A Nearly One-to-One Method to Convert Analog Signals into a Small Volume of Data: Second Part

Top

Introduction

Sounds are considered as 1-D signals but the images as 2-D signals. There exist many manners to extend 1-D to 2-D. A geometrical manner to conceive the extension to more dimensions is to start from a dimensionless point placed at the “origin” of the space coordinates. Physically, the origin point of the space coordinates may be considered the point where the observer is placed. Now, we choose a given direction in space and move this dimensionless point in that direction. It results a line that may be considered as a “support” for 1-D signals if these signals are send in this direction. Let now consider an orthogonal direction and move the line on this new direction. It results a plane in which 2-D signals may be placed. Further, if will consider now a new orthogonal direction and move the plane on this new direction it results a volume where 3-D signals may be placed. Of course, this procedure may be extended to more than 3 dimensions. Figure 1 represents these line, plane and volume and Figure 2, the manner to pass to 4-D.

It may be remarked that:

• •

  All the considered directions have to be orthogonal. Then, they formed an equivalent orthogonal system of coordinates.

• •

  Only the first direction may be arbitrarily chosen; all the other directions results by orthogonal properties.

• •

  For 1-D signals, the signal direction in space may be considered also as its time-support and then, mathematically, correspond to (one of) the independent variable of the function that represent it. Then, the actual equivalent position in space of the signal may be deduced by considering its velocity.

• •

  For more than 1-D, the movement is “collective”: all the points of the plane, volume and so on are in move together. As a simple physical example, a scanner or a photocopying machine process the 2-D signals (images) by moving a light line.

• •

  The 2-D signals may be “still”, like the photos and/or depending of the time like for “movies”. In this last situation, the direction of the time support may be independent of the space coordinates of any frame of the movie but all the “pixels” of any frame (image) move collectively at the same instant. Chemical sensor process at a same time the whole image but electronic sensors uses generally a given scanning procedure. This last procedure is equivalent to transform the 2-D input signal into a 1-D signal.

• •

  It can be remarked that a 3-D cube may look very different in a 2-D projection and similarly for a 4-D cube in 3-D and/or 2-D projections.

The “object dimension” in space has been defined here intuitively. Mathematically, the dimensions “d” were defined informally by Hausdorff (Wikipedia, 2009) as:

(1) for self-similar objects for which the parameters “N” and “s” are described by a power law such as:

(2)

Intuitively, for a given space “X”, “N” balls of radius “s” may cover it completely. If N(s) grows in the same way as when , the dimension of this space is “d”.

Formally, let “A” be a subset of a metric space “X”. Then the Hausdorff dimension “D(A)” of “A” is the infinum of such that the “d-dimensional Hausdorff measure” of “A” is 0. It can be remarked that D(A) need not be an integer.