Image Pyramids and Blending

Smoothing Filter

Assume the smoothing weights \(\hat w\) is \(1\times 5\). To make such weights a proper smoothing filters. \(\hat w\) is symmetric, sum to \(1\), and have equal contribution

\[\hat w = [c, b, a, b, c], b = 1/4, c = \frac{1-2a}{4}\]

\[a+2b+2c = 1\]

\[a+2c = 2b=1/2\]

then the convolution filter is \(W = \hat w\hat w^T\) is a \(5\times 5\) filter.

Reduce Function

Define \(Reduce: g_l\Rightarrow g_{l+1}\) where

\[g_{l+1}(i,j) = \sum_{-2}^2 \sum_{-2}^2 W(m,n)g_i(2i-m,2j-n)\]

\[D_l = \begin{bmatrix} 1&0&0&...&0\\ 0&0&1&...&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&...&1 \end{bmatrix}, C_l = \begin{bmatrix} c&b&a&b&c&0&...&0\\ 0&c&b&a&b&c&...&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&0&0&0&0&...&c \end{bmatrix}\]

so that \(g_{l+1} = D_L \cdot C_l\cdot g_l\) where \(g\) is vectorized image.

Laplacian Pyramid

Let \(L_i = g_i - expand(g_{i+1})\) be the difference between levels \(g_l, g_{l+1}\), since \(g_{l+1}\) has different size. We need to expand.

Expand Function

\[expand(g_l) = 4\sum^2_{-2}\sum^2_{-2} W(m,n)g_l(\frac{i-m}{2}, \frac{j-n}{2})\]

Blending

Given source images \(A,B\) and binary matte \(M\), compute Laplacian Pyramids for \(A, B\) i.e. \(AL_0, ..., AL_{N-1}, Ag_N, BL_0, ..., BL_{N-1}, Bg_N\), Gaussian pyramid for \(M\), i.e. \(Mg_0, ..., Mg_N\). Then, using matting equation, using \(M\) as alpha channel.

\[SL_i = Mg_l AL_l + (1-Mg_l)BL_l\]

Finally, reconstruct Laplacian pyramid \(SL\), which is desired.