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VPA Compatible memory representations

The VPA scheme clearly resolves the problem of interlaced images (an example is included).
It would, in fact, also describe an interlaced image, where the odd field is upside down and the even field rightside up or more pathological cases.

It is interesting to identify the class of memory representations which can be represented by VPA tables.
The class evidently includes all linear representations. Which are excluded? The equation

M(X, Y) = XTable ( X ) + YTable ( Y ),

evidently implies that

M(X1, Y) - M(X2, Y) = XTable ( X1) - XTable(X2) is independent of Y, so that M(X1, Y1) - M(X2, Y1) = M(X1, Y2) - M(X2, Y2)

for all X1, X2, Y1, Y2 which is equivalent to M(X1, Y1) + M(X2, Y2) = M(X1, Y2) + M(X2, Y1), for all X1, X2, Y1, Y2.

This equation exactly describes image planes which can be described by VPA tables.

It must hold if the tables exist. On the other hand if the formula is valid then the required tables are easily seen to be

XTable(X) = M(X, 0) - M(0, 0)

YTable(Y) = M(0, Y)

The asymmetry is only superficial. We could have subtracted M(0,0) from the Ytable entries just as well.

In more intuitive terms the basic condition is that in memory, all scan lines (chaotic as they may be) are translations of each other, and that (equivalently) all scan columns are also translations of each other.
Excluded memory representations therefore change the representation scheme between scan lines.

An example of an excluded representation is a four-quadrant type of frame grabber where the quadrants are mapped to essentially random offsets in memory.
If the quadrants are mapped to equidistant offsets in memory (e.g. sequentially) the representation is already covered by VPA (bizarre as it may be).

This discussion is somewhat academic. We may simply summarize: Almost all practically relevant memory formats can be represented by VPA.
Examples for the most common ones will follow.