 # HMM and some of its applications

 Application to an image processing problem. We wish to compress a dynamically changing image field. In the continuous spatial domain, the image intensity field is modeled by the pde

p

57 Dab(t)d2(t, x, y)/dxadxb + w(t, x, y) = x, y)/dt a,b=l

ie a diffusion equation with slowly time varying diffusion matrix D(t) = ((DH(,(t))) We assume that the diffusion matrix D(t) is slowly varying in the sense that it is a constant over the time slot |mT, (m + 1)T) and then makes a transition to another matrix in the next time slot [(m + 1)T. (m + 2)T). The aim is firstly to estimate the diffusion matrix values over each slot and then to reestimate their transition probabilities from the measurements of over the whole time duration [0, NT], Assume that the diffusion matrix assumes values in a finite set £ = {Di,Da} and that the Markov transition probabilities are Tr(k,m) = P(D(nT + 0) = DmD(nT — 0) = Db). Assuming then that the image field has been discretized in space and time, we replace the above equation by its discretized version

(t+l,x,y) = (t,z,y)+6w(t,x,y)+60^ Dab ]AaAb(t,x,y),nT < t < (n+l)T a,b

and the probability of making these observations can be expressed as

# Quantum Image Processing

After introducing concepts such as diffusion equation based edge smoothing and image enhancement and histogram equalization of classical images (ie, constructing a probability density for image pixel grey-scale levels by maximizing the entropy of the pdf over each image patch subject to moment constraints) the course should aim at dealing with problems of modeling the electromagnetic field coming from the image as a quantum em field described in terms of creat ion and annihilation operators. When such a quantum em field hits an object that is moving or rotating, some of it gets scattered and some of it passes through the object. If we model the dynamically varying object parameters as > 0, and the quantum em field as

¿(V) = + a*kh(t, r)), [at, a*] = 6kj

k

then after passing through the moving object, this received em field has the form

Al (t,r) = : s < t) + akxk(0(s) :s

k

and after this field interacts with an atomic receiver or a lattice of atomic receivers, each of whichhas a free valence electron, the interaction Hamiltonian can be expressed as

V(t) = :sk : s < t)]

k

where are atomic observables depending on the parameter history in some specific way. Now let |p > be the initial state of the lattice of atomic electrons and let the coherent state |<£(u) > be the initial state of the quantum em field. Thus,

ak|<£(u) >= ukphi(u) >,

«kl0(u) >= - Wfc)|d>(«) >

The initial state of the atomic lattice and the field is the tensor product |(w)p > and the final state is |d>(w)<7 >• The first order transition probability from the former to the latter in time T is obtained using quantum mechanical time dependent perturbation theory as

Pt(p >-» | lw) =

| [ < (u)qV(t)(u)p > exp(i(E(q) - E(p))t)dt|2 Jo

Now,

< u)qv(t)=

52(< <^(«)|afc|^(«) >< qU-p > + < <^(«)|afc|^(«) >< g|Cfc|p >) k

= 27?e[^2«fc < q£k(0(s) : s < t)|p >] k

which means that by measuring the transition probabilities P-j . T > 0 between two atomic states with the quantum image field remaining in the same coherent state, we get information about the object motion parameters 0(t), t > 0.

If for example, the moving object is a fan, then 0(t) satisfies a second order differential equation with noise present and we can use the EKF to estimate it dynamically based on noisy measurements of the transition probabilities. More generally, if the object is also of quantum size, then its parameter #(t) will be an observable in another Hilbert space which may satisfy a quantum stochastic differential equation like the Hudson-Parthasarathy equation and then we must use the Belavkin quantum filter to estimate this observable dynamically from our measurements.

Other problems in quantum image processing:

[a] How to teach the basic concepts in quantum image processing based on static filters. The quantum image field is a state of the electromagnetic field that can be expressed either as a pure state

|I >= y~^c(fe,m)|Ar,m > k,m

or more generally as a mixed state of the form

PI = 5? |A:,m > p(fc,m,J,Z) < j,Z|

In these expressions, |fc,m > denotes the state of the kth image pixel when its grey scale level is specified by the index m. Thus, if |I > is the quantum image and a measurement in the standard basis {|fc,m >< k, m| : k,m} is made, then the probability that the kth pixel will be selected and that this pixel is has the grey scale level m is given by c(k, m)|2. On the other hand, if the image field is in the mixed state pi, then if the above measurement is made, the probability that the kth pixel is selected and that it has the grey scale level m is given by

< k, m >= p(k, m, k, m)

By processing a quantum image, we mean that we apply a quantum operation £ to it, ie, f is a TPCP map and hence admits the Stinespring-Choi-Kraus representation,

(P) = ^ErpE*r.^E*rEr = I r r

Thus, the output image state after processing and adding quantum noise is given by

po = 5? Erp¡E*

r

The problem is to recover pi from po by the use of recovery operators. This problem is solved completely by the Knill-Laflamme theorem which states that

[b] If we are to apply Belavkin’s theory of filtering based on non-demolition measurements, we must first couple the image field to a bath described by Boson-Fock space and introduce creation, annihilation and conservation quantum noise processes into the joint unitary evolution of the image field and bath. After unitarily evolving for time t, we partially trace out over the bath state to obtain the output image state. This state is of the form (1) to which the Knill-Laflamme theorem can be applied:

Po = Tr2(U(T)(Pl ® |d>(u) >< ^(U)|)[/(T)*)

= ^2 ErpjEr

r

where the system operators Er are defined in terms of the system operators that drive the Hudson-Parthasarathy unitary evolution U(t), ie, in terms of II. Li, L2,S, P where

dU(t) = (~(iH + P)dt + Li ® dA(t) - L-2 ® dA(t)* + S(t) ® dA(t))i/(t)

Exercise, by using time dependent perturbation theory with the perturbation parameter 8 attached to Li, L2 and <52 attached to P, give approximate formulae for the operators Ek in terms of Li, L2, P, H. Assume that the bath is in the coherent state |(w) >•