TABLE OF CONTENTS A basis is a projection onto a coordinate axis. If we have a point on a graph, we can talk of its distance from the origin, another point used for reference. But this only narrows it down to a circle, so we need more information: the angle between the vector made by the origin to the point of interest, and another line for reference (i.e., the x-axis). For a point in 2D space, these polar coordinates are sufficient information to unambiguously locate the point with respect to the origin. But we often use the Cartesian basis {1, 0} and {0, 1} to represent this point, which tells about the point in terms of a unit vector along the x-axis and another unit vector along the y-axis (they are normalized). We know that the y-axis intersects the x-axis at the origin, and that there is a 90-degree angle between them (they are orthogonal). Though, generally speaking, we could use a different set of axes (i.e., a different basis).
The Haar basis is { 1/ ?2, 1/ ?2} and {1/ ?2, ?1/ ?2}, instead of the normal Cartesian basis. This section answers the question of what the points look like in the Haar-domain instead of the Cartesian-domain. What we will see is that the points have the same relationship to one another, but the coordinate system moves by a rotation of 45 degrees.
First, consider the...
