For image in painting and blending a pixel its new value is dependent on its neighboring pixels. These neighbors may or may not lie on a border. The aim is to interpolate and create smooth transitions. It is thus not only dependent on a border condition, nor does a value on the border get replicated exactly to inside pixels. Should the area between boundaries be exactly 1 pixel wide, this pixel will most likely be the average of the boundary values.
A simple blender algorithm will work like this. Given:
- A background image $\textbf{b}$
- A foreground image $\textbf{f}$
- A mask $\textbf{m}$
where we will put the foreground image $\textbf{f}$ onto the background $\textbf{g}$, where the mask $\textbf{m}$ is true.
To avoid a visible seam we will make a correction image $\textbf{c}$. The run-of-the-mill patching to form new image $\textbf{g}$ then is:
$g(x, y) = \left \{\begin{array}{cc}
b(x, y) & \textrm{if } m(x, y) = 0 \\
f(x, y) + c(x, y) & \textrm{if } m(x, y) = 1
\end{array} \right .$
This image $\textbf{c}$ is obviously some form of interpolation, but informally we want two things from $\textbf{c}$:
- $\textbf{f}$+$\textbf{c}$ and $\textbf{b}$ should match on the edges
- It should be 'smooth' as to not create disturbances
Smooth transitions can be attained by simply stipulating that a pixel its new value is the weighted average of its four neighbors (up, down, left and right):
$c(x,y) = \frac{c(x-1, y)+c(x+1, y)+c(x, y-1)+c(x-1, y+1)}{4}$
And for boundary pixels the corresponding values of $\textbf{c}$ are replaced by $\textbf{b}-\textbf{f}$. If for instance for $c(x,y)$ the right neighbour is on de boundary:
$c(x,y) = \frac{c(x-1, y)+c(x+1, y)+c(x, y-1)+(b(x-1, y+1)-f(x-1, y+1))}{4}$
This means that $\textbf{every}$ pixel of $\textbf{f}$ under mask $\textbf{m}$ gives a linear equation, and each pixel in the patch is also a variable.
The result is a square linear system: $\textbf{A}\textbf{c} = \textbf{d}$, which can be solved for $\textbf{c}$.
Note that
- For boundary pixels, the values of $\textbf{b}$ and $\textbf{f}$ are $\textbf{constants}$ in this system, not variables. They thus go `to the right' of the equals sign.
- Pixels outside the masked region, as defined by $\textbf{m}$ are not variables.
See also
- J. Jia, J. Sun, C.K. Tang, and H.Y. Shum. Drag-and-drop pasting. In ACM
Transactions on Graphics (TOG), volume 25, pages 631-637. ACM, 2006.
- P. Perez, M. Gangnet, and A. Blake. Poisson image editing. In ACM
Transactions on Graphics (TOG), volume 22, pages 313-318. ACM, 2003.
