Fooling the sniffer

Let us consider first the problem of “fooling" the sniffer. We want to find a way of presenting an e-Nose S with a palette mixture that mimics the original odor it was given. Formally, let (o; c) be an odorant, represented by the m-dimensional odorant vector d S(o; c). We want to find a mixing vector v such that when given PS _ v the sniffer S will produce a fingerprint as similar as possible to the one elicited by (o; c) itself. This is a simplified version of the mixing problem. First, it does not require any space-to-space mapping, since we are working in a single space, the sniffer space. Second, fooling an eNose, whose fingerprints are relatively controllable and are easily measured and studied, seems on the face of it to be simpler than fooling the human perception. Dealing this problem first will provides us with insight regarding the solution of the more general problem.



In analogy with (1), our task is to find a vector v that satisfies

Notice that unlike (1), here the odorant vectors are taken to be in the sniffer space too.
Let us now discuss such a PS in a relatively simple special case. An m-dimensional sniffer space for a sniffer S is called linear if it has the following properties:
(1) Linearity of response: For an odorant (o; c), each of the elements
is proportional to the odorant's concentration. That is,
where is an odorant-dependent constant. Denoting we can write this property in the compact form


(2) Additivity of mixtures: The odorant vector describing the mixture
is the vector sum of the odorant vectors of the individual elements,



For a linear sniffer, the operators pSi are simply multiplications by constant vectors,
Similarly, the operator PS is just a multiplication
by a matrix, If we take ||.||to be the standard
Euclidean norm, then finding v is equivalent to solving the well-known least squares problem,

Actually, v is constrained to be a non-negative vector.
Thus, had the sniffer space been linear, the mixing vector would have been easily calculated as the minimizer of a constrained least squares problem.