The MTM Algorithm

Now that we are equipped with notions of odor space, we can redefine the algorithmic scheme in more accurate terms. Let the whiffer contain n palette odorants, and let ti stand for the i'th of these. We use the generic term to denote an odorant vector that constitutes a representation of palette odorant i in concentration vi in some odor space E. For example, if E is the sniffer space S, then would be the m dimensional odorant vector . If E is the psychophysical space P, then pPi .vi would be l-dimensional odorant vector d P (ti; vi). In this way, p Ei can be viewed as an operator that is applied to the concentration vi to yield some representation of the i'th palette odorant in concentration vi. Notice that we use the symbol vi, rather than c, to denote the concentration of the i'th palette odorant; this is to distinguish the palette odorants from other odorants, for which we use c. We define the mixing vector v = (v1……. vn) T to be the list of palette odorant concentrations in a particular mixture. In accordance with our earlier notations, we represent a palette mixture in the odor space E by PE .v, with v being the mixing vector and PE being an as-of-yet unspecified operator.

Let (o; c) be an arbitrary odorant. The role of the mixing algorithm is to find a mixing vector v, such that the perception of PE _v is as similar as possible to that of (o; c). More formally, we would like d P (o; c) to be as close as possible to PP .v; i.e., we are seeking

with ||.|| some appropriately chosen norm. The general scheme of the mixing algorithm discussed above is described in Figure 2. The sniffer provides the algorithm with a measured odorant vector d S (o; c). The mapping algorithm then transforms this vector into the odorant vector d P (o; c) in the psychophysical space. Following this, based on the specific palette that resides in the whiffer, the algorithm calculates from (1) the mixing vector v, and transmits it to the whiffer. The whiffer then prepares the corresponding mixture and releases it.

We are now in a position to describe our algorithm. In the interest of clarifying its dynamics, we have chosen to describe its development in three stages, each adding a further complication.