MAPaint Technical Report:
Look up Tables for Display to Image Coordinates

Look up Tables for Display to Image Coordinates

To make the calculation more efficient we can minimize the number of floating point operations by pre-calculating the trigonometric terms and storing them as look-up tables (LUT). In this case we are transforming from the 2D display frame back to 3D image coordinates and to avoid round-off error we require 6 LUTs for each of x' and y' to (x,y,z). Note x' and y' are always integer coordinates in the display frame. In our coordinate system we want the transform from (x',y',d) to the original image coordinates therefore using equation 6 we get

$$\frac{1}{s}(R^{-1}_{11} x' + R^{-1}_{12} y' + R^{-1}_{13} d) + f_x\;,$$ x =

$$\frac{1}{s}(R^{-1}_{21} x' + R^{-1}_{22} y' + R^{-1}_{23} d) + f_y\;\;\mbox{and}$$ y = (13)

$$\frac{1}{s}(R^{-1}_{31} x' + R^{-1}_{32} y' + R^{-1}_{33} d) + f_z\;.$$ z =

We define LUTs for x' to (x,y,z) which include the constant terms:

$$\frac{1}{s}(R^{-1}_{11} x' + R^{-1}_{13} d) + f_x\;,$$ T_x'x(x') =

$$\frac{1}{s}(R^{-1}_{21} x' + R^{-1}_{23} d) + f_y\;,$$ T_x'y(x') = (14)

$$\frac{1}{s}(R^{-1}_{31} x' + R^{-1}_{33} d) + f_z\;.$$ T_x'z(x') =

and similarly for y' to (x,y,z):

$$\frac{1}{s}R^{-1}_{12} y'\;,$$ T_y'x(x') =

$$\frac{1}{s}R^{-1}_{22} y'\;,$$ T_y'y(x') = (15)

$$\frac{1}{s}R^{-1}_{32} y'\;.$$ T_y'z(x') =

To determine the new section image the transformations can be calculated by look-up plus one addition:

$$T_{x'x}(x') + T_{y'x}(y')\;,$$ x =

$$T_{x'y}(x') + T_{y'y}(y')\;,$$ y = (16)

$$T_{x'z}(x') + T_{y'z}(y')\;.$$ z =

If the viewing direction does not change and only the distance parameter is incremented by $\delta d$ then the y' LUTs are unchanged and the x' LUTs become:

$$T_{x'x}(x', d + \delta d) T_{x'x}(x',d) + \frac{1}{s} R^{-1}_{13} \delta d\;,$$ =

$$T_{x'y}(x', d + \delta d) T_{x'y}(x',d) + \frac{1}{s} R^{-1}_{23} \delta d\;,$$ = (17)

$$T_{x'z}(x', d + \delta d) T_{x'z}(x',d) + \frac{1}{s} R^{-1}_{33} \delta d\;.$$ =

Note R^-1 = R^T. The LUTs are calculated over the maximum range possible for x' and y' which is determined by forward transform (equation 5) of the bounding box of the original image.