd9\u91cc\u662f\u4e00\u4e9b\u5b8c\u5168\u201c\u72ec\u7acb\u201d\u7684\u968f\u673a\u53d8\u91cf\uff0c\u8fd8\u6bd4\u8f83\u597d\u7406\u89e3\u3002\u6211\u4eec\u5148\u770b\u770b\u8fd9\u91cc\u7684\uff1a \\(\\epsilon \\sim \\mathcal{N}(0, \\mathbf{I}) \\)\u3002<\/p>\n\n\n\n<p>\u7b2c\u4e00\u6b21\u770b\u5230\u8fd9\u4e9b\u4e2a\u7b26\u53f7\u7684\u65f6\u5019\uff0c\u4e5f\u662f\u88ab\u201c\u6014\u201d\u4e86\u4e00\u4e0b\u7684\uff0c\u4ed4\u7ec6\u4e00\u770b\u8fd8\u597d\u3002<\/p>\n\n\n\n<p>\u9996\u5148\uff0c\u8fd9\u91cc\u516c\u5f0f\u4e2d\u7684 \\(x_t \\) \u662f\u4e00\u5f20\u56fe\u7247\u6240\u6709\u7684\u50cf\u7d20\u4fe1\u606f\uff0c\u4f8b\u5982\uff0c\u5982\u679c\u662f\u4e00\u5f20 128&#215;128 \u7684\u56fe\u7247\uff0c\u90a3\u4e48\u6240\u6709\u7684\u50cf\u7d20\u4fe1\u606f\u5219\u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 1 x 3 x 128 x 128 \u7684\u5411\u91cf\u3002\u5373\uff0c\\(x_t \\) \u662f\u4e00\u4e2a 3 x 128 x 128 \u7684\u5411\u91cf\u3002<\/p>\n\n\n\n<p>\u5bf9\u5e94\u7684\uff0c\u8fd9\u91cc\u7684 \\(\\epsilon \\) \u4e5f\u662f\u4e00\u4e2a\u8fd9\u6837\u7684\u5411\u91cf\uff08\u4f8b\u5982\uff0c 3 x 128 x 128 \u7684\u5411\u91cf\uff0c\u800c\u4e0d\u662f\u6570\u5b66\u4e2d\u5e38\u89c1\u7684\u8868\u793a\u4e00\u4e2a\u5f88\u5c0f\u7684\u503c\uff09\uff0c\u53ef\u4ee5\u8fd9\u6837\u7406\u89e3\uff0c\u8fd9\u4e2a\u5411\u91cf\u7684\u6bcf\u4e00\u4e2a\u53d6\u503c\u90fd\u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff08\u4f8b\u5982\u4e00\u5171 3 x 128 x 128 \u4e2a\u968f\u673a\u53d8\u91cf\uff09\uff0c\u6bcf\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u90fd\u662f\u72ec\u7acb\u7684\uff0c\u5373\u534f\u65b9\u5dee\u77e9\u9635\u4e3a\u5355\u4f4d\u77e9\u9635\uff08\u8fd9\u91cc\u7684 \\(\\mathbf{I} \\)\uff09\uff0c\u5e76\u4e14\u6bcf\u4e2a\u968f\u673a\u53d8\u91cf\u7b26\u5408\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u5373 \\(\\mathcal{N}(0, 1) \\)\u3002<\/p>\n\n\n\n<p>\u518d\u56de\u5934\u770b\u770b\u539f\u516c\u5f0f\uff0c\u662f\u4e0d\u662f\u7b80\u5355\u4e86\u5f88\u591a\uff1a<\/p>\n\n\n\n<p>$$x_t = \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I})  $$<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-10\"><\/span>5.3 \u7ebf\u6027\u8c03\u5ea6<\/h4>\n\n\n\n<p>\u518d\u6765\u770b\u516c\u5f0f\u4e2d\u7684 \\(\\beta_{t} \\)\uff1a<\/p>\n\n\n\n<p>$$x_t = \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I}) $$<\/p>\n\n\n\n<p>\u5728\u539f\u59cb\u7684 <a href=\"https:\/\/arxiv.org\/pdf\/2006.11239\">Denoising Diffusion Probabilistic Models<\/a> \u8bba\u6587\u4e2d\uff0c\u53d6\u503c\u5982\u4e0b\uff1a\\( \\beta_1 = 10^{-4} ,\\, \\beta_{T} = 0.02 \\)\uff0c\u5373\u5747\u5300\u7ebf\u6027\u7684\u57281000\u6b21\u566a\u58f0\u6dfb\u52a0\u4e2d\uff0c\\(\\beta \\)\u5747\u5300\u7684\u4ece\\(0.0001 \\) \u589e\u957f\u5230 \\(0.02 \\)\uff0c\u5373\uff1a<\/p>\n\n\n\n<p>$$  \\beta_1 = 0.0001,   \\beta_2 = 0.0001199, \\beta_3 = 0.0001398 , &#8230; , \\beta_{1000} = 0.02  $$<\/p>\n\n\n\n<p>\u5728 Python  \u4e2d\u5c31\u662f\u5982\u4e0b\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">T = 1000  # \u603b\u6b65\u6570\nbetas = torch.linspace(0.0001, 0.02, T) # \u7ebf\u6027\u8c03\u5ea6\uff1a\u566a\u58f0\u65b9\u5dee\u9010\u6e10\u589e\u5927<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-11\"><\/span>5.4 \u201c\u7ebf\u6027\u8c03\u5ea6\u201d\u7684\u771f\u5b9e\u8ba1\u7b97\u5f0f<\/h4>\n\n\n\n<p>\u4f46\u662f\uff0c\u5728\u5b9e\u9645\u7684\u8fd0\u7b97\u4e0d\u4f1a\u7528\u4e0a\u9762\u7684\u516c\u5f0f\u3002\u800c\u662f\uff0c\u4f7f\u7528\u4e86\u8fd9\u4e2a\u7248\u672c\u7684\u63a8\u5bfc\uff0c\u4ece\u800c\u66f4\u52a0\u9ad8\u6548\uff0c\u66f4\u52a0\u76f4\u89c9\uff0c\u540c\u65f6\uff0c\u770b\u8d77\u6765\u66f4\u52a0\u201c\u62bd\u8c61\u201d\uff0c\u5373\u5728\u8bba\u6587\u4e2d\u7684\u5982\u4e0b\u516c\u5f0f\uff1a<\/p>\n\n\n\n<p>$$ q(x_t|x_0) = \\mathcal{N}(x_t; \\sqrt{\\bar{\\alpha}}x_0,(1-\\bar{\\alpha_t}\\mathbf{I}) ) \\tag{3}$$<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u7248\u672c\u770b\u8d77\u6765\u5c31\u5f88\u201c\u552c\u4eba\u201d \uff0c\u4f46\u7406\u89e3\u4e86\u5176\u610f\u601d\u8fd8\u662f\u611f\u89c9\u6bd4\u8f83\u201c\u7b80\u6d01\u201d\u7684\uff0c\u518d\u7406\u89e3\u4e4b\u540e\uff0c\u5c31\u89c9\u5f97\u4e5f\u8fd8\u6bd4\u8f83\u201c\u7b80\u5355\u201d\u3002<\/p>\n\n\n\n<p>\u672c\u8d28\u4e0a\uff0c\u8fd9\u516c\u5f0f\u662f\u524d\u9762\u201c\u516c\u5f0f(1)\u201d\u7684\u63a8\u5bfc\u4e0e\u5feb\u901f\u8ba1\u7b97\u7248\u672c\uff0c\u8be5\u516c\u5f0f\u63d0\u4f9b\u4e86\u4e00\u4e2a\u65e0\u9700\u8fed\u4ee3\u8ba1\u7b97\uff0c\u800c\u76f4\u63a5\u6839\u636e\\(x_0 \\) \u8ba1\u7b97 \\(x_t \\) \u7684\u65b9\u6cd5\u3002\u5176\u4e2d\u7684 \\(\\alpha_t := 1-\\beta_t ,\\, \\bar{\\alpha}_t = \\prod_{s=1}^{t}\\alpha_s \\) \u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-12\"><\/span>5.5 \u201c\u8c03\u5ea6\u516c\u5f0f\u201d\u7684\u63a8\u5bfc<\/h4>\n\n\n\n<p>\u8fd9\u91cc\u6765\u5c1d\u8bd5\u505a\u4e00\u4e0b\u201c\u516c\u5f0f(1)\u201d\u5230\u201c\u516c\u5f0f(2)\u201d\u7684\u63a8\u5bfc\uff0c\u5c1d\u8bd5\u7406\u89e3\u4ee5\u4e0b\u7814\u7a76\u8005\u4eec\u8fd9\u90e8\u5206\u5de5\u4f5c\uff08\u4e5f\u53ef\u4ee5\u53c2\u8003\u8fd9\u91cc\uff1a<a href=\"https:\/\/arxiv.org\/pdf\/2511.11746\">Diffusion Models: A Mathematical Introduction<\/a>\u7684\u7b2c14\u9875\uff09\u3002<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nx_t  &#038;=  \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon \\\\[0.5em]<br \/>\nx_{t-1} &#038;=  \\sqrt{1 &#8211; \\beta_{t-1}} x_{t-2} + \\sqrt{\\beta_{t-1}} \\epsilon \\\\[0.5em]<br \/>\nx_{t-2} &#038;=  \\sqrt{1 &#8211; \\beta_{t-2}} x_{t-3} + \\sqrt{\\beta_{t-2}} \\epsilon \\\\[0.5em]<br \/>\n \\quad &#038;\\vdots &#038; \\\\[0.5em]<br \/>\nx_{1} &#038;=  \\sqrt{1 &#8211; \\beta_{1}} x_{0} + \\sqrt{\\beta_{1}} \\epsilon \\\\[0.5em]<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u6240\u4ee5\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nx_t  &#038;=  \\sqrt{1 &#8211; \\beta_t} (\\sqrt{1 &#8211; \\beta_{t-1}} x_{t-2} + \\sqrt{\\beta_{t-1}} \\epsilon) + \\sqrt{\\beta_t} \\epsilon \\\\[0.5em]<br \/>\n&#038;=  ( \\sqrt{1 &#8211; \\beta_t}\\sqrt{1 &#8211; \\beta_{t-1}} )x_{t-2} + ( \\sqrt{1 &#8211; \\beta_t}\\sqrt{\\beta_{t-1}}  )\\epsilon + \\sqrt{\\beta_t} \\epsilon<br \/>\n\\end{aligned} \\tag{4}<br \/>\n$$<\/p>\n\n\n\n<p>\u63a5\u4e0b\u53bb\u770b\u770b\u4e0a\u9762\u7b49\u5f0f\u7684\u540e\u9762\u4e24\u90e8\u5206\uff1a<\/p>\n\n\n<p>$$ ( \\sqrt{1 &#8211; \\beta_t}\\sqrt{\\beta_{t-1}}  )\\epsilon + \\sqrt{\\beta_t} \\epsilon \\tag{5}$$<\/p>\n\n\n\n<p>\u8fd9\u91cc\u5e76\u4e0d\u662f\u7b80\u5355\u7684\u52a0\u6cd5\uff0c\u800c\u662f\u4e24\u4e2a\u72ec\u7acb\u6982\u7387\u5206\u5e03\u7684\u52a0\u6cd5\uff1a\u5982\u679c\u6982\u7387\u57fa\u7840\u8fd8\u5728\u7684\u8bdd\uff0c\u5c31\u6709\u5982\u4e0b\u7684\u516c\u5f0f\uff0c\u4e24\u4e2a<span style=\"text-decoration: underline;\">\u72ec\u7acb<\/span>\u7684\u9ad8\u65af\u5206\u5e03\uff08\u4f8b\u5982\uff1a\\(X \\sim \\mathcal{N} (\\mu_X ,\\sigma^2_X ) \\quad Y \\sim \\mathcal{N} (\\mu_Y ,\\sigma^2_Y ) \\)\uff09\u7684\u968f\u673a\u53d8\u91cf\u4e4b\u548c\uff0c\u5176\u7ed3\u679c\u4f9d\u65e7\u662f\u9ad8\u65af\u5206\u5e03\uff0c\u5e76\u4e14\u5747\u503c\u4f9d\u636e\u662f\u4e24\u4e2a\u5747\u503c\u7684\u548c\u3001\u65b9\u5dee\u4e5f\u662f\u4e24\u4e2a\u65b9\u5dee\u7684\u548c\uff08 \\(X+Y \\sim \\mathcal{N} (\\mu_X + \\mu_Y ,\\sigma^2_X + \\sigma^2_Y ) \\) \uff09\u3002<\/p>\n\n\n\n<p>\u6ce8\u610f\u4e0a\u9762\u7684\u8868\u793a \\(\\sqrt{\\beta_t}\\epsilon \\) \u4e2d\uff0c\\(\\sqrt{\\beta_t} \\) \u662f\u6807\u51c6\u5dee\uff0c\u65b9\u5dee\u5373 \\(\\beta_t \\)\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\u4e0a\u9762\u201c\u516c\u5f0f(5)\u201d\u4e24\u4e2a\u5206\u5e03\u7684\u548c\uff0c\u4f9d\u65e7\u662f\u9ad8\u65af\u5206\u5e03\uff0c\u4e14\u5747\u503c\u4f9d\u65e7\u662f 0\uff0c\u65b9\u5dee\u5219\u4e3a \\((1-\\beta_t)\\beta_{t-1}+\\beta_t \\)\uff0c\u5373\u6709\u4e86\u5982\u4e0b\u770b\u4f3c\u9519\u8bef\u7684\uff0c\u4f46\u662f\u5374\u662f\u6b63\u786e\u7684\u63a8\u5bfc\uff1a<\/p>\n\n\n<p>$$<br \/>\n( \\sqrt{1 &#8211; \\beta_t}\\sqrt{\\beta_{t-1}}  )\\epsilon + \\sqrt{\\beta_t} \\epsilon  = \\sqrt{(1-\\beta_t)\\beta_{t-1}+\\beta_t} \\epsilon<br \/>\n$$<\/p>\n\n\n\n<p>\u5176\u5b9e\u4e0a\u9762\u5e76\u4e0d\u662f\u4e00\u4e2a\u4e00\u822c\u610f\u4e49\u7684\u201c\u7b49\u5f0f\u201d\uff0c\u800c\u662f\u8868\u8fbe\u4e86\u5982\u4e0b\u7684\u542b\u4e49\uff1a<\/p>\n\n\n<p>$$<br \/>\n( \\sqrt{1 &#8211; \\beta_t}\\sqrt{\\beta_{t-1}}  )\\epsilon + \\sqrt{\\beta_t} \\epsilon  \\sim \\mathcal{N}(0,(1-\\beta_t)\\beta_{t-1}+\\beta_t)<br \/>\n$$<\/p>\n\n\n\n<p>\u6709\u4e86\u8fd9\u91cc\u7684\u7406\u89e3\uff0c\u5c31\u4ee5\u7ee7\u7eed\u4e0a\u9762\u516c\u5f0f(4)\u7684\u63a8\u5bfc\u5c31\u975e\u5e38\u5bb9\u6613\u6709\u5982\u4e0b\u7684\u7ed3\u8bba\u4e86\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nx_t  &#038;=  \\sqrt{1 &#8211; \\beta_t} (\\sqrt{1 &#8211; \\beta_{t-1}} x_{t-2} + \\sqrt{\\beta_{t-1}} \\epsilon) + \\sqrt{\\beta_t} \\epsilon \\\\[0.5em]<br \/>\n&#038;=  ( \\sqrt{1 &#8211; \\beta_t}\\sqrt{1 &#8211; \\beta_{t-1}} )x_{t-2} + ( \\sqrt{1 &#8211; \\beta_t}\\sqrt{\\beta_{t-1}}  )\\epsilon + \\sqrt{\\beta_t} \\epsilon \\\\[0.5em]<br \/>\n&#038;= \\sqrt{\\alpha_t\\alpha_{t-1}}x_{t-2} +  \\sqrt{1-\\alpha_t\\alpha_{t-1}} \\epsilon \\\\[0.5em]<br \/>\n&#038;\\vdots \\\\[0.5em]<br \/>\n&#038;=\\sqrt{\\alpha_t\\alpha_{t-1}\\cdots\\alpha_1}x_{0} +  \\sqrt{1-\\alpha_t\\alpha_{t-1}\\cdots\\alpha_{1}} \\epsilon \\\\[0.5em]<br \/>\n&#038;= \\sqrt{\\bar{\\alpha_t}}x_0 + \\sqrt{1-\\bar{\\alpha_t}} \\epsilon \\\\[0.5em]<br \/>\n\\text{where}\\, \\alpha_t &#038;:= 1-\\beta_t ,\\, \\bar{\\alpha_t} = \\prod_{s=1}^{t}\\alpha_s<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u5373\u6709\u4e86\u6700\u7ec8\u7684\u516c\u5f0f\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_t = \\sqrt{\\bar{\\alpha_t}}x_0 + \\sqrt{1-\\bar{\\alpha_t}} \\epsilon \\\\<br \/>\n\\text{where}\\, \\alpha_t := 1-\\beta_t ,\\, \\bar{\\alpha_t} = \\prod_{s=1}^{t}\\alpha_s<br \/>\n$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span id=\"bppb-heading-anchor-13\"><\/span>5.6 \u5728Python\u4e2d\u7684\u5b9e\u73b0<\/h4>\n\n\n\n<p>\u4e86\u89e3\u4e86\u4e0a\u9762\u8fd9\u4e9b\uff0c\u518d\u770b\u8fd9\u4e9b\u4ee3\u7801\u5c31\u5f88\u7b80\u5355\u4e86\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">T = 1000  # \u603b\u6b65\u6570\nbetas = torch.linspace(0.0001, 0.02, T) # \u7ebf\u6027\u8c03\u5ea6\uff1a\u566a\u58f0\u65b9\u5dee\u9010\u6e10\u589e\u5927\n\n# \u8ba1\u7b97\u4e2d\u95f4\u53d8\u91cf\nalphas = 1. - betas\nalphas_cumprod = torch.cumprod(alphas, axis=0) # \u5bf9\u5e94\u516c\u5f0f\u4e2d\u7684 alpha_bar<\/code><\/pre>\n\n\n\n<p>\u4ee5\u53ca\u6700\u540e\u7684 Forward Diffusion Process \u8ba1\u7b97\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code lang=\"python\" class=\"language-python\">    sqrt_alphas_cumprod_t = torch.sqrt(alphas_cumprod[t]) # \u4fe1\u53f7\u7cfb\u6570\n    sqrt_one_minus_alphas_cumprod_t = torch.sqrt(1. - alphas_cumprod[t]) # \u566a\u58f0\u7cfb\u6570\n\n    # \u6838\u5fc3\u516c\u5f0f\u5b9e\u73b0\n    return sqrt_alphas_cumprod_t * x_0 + sqrt_one_minus_alphas_cumprod_t * noise, noise<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">6. \u5fae\u5206\u65b9\u7a0b\u89d2\u5ea6\u7684\u8003\u8651<\/h3>\n\n\n\n<p>\u4ece\u968f\u673a\u5fae\u5206\u65b9\u7a0b\u7684\u89d2\u5ea6\u8003\u8651 \u201cDDPM\u201d \u6a21\u578b\u662f\u8be5\u6a21\u578b\u53d1\u5e03\u4e4b\u540e\u7684\u4e8b\u60c5\u4e86\uff0c\u6211\u4eec\u8fd9\u91cc\u5148\u770b\u770b\u8fd9\u4e2a\u968f\u673a\u5fae\u5206\u65b9\u7a0b\u7684\u201c\u6f02\u79fb\/Drift\u201d\u90e8\u5206\u4e0e\u4e0a\u8ff0\u8fed\u4ee3\u5f0f\u5b50\u7684\u5173\u7cfb\u3002<\/p>\n\n\n\n<p>\u5148\u4e0d\u8003\u8651\u201c\u968f\u673a\u9879\u201d\u7684\u589e\u52a0\uff0c\u90a3\u4e48\u5728\u8bbe\u8ba1\u65f6\uff0c\u5e0c\u671b\u968f\u7740\u65f6\u95f4\u6b65\u9aa4\u7684\u8fed\u4ee3\uff0c\u53d8\u5316\u7684\u901f\u7387\u9010\u6e10\u589e\u52a0\uff0c\u5373\u8fc1\u79fb\u53d8\u5316\u6162\uff0c\u540e\u671f\u53d8\u5316\u5feb\uff0c\u5373\u8003\u8651\u5728\u5fae\u5206\u65b9\u7a0b\u7684\u53f3\u4fa7\u589e\u52a0\u4e00\u4e0b \\(\\beta(t) \\)\uff0c\u8be5\u51fd\u6570\u968f\u7740\u65f6\u95f4\u589e\u52a0\u800c\u589e\u52a0\uff0c\u6700\u4e3a\u5e38\u89c1\u7684\u5373\u4e3a\u7ebf\u6027\u589e\u957f\uff08\u5bf9\u5e94\u4e8e\u201c\u7ebf\u6027\u8c03\u5ea6\u201d\uff09\u3002\u6b64\u5916\uff0c\u8be5\u53d8\u5316\u7387\u5e94\u8be5\u4e0e\u5f53\u524d\u503c\u6709\u5173\uff0c\u5f53\u524d\u503c\u8d8a\u5927\uff0c\u5219\u53d8\u5316\u7387\u5e94\u8be5\u8d8a\u5927\uff1b\u5e76\u4e14\u671f\u671b\u6700\u7ec8\u8fed\u4ee3\u7ed3\u679c\u8d8b\u5411\u4e8e\u96f6\uff08\u5373\u5747\u503c\u6700\u7ec8\u4e3a\u96f6\u7684\u6b63\u6001\u5206\u5e03\uff09\uff0c\u5c31\u6709\u6700\u7ec8\u7684\u5fae\u5206\u65b9\u7a0b\u8bbe\u8ba1\uff1a<\/p>\n\n\n<p>$$ \\frac{dx}{dt} = -\\frac{1}{2}\\beta(t)x \\tag{5} $$<\/p>\n\n\n\n<p>\u8bf4\u660e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u65b9\u7a0b\u53f3\u4fa7\u7684\u8d1f\u53f7\uff0c\u8868\u793a\u603b\u662f\u671d\u7740x\u53d6\u503c\u76f8\u53cd\u7684\u65b9\u5411\u79fb\u52a8\uff0c\u5373\u603b\u662f\u671d\u7740\u539f\u70b9\u65b9\u5411\u79fb\u52a8<\/li>\n\n\n\n<li>\\(\\beta(t)x \\)\u5219\u8868\u8fbe\u4e86\u4e0a\u8ff0\u7684\u4e24\u4e2a\u5173\u4e8e\u201c\u53d8\u5316\u7387\u201d\u5927\u5c0f\u7684\u610f\u56fe<\/li>\n<\/ul>\n\n\n\n<p>\u8be5\u5fae\u5206\u65b9\u7a0b\u7684\u8fed\u4ee3\u89e3\u5c31\u6709\u5982\u4e0b\u7684\u8fed\u4ee3\u8868\u8fbe\u5f0f\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n&#038;x_{t} &#8211; x_{t-1} = -\\frac{1}{2}\\beta(t)x_{t-1}  \\\\[0.5em]<br \/>\n&#038;x_{t}  = x_{t-1} -\\frac{1}{2}\\beta(t)x_{t-1}  \\\\[0.5em]<br \/>\n&#038;x_{t}  = (1 -\\frac{1}{2}\\beta(t))x_{t-1}  \\\\[0.5em]<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u5f88\u795e\u5947\u7684\u662f\uff0c\u5728\\(\\beta \\)\u5f88\u5c0f\u7684\u65f6\u5019\uff0c\u6839\u636e\u6cf0\u52d2\u5c55\u5f00\u6709\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\sqrt{1-\\beta} = 1 &#8211; \\frac{1}{2}\\beta &#8211; \\frac{1}{8}\\beta^2 + \\cdots<br \/>\n$$<\/p>\n\n\n\n<p>\u6700\u7ec8\u5c31\u6709\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_{t}  = \\sqrt{1-\\beta_t}x_{t-1}<br \/>\n$$<\/p>\n\n\n\n<p>\u8fd9\u91cc\u5c31\u53ef\u4ee5\u4ece\u201c\u5fae\u5206\u65b9\u7a0b\u201d\u7684\u89d2\u5ea6\u53bb\u7406\u89e3\u4e0a\u8ff0\u7684 stable diffusion \u4e2d Forward Diffusion Process \u4e2d\u524d\u534a\u90e8\u5206\u4e86\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">7. \u5c0f\u7ed3 FDP<\/h3>\n\n\n\n<p>\u5728\u4e86\u89e3 What \u4e2d\uff0c\u4e5f\u5728\u6162\u6162\u7406\u89e3 How \u4ee5\u53caWhy \u3002\u8fd9\u91cc\u518d\u6b21\u4ece\u5b8f\u89c2\u4e0a\u6982\u8ff0 FDP \u7684\u8fc7\u7a0b\uff0c\u4ece\u800c\u8df3\u51fa\u4e0a\u8ff0\u7684 What \u7ec6\u8282\uff0c\u518d\u6b21\u5ba1\u89c6\u8fd9\u4e2a\u8fc7\u7a0b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">7.1 \u9996\u5148\uff0c\u4e3a\u4ec0\u4e48\u9700\u8981 \u201cForward Diffusion Process\u201d \uff1f<\/h4>\n\n\n\n<p>\u7b80\u5355\u56de\u7b54\uff1a\u7ed9\u6837\u672c\u6dfb\u52a0\u566a\u58f0\uff0c\u6784\u5efa\u8bad\u7ec3\u6570\u636e\u3002<\/p>\n\n\n\n<p>\u201cForward Diffusion Process\u201d \u8fc7\u7a0b\u7684\u6570\u636e\u4e3b\u8981\u7528\u4e8e\u8bad\u7ec3\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u7ed9\u5b9a\u7684\u56fe\u7247\uff0c\u9010\u6b65\u6dfb\u52a0\u566a\u58f0\uff0c\u6700\u540e\u8ba9\u5176\u53d8\u6210\u4e00\u5f20\u7eaf\u7cb9\u7684\u3001\u9ad8\u65af\u5206\u5e03\u7684\u566a\u58f0\u3002\u800c\u8fd9\u4e2a\u8fc7\u7a0b\u7684\u6570\u636e\uff0c\u5219\u53ef\u4ee5\u7528\u4e8e\u8bad\u7ec3 U-Net \u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u8ba9\u8be5U-Net\u5177\u5907\u4e00\u4e2a\u795e\u5947\u7684\u80fd\u529b\uff1a\u5373\u7ed9\u51fa\u4e00\u5f20\u56fe\u7247\uff08\u5e26\u6709\u566a\u58f0\u7684\uff09\uff0c\u8be5 U-Net \u53ef\u4ee5\u9884\u6d4b\u51fa\u8fd9\u5f20\u56fe\u7247\u4e2d\u6709\u54ea\u4e9b\u662f\u566a\u58f0\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">7.2 \u201cForward Diffusion Process\u201d \u64cd\u4f5c\u7684\u6570\u5b66\u8ba1\u7b97<\/h4>\n\n\n\n<p>\u5176\u6838\u5fc3\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_t = \\sqrt{1 &#8211; \\beta_t} x_{t-1} + \\sqrt{\\beta_t} \\epsilon, \\quad \\epsilon \\sim \\mathcal{N}(0, \\mathbf{I})<br \/>\n$$<\/p>\n\n\n\n<p>\u7ecf\u8fc7\u63a8\u5bfc\uff0c\u7b49\u4ef7\u4e0e\u5982\u4e0b\u516c\u5f0f\uff08\u5173\u4e8e\u516c\u5f0f\u7684\u63a8\u5bfc\uff1a<a href=\"https:\/\/www.orczhou.com\/?p=22092&amp;preview=true\">Forward Diffusion Process<\/a>\uff09\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_t = \\sqrt{\\bar{\\alpha_t}} x_0 + \\sqrt{1-\\bar{\\alpha_t}}\\epsilon   \\quad \\epsilon \\sim \\mathcal{N}(0,1) \\\\<br \/>\n\\text{Where} \\quad \\alpha_t := 1-\\beta_t ,\\, \\bar{\\alpha_t} = \\prod_{s=1}^{t}\\alpha_s \\tag{a}<br \/>\n$$<\/p>\n\n\n\n<p>\u5728\u8bba\u6587\u4e2d\u53ef\u80fd\u770b\u5230\u7684\u5f62\u5f0f\uff1a<\/p>\n\n\n<p>$$<br \/>\nq(x_t|x_0) = \\mathcal{N}(x_t; \\sqrt{\\bar{\\alpha}}x_0,(1-\\bar{\\alpha_t}\\mathbf{I}) )<br \/>\n$$<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u4e00\u5f20\u7167\u7247\u5b9e\u9645\u505a\u4e0a\u8ff0\u64cd\u4f5c\u5219\u6709\u5982\u4e0b\u6548\u679c\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"180\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-1024x180.png\" alt=\"\" class=\"wp-image-22090\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-1024x180.png 1024w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-300x53.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11-768x135.png 768w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/11.png 1182w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\u5173\u4e8e\u4e0a\u8ff0\u516c\u5f0f\u7684\u4e00\u4e9b\u91cd\u8981\u7279\u6027\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7ecf\u8fc7\u82e5\u5e72\u6b21\u8fed\u4ee3\u540e\uff0c\u4e00\u5f20\u6e05\u6670\u7684\u56fe\u7247\u6700\u7ec8\u53d8\u6210\u4e00\u5f20\u201c\u968f\u673a\u201d\u566a\u58f0\u56fe\u7247\uff0c\u8fd9\u91cc\u7684\u201c\u968f\u673a\u201d\u662f\u6307\u7684\u6b63\u6001\u5206\u5e03<\/li>\n\n\n\n<li>\u4ece\u201c\u516c\u5f0f(a)\u201d\u53ef\u4ee5\u770b\u5230\uff0c\u9010\u6b65\u8fed\u4ee3\u548c\u591a\u6b65\u5408\u5e76\u8fed\u4ee3\u6709\u4e00\u6837\u7684\u6548\u679c\u3002\u5f53\u7136\uff0c\u4e3a\u4e86\u83b7\u5f97\u8bad\u7ec3\u6570\u636e\uff0c\u603b\u662f\u9010\u6b65\u8fed\u4ee3<\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1 midjourney \u7528\u7684\u6bd4\u8f83\u591a\uff0c\u5f88\u597d\u5947\uff0c\u8fd9\u80cc\u540e\u7684\u6280\u672f\u5927\u6982\u662f\u600e\u6837\u7684\u3002\u597d\u4e86\uff0c\u90a3\u5c31\u53bb\u505a\u4e00\u4e9b\u4e86\u89e3\u5427\uff08\u8865\u5145\u6ce8\uff1a\u540e\u6765\u6211\u624d\u53d1\u73b0\u8fd9\u4e2a\u5751\u6709\u591a\u5927&#8230;\uff09\u3002 \u9664\u4e86\u5927\u8bed\u8a00\u6a21\u578b\u4e4b\u5916\uff0c\u53e6\u4e00\u4e2a\u975e\u5e38\u6d3b\u8dc3\u7684\u9886\u57df\u5c31\u662f\u56fe\u7247\u5904\u7406\u6280\u672f\uff0c\u4f8b\u5982\u6587\u751f\u56fe\uff08Text-to-Image\uff09\u3001\u591a\u6a21\u6001\u6a21\u578b\u7b49\u3002\u9664\u4e86\u5728\u539f\u6709\u7684 CNN \u6280\u672f\u67b6\u6784\u4e0a\uff0c\u4e5f\u51fa\u73b0\u4e86\u5f88\u591a\u65b0\u7684\u7a81\u7834\u3002\u867d\u7136\u540c\u6837\u4f7f\u7528\u7684\u662f\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\uff0c\u4f46\u56fe\u50cf\u751f\u6210\u6280\u672f\u4e0e\u5927\u8bed\u8a00\u6a21\u578b\u6280\u672f\uff08LLM\uff09\u662f\u975e\u5e38\u4e0d\u540c\u7684\u3002\u5176\u6a21\u578b\u67b6\u6784\u4e0d\u540c \uff0c\u5e95\u5c42\u7684\u6570\u5b66\u3001\u7269\u7406\u539f\u7406\u4e5f\u975e\u5e38\u4e0d\u4e00\u6837\u3002\u5728\u5bf9\u5927\u8bed\u8a00\u6a21\u578b\u6709\u4e00\u4e2a\u6846\u67b6\u6027\u7684\u4e86\u89e3 \u4e4b\u540e\uff0c\u73b0\u5728\u6253\u7b97\u5f00\u4e00\u4e2a\u65b0\u7684\u201c\u5751\u201d\uff0c\u5f53\u7136\uff0c\u4e5f\u5c31\u6ca1\u6709\u6253\u7b97\u722c\u51fa\u6765\uff0c\u53ef\u80fd\u5c31\u201c\u6d45\u5c1d\u8f84\u6b62\u201d\u505a\u4e2a\u6700\u4e3a\u57fa\u7840\u7684\u4e86\u89e3\u3002\u4e00\u822c\u5b66\u4e1c\u897f\uff0c\u90fd\u662f\u4ece \u201cWhy\/How\/What\u201d \u7684\u987a\u5e8f\uff0c\u4f46\u5728\u5c1d\u8bd5\u4e86\u4e00\u4e0b\uff0c\u65e0\u6cd5\u7406\u89e3\u201cWhy\u201d\u540e\uff0c\u4e8e\u662f\u5c31\u6253\u7b97\u7ed5\u9053\uff0c\u770b\u770b How \u548c What \u4e86\u3002\u90a3\u4e48\uff0c\u201cWhat\u201d\u6253\u7b97\u5c31\u4ece\u201cFoward Diffusion Process\u201d\u5f00\u59cb\u5427\u3002 \u6574\u4e2a\u6587\u751f\u56fe\uff08Text-to-Image\uff09\u7684\u67b6\u6784\u662f\u6bd4\u8f83\u590d\u6742\u7684\uff0c\u5176\u4e2d\u8f83\u4e3a\u201c\u7b80\u5355\u201d\u3001\u57fa\u7840 \u7684\u4e00\u6b65\u5373\u4f4d\u201cForward Diffusion Process\u201d\uff0c\u672c\u6587\u4ece\u201cDenoising Diffusion Probabilistic Models\u201d\u8bba\u6587\u4e2d\u4e3a\u4f8b\uff0c\u8bf4\u660e\u8fd9\u4e2a\u8fc7\u7a0b\u3002 1. \u6982\u8ff0\uff1a\u201cFDP\u201d \u662f\u4e00\u4e2a\u52a0\u968f\u673a\u566a\u58f0\u7684\u8fc7\u7a0b \u5f88\u591a\u5730\u65b9\u90fd\u4f1a\u8bf4\uff1a\u201cForward Diffusion Process\u201d \u5c31\u662f\u4e00\u4e2a\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u7684\u8fc7\u7a0b\u3002\u8fd9\u8bdd\u5bf9\u3001\u4e5f\u4e0d\u5bf9\u3002\u9996\u5148\uff0c\u201cFDP\u201d \u786e\u5b9e\u662f\u4e00\u4e2a\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u7684\u8fc7\u7a0b\uff0c\u4f46\u662f\u8fd9\u4e2a\u968f\u673a\u566a\u58f0\u6dfb\u52a0\u5f97\u975e\u5e38\u6709\u201c\u8bb2\u7a76\u201d\u3002\u6211\u4eec\u5c31\u4ece\u201c\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u201d\u548c\u201c\u8bb2\u7a76\u201d\u4e24\u4e2a\u89d2\u5ea6\u53bb\u4ecb\u7ecd\u3002 \u201cForward Diffusion Process\u201d \u8fc7\u7a0b\u7684\u6570\u636e\u662f\u7528\u4e8e\u8bad\u7ec3\u795e\u7ecf\u7f51\u7edc\uff08\u901a\u5e38\u662f\u4e00\u4e2aU-Net\u67b6\u6784\uff09\u7684\u53c2\u6570\u7684\uff0c\u4ece\u800c\u6700\u7ec8\u5b9e\u73b0\u5176\u9006\u8fc7\u7a0b\uff08\u5373\u201cReverse Diffusion Process\u201d\uff09\uff0c\u800c\u751f\u6210\u56fe\u7247\u3002 1.1 \u6dfb\u52a0\u968f\u673a\u566a\u58f0 \u201cFDP\u201d \u505a\u5982\u4e0b\u7684\u4e8b\u60c5\uff1a \u4f8b\u5982\uff0c\u6211\u4eec\u4f7f\u7528\u8bba\u6587 DDPM \u4e2d\u7684\u8bbe\u5b9a\u5bf9\u5982\u4e0b\u56fe\u7247\u6dfb\u52a0\u566a\u58f0\uff0c\u5c31\u53ef\u4ee5\u89c2\u5bdf\u5230\u5982\u4e0b\u8fc7\u7a0b\uff1a \u66f4\u4e3a\u8be6\u7ec6\u7684\uff1a 2. \u201c\u7ebf\u6027\u201d\u6dfb\u52a0\u566a\u58f0 \u5728 Denoising Diffusion Probabilistic Models \u8bba\u6587\u4e2d\u4f7f\u7528\u4e86\u201c\u7ebf\u6027\u8c03\u5ea6\u201d\u7684\u65b9\u5f0f\u6dfb\u52a0\u566a\u58f0\u3002\u5373\u6dfb\u52a0\u566a\u58f0\u7684\u5f3a\u5ea6\u201c\u7ebf\u6027\u201d\u7684\u9010\u6e10\u589e\u5f3a\uff0c\u8fd9\u91cc\u7684\u201c\u7ebf\u6027\u201d\u662f\u6307\u589e\u52a0\u7684\u566a\u58f0\u7684\u201c\u65b9\u5dee\u201d\u7ebf\u6027\u589e\u52a0\u3002 \u5148\u7528\u66f4\u52a0\u5f62\u5f0f\u5316\u7684\u6570\u5b66\u8bed\u8a00\u63cf\u8ff0\u4e0a\u8ff0\u7684\u566a\u58f0\u6dfb\u52a0\uff0c\u5373\uff1a $$ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":22172,"comment_status":"open","ping_status":"closed","sticky":false,"template":"wp-custom-template-a-1440-px-width-template","format":"standard","meta":{"_eb_attr":"","inline_featured_image":false,"_tocer_settings":[],"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22092","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-simplelife"],"_links":{"self":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/22092","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/comments?post=22092"}],"version-history":[{"count":209,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/22092\/revisions"}],"predecessor-version":[{"id":23427,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/22092\/revisions\/23427"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media\/22172"}],"wp:attachment":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media?parent=22092"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/categories?post=22092"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/tags?post=22092"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}