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<h1 id="forensics_3_3">数字图像取证 - 基于像素的取证 3</h1>
<p>
Posted on Thu 25 February 2021 in <a href="/category/digital-image-forensics.html">Digital Image Forensics</a>
• 1 min read
</p>
</header>
<div>
<h2>缩略图</h2>
<p>在JPEG文件格式中, 会在头部嵌入经过滤波，对比度调整，压缩的缩略图.这 些特定的操作就导致了不同的相机厂商或软件厂商的细微差异。
设一个原图为<span class="math">\(f(x, y)\)</span>，那么一个图像经过下面6步骤进行制作</p>
<ol>
<li>裁剪/填充</li>
<li>预滤波</li>
<li>降采样</li>
<li>对比度</li>
<li>亮度调整</li>
<li>JEPG压缩</li>
</ol>
<h2>裁剪/填充</h2>
<p>当缩略图与原图的分辨率比例不相同时, 需要对图像进行裁剪/填充处理. 这一阶段会产生四个参数<span class="math">\(c_l, c_r, c_t, c_b\)</span>分别表示左右上下的裁剪/填充系数</p>
<h2>抽象预滤波, 降采样, 对比度以及亮度调整</h2>
<p>完成裁剪/填充操作后, 后续的四个操作可以将其抽象为一组模型
</p>
<div class="math">$$
t(x, y) = \alpha(D\{\hat{f}(x, y) * h_1(x, y)\} * h_2(x, y)) + \beta
$$</div>
<p>
其中<span class="math">\(t(x, y)\)</span>是缩略图,<span class="math">\(D\)</span>表示降采样操作, <span class="math">\(h_1\)</span>为预滤波, <span class="math">\(h_2\)</span>为后滤波(可选, 一般用于锐化处理), <span class="math">\(*\)</span>为卷积, <span class="math">\(\alpha, \beta\)</span>则是对比度系数以及亮度系数.为了简化模型, 做出如下假设 </p>
<ol>
<li>预滤波器为圆对称高斯模型, <span class="math">\(exp(-(x^2 + y^2)/\theta^2), 其中\)</span>\theta$为宽度.</li>
<li>预滤波器是单元和.</li>
<li>后滤波器的大小为3x3.</li>
<li>后滤波器是对称的, 即<span class="math">\((h_2(x, y) = h_2(-x, y),h_2(x, y) = h_2(x, -y))\)</span>. 产生如下形式的滤波器(a b a; b c b; a b a)</li>
<li>后滤波器也是单元和, 约束$c = 1 - (4a + 4b)</li>
</ol>
<p>在以上的约束下, 缩略图模型将会产生11个参数, 2个缩略图尺寸参数, 4个填充/裁剪参数, 1个预滤波参数<span class="math">\(\theta\)</span>, 2个后滤波参数<span class="math">\(a, b\)</span>以及对比度参数<span class="math">\(\alpha\)</span>和亮度参数<span class="math">\(\beta\)</span>. 考虑到JPEG图像格式的特有参数128个, 即2个8x8的对亮度以及色彩通道的量化表. 故对于一个Jpeg文件总共可以产生139个参数对其进行描述.</p>
<h2>求解剪切/填充参数</h2>
<p>缩略图处理流程中剪切/填充的边界正相关于全尺寸图像的分辨率(已知), 但具体的比例仍然需要估算. 首先设定一个初始的符合分辨率比例关系的初始缩略图<span class="math">\(\hat{f}_0(x, y)\)</span>. 故初始缩率图可以被建模为
</p>
<div class="math">$$
\hat{t}_0(x, y) = D\{\hat{f}(x, y)\}
$$</div>
<p>
其中<span class="math">\(\hat{f}_0(x, y)\)</span>为原始图像, D为降采样操作, 其比例为<span class="math">\(max(N_x/n_x, N_y/n_y)\)</span>, <span class="math">\((N_x, N_y)\)</span>以及<span class="math">\((n_x, n_y)\)</span>为图像和缩略图的尺寸. 随后初始缩率图经过各项异性(各个维度变换系数不相同)的缩放(scaling)和变换(translation)得到提取出的缩略图<span class="math">\(t(x, y)\)</span>
</p>
<div class="math">$$
t(x, y) = \hat{t}(s_xx + \delta_x, s_yy + \delta_y)
$$</div>
<p>
其中<span class="math">\((s_xx, s_yy), (\delta_x, \delta_y)\)</span>分别为缩放和变换系数.
在灰度图上基于粗粒度至细粒度的差分匹配算法(coarse-to-fine differential registration)求解上述参数</p>
<h2>求解滤波器以及对比度和亮度参数</h2>
<p>求解各滤波器以及对比度和亮度参数, 可以通过减少真实图像与预估图像之间的误差进行求解. 即
</p>
<div class="math">$$
E_1(h_1, h_2) = \sum_{x, y}[t(x, y) - \alpha(D\{\hat{f}(x, y) * h_1(x, y)\} * h_2(x, y)) - \beta]^2
$$</div>
<p>
但上述公式仅根据预滤波和后滤波得出, 无法求解对比度以及亮度. 故对于亮度与对比度系数, 给定滤波器的参数, 定义误差函数为
</p>
<div class="math">$$
E_2(\alpha, \beta) = \sum_{x, y}[t(x, y) - \alpha(D\{\hat{t}_h(x, y) + \beta)]^2
$$</div>
<p>
其中<span class="math">\(\hat{t}_h(x, y) = D\{\hat{f}(x, y) * h_1(x, y)\} * h_2(x, y))\)</span>
利用最小二乘法即可<span class="math">\(E_2\)</span>进行求解. 对于<span class="math">\(E_1\)</span>则需要使用暴力搜索求解最小值, 或使用Nelder-Mead 最小化算法加速求解过程.
在实验中, 作者发现求解<span class="math">\(E_1\)</span>时在傅里叶域中效果会更好, 即最小化
</p>
<div class="math">$$
E_1(h_1, h_2) = \sum_{\omega_x, \omega_y}[||T(\omega_x, \omega_y)|| - ||\hat{T}(\omega_x, \omega_y)||]^2
$$</div>
<p>
其中<span class="math">\(T(\cdot)\)</span>表示傅里叶变换后的真实缩率图<span class="math">\(t(\cdot)\)</span>,<span class="math">\(\hat{T}(\cdot)\)</span>表示傅里叶变换后的预估缩率图<span class="math">\(D\{\hat{f}(x, y) * h_1(x, y)\} * h_2(x, y))\)</span>, <span class="math">\(||\cdot||\)</span>表示傅里叶变换的量级. 由于经过<span class="math">\(
t(x, y) = \hat{t}(s_xx + \delta_x, s_yy + \delta_y)\)</span>的初始校准, 低频特征时十分均衡的, 故该损失函数在高频特征上进行计算</p>
<p>最后是, 128个量化表系数可以直接从量化表中获取. </p>
<h2>参考文档</h2>
<p>1.E. Kee and H. Farid. Digital image authentication from thumbnails. In SPIE Symposium on Electronic Imaging, San Jose, CA, 2010.</p>
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