Basic operators and adjoints

Zero padding is the transpose of truncation

Surrounding a dataset by zeros (zero padding) is adjoint to throwing away the extended data (truncation). Let us see why. Set a signal in a vector $\bold x$, and then to make a longer vector $\bold y$, append some zeros to $\bold x$. This zero padding can be regarded as the matrix multiplication

$$\bold y\eq \left[ \begin{array}{c} \bold I \\ \bold 0 \end{array} \right] \ \bold x$$ (11)

The matrix is simply an identity matrix $\bold I$ above a zero matrix $\bold 0$. To find the transpose to zero-padding, we now transpose the matrix and do another matrix multiply:

$$\tilde {\bold x} \eq \left[ \begin{array}{cc} \bold I & \bold 0 \end{array} \right] \ \bold y$$ (12)

So the transpose operation to zero padding data is simply truncating the data back to its original length. Module zpad1 pads zeros on both ends of its input. Modules for two- and three-dimensional padding are in the library named zpad2() and zpad3(). user/gee/zpad1.c

    for (d=0; d < nd; d++) { 
	p = d + (np-nd)/2;
	if (adj) data[d] += padd[p];
	else     padd[p] += data[d];
    }

Basic operators and adjoints