\documentclass{article}

\begin{document}

\title{Some image transform math}

\author{Owen Taylor}

\date{18 February 2003}

\maketitle

\section{Basics}

The transform process is composed of three steps;

first we reconstruct a continuous image from the 

source data \(A_{i,j}\):

\[a(u,v) = \sum_{i = -\infty}^{\infty} \sum_{j = -\infty}^{\infty} A_{i,j}F\left( {u - i \atop v - j} \right) \]

Then we transform from destination coordinates to source coordinates:

\[b(x,y) = a\left(u(x,y) \atop v(x,y)\right)

         = a\left(t_{00}x + t_{01}y + t_{02} \atop t_{10}x + t_{11}y + t_{12} \right)\]

Finally, we resample using a sampling function \(G\):

\[B_{x_0,y_0} = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} b(x,y)G\left( {x - x_0 \atop y - y_0} \right) dxdy\]

Putting all of these together:

\[B_{x_0,y_0} = 

\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}

\sum_{i = -\infty}^{\infty} \sum_{j = -\infty}^{\infty} A_{i,j}

F\left( {u(x,y) - i \atop v(x,y) - j} \right)

G\left( {x - x_0 \atop y - y_0} \right) dxdy\]

We can reverse the order of the integrals and the sums:

\[B_{x_0,y_0} = 

\sum_{i = -\infty}^{\infty} \sum_{j = -\infty}^{\infty} A_{i,j}

\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}

F\left( {u(x,y) - i \atop v(x,y) - j} \right)

G\left( {x - x_0 \atop y - y_0} \right) dxdy\]

Which shows that the destination pixel values are a linear combination of the 

source pixel values. But the coefficents depend on \(x_0\) and \(y_0\). 

To simplify this a bit, define:

\[i_0 = \lfloor u(x_0,y_0) \rfloor = \lfloor {t_{00}x_0 + t_{01}y_0 + t_{02}} \rfloor \]

\[j_0 = \lfloor v(x_0,y_0) \rfloor = \lfloor {t_{10}x_0 + t_{11}y_0 + t_{12}} \rfloor \]

\[\Delta_u = u(x_0,y_0) - i_0 = t_{00}x_0 + t_{01}y_0 + t_{02} - \lfloor {t_{00}x_0 + t_{01}y_0 + t_{02}} \rfloor \]

\[\Delta_v = v(x_0,y_0) - j_0 = t_{10}x_0 + t_{11}y_0 + t_{12} - \lfloor {t_{10}x_0 + t_{11}y_0 + t_{12}} \rfloor \]

Then making the transforms \(x' = x - x_0\), \(y' = y - x_0\), \(i' = i - i_0\), \(j' = j - x_0\)

\begin{eqnarray*}

F(u,v) & = & F\left( {t_{00}x + t_{01}y + t_{02} - i \atop t_{10}x + t_{11}y + t_{12} - j} \right)\\

       & = & F\left( {t_{00}(x'+x_0) + t_{01}(y'+y_0) + t_{02} - (i'+i_0) \atop 

                      t_{10}(x'+x_0) + t_{11}(y'+y_0) + t_{12} - (j'+j_0)} \right) \\

       & = & F\left( {\Delta_u + t_{00}x' + t_{01}y' - i' \atop 

                      \Delta_v + t_{10}x' + t_{11}y' - j'} \right)

\end{eqnarray*}

Using that, we can then reparameterize the sums and integrals and

define coefficients that depend only on \((\Delta_u,\Delta_v)\),

which we'll call the \emph{phase} at the point \((x_0,y_0)\):

\[

B_{x_0,y_0} = 

\sum_{i = -\infty}^{\infty} \sum_{j = -\infty}^{\infty} A_{i_0+i,j_0+j} C_{i,j}(\Delta_u,\Delta_v)

\]

\[

C_{i,j}(\Delta_u,\Delta_v) =

\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}

F\left( {\Delta_u + t_{00}x + t_{01}y - i \atop 

         \Delta_v + t_{10}x + t_{11}y - j} \right)

G\left( {x \atop y} \right) dxdy

\]

\section{Separability}

A frequent special case is when the reconstruction and sampling functions

are of the form:

\[F(u,v) = f(u)f(v)\]

\[G(x,y) = g(x)g(y)\]

If we also have a transform that is purely a scale and translation;

(\(t_{10} = 0\), \(t_{01} = 0\)), then we can separate 

\(C_{i,j}(\Delta_u,\Delta_v)\) into the product of a \(x\) portion

and a \(y\) portion:

\[C_{i,j}(\Delta_u,\Delta_v) = c_{i}(\Delta_u) c_{j}(\Delta_v)\]

\[c_{i}(\Delta_u) = \int_{-\infty}^{\infty} f(\Delta_u + t_{00}x - i)g(x)dx\]

\[c_{j}(\Delta_v) = \int_{-\infty}^{\infty} f(\Delta_v + t_{11}y - j)g(y)dy\]

\section{Some filters}

gdk-pixbuf provides 4 standard filters for scaling, under the names ``NEAREST'',

``TILES'', ``BILINEAR'', and ``HYPER''. All of turn out to be separable

as discussed in the previous section. 

For ``NEAREST'' filter, the reconstruction function is simple replication

and the sampling function is a delta function\footnote{A delta function is an infinitely narrow spike, such that:

\[\int_{-\infty}^{\infty}\delta(x)f(x) = f(0)\]}:

\[f(t) = \cases{1, & if \(0 \le t \le 1\); \cr 

                0, & otherwise}\]

\[g(t) = \delta(t - 0.5)\]

For ``TILES'', the reconstruction function is again replication, but we

replace the delta-function for sampling with a box filter:

\[f(t) = \cases{1, & if \(0 \le t \le 1\); \cr 

                0, & otherwise}\]

\[g(t) = \cases{1, & if \(0 \le t \le 1\); \cr 

                0, & otherwise}\]

The ``HYPER'' filter (in practice, it was originally intended to be 

something else) uses bilinear interpolation for reconstruction and

a box filter for sampling:

\[f(t) = \cases{1 - |t - 0.5|, & if \(-0.5 \le t \le 1.5\); \cr 

                0, & otherwise}\]

\[g(t) = \cases{1, & if \(0 \le t \le 1\); \cr 

                0, & otherwise}\]

The ``BILINEAR'' filter is defined in a somewhat more complicated way;

the definition depends on the scale factor in the transform (\(t_{00}\)

or \(t_{01}]\). In the \(x\) direction, for \(t_{00} < 1\), it is

the same as for ``TILES'':

\[f_u(t) = \cases{1, & if \(0 \le t \le 1\); \cr 

                  0, & otherwise}\]

\[g_u(t) = \cases{1, & if \(0 \le t \le 1\); \cr 

                  0, & otherwise}\]

but for \(t_{10} > 1\), we use bilinear reconstruction and delta-function

sampling:

\[f_u(t) = \cases{1 - |t - 0.5|, & if \(-0.5 \le t \le 1.5\); \cr 

                  0, & otherwise}\]

\[g_u(t) = \delta(t - 0.5)\]

The behavior in the \(y\) direction depends in the same way on \(t_{11}\).

\end{document}