markPoisson: Marked Poisson Process
In PET: Simulation and Reconstruction of PET Images

Description Usage Arguments Details Value Author(s) References See Also Examples

The function implements the Acceptance Rejection (AR) procedure to simulate a marked Poisson process with spatial varying intensity. Therewith it is possible to create noisy data from a phantom, as generated in Positron Emission Tomography (PET).

markPoisson(DataInt, nSample = 200000, ThetaSamples = 181, 
            RhoSamples = 2*round(sqrt(sum((dim(DataInt))^2))/2)+1,
            RhoMin = -0.5*sqrt(2),
            vect.length = 100000, image = TRUE, DebugLevel = "Normal")

`DataInt`	(matrix) A two dimensional image in which the matrix elements comply with intensities. That means the intensity of the decay of positrons in a certain tissue.
`nSample`	(integer) `nSample` determines the number of accepted events that will be generated with AR method. Defaults to `nSample = 200000`.
`ThetaSamples`	(integer) Specifies the number of samples in the angular parameter theta in the sinogram. The sinogram is sampled linearly from `0` to (approximately) pi radians. Defaults to `ThetaSamples=181`.
`RhoSamples`	(integer) Specifies the number of samples in the distance parameter rho in the sinogram. Defaults to `RhoSamples=2*round(sqrt(sum((dim(DataInt))^2))/2)+1`.
`RhoMin`	(double) Specifies the minimum sample position in the sinogram on the second axis. It is assumed that `DataInt` is scaled to [-0.5,0.5]^2. Defaults to `RhoMin=-0.5*sqrt(2)`.
`vect.length`	(integer) Determines a bound of number of generated accepted events in each iteration. That means, if `nSample > vect.length` then in each iteration `vect.length/(0.1217)` events will be generated and roughly `vect.length` accepted events. If you scale down `vect.length` the computation will be more time-consuming and if you increase the value of `vect.length`, more memory will be necessary. Defaults to `vect.length = 100000`.
`image`	(logical) If `image=FALSE`, only a generated sinogram will be returned, i.e. a noisy Radon transformation of the phantom will be returned. Otherwise, if `image=TRUE` an additional image will be returned, where each pixel of the image corresponds to a number of accepted events generated with the AR method. Defaults to `image=TRUE`.
`DebugLevel`	(character) This parameter controls the level of output. Available are: The default `DebugLevel="Normal"` for standard level output or alternative `"Detail"` if it desirable to logged almost all output to screen or `"HardCore"` for no information at all.

The function implements the Acceptance Rejection (AR) method to simulate a marked Poisson process with spatial varying intensity. The function was developed to simulate data of a PET scanner. Therefore, new random events are generated with the AR method from DataInt. An angle theta will be generated to each event (uniform and iid. in [0,pi]), because a PET scanner detects two photons, who will travel in (nearly) opposite directions. The line between the two detectors has the line parameters (rho,theta), whereas rho is the shortest distance from the origin of the coordinate system to the line, and theta an angle corresponding to the angular orientation of the line. The only obtainable information from the two photons is the fact, that the photon emission took place somewhere along that line.

`rData`	A matrix, that contains the Radon transformed data.
`Data`	Will be returned only in case of `image = TRUE`. A matrix, where each pixel of the image corresponds to a number of accepted events generated with the AR method.
`Header`	A list of following values: SignalDim The dimension of the `rData`. XYmin The minimum x- and y-position in `rData`. DeltaXY Sampling distance on the x- and y-axis in `rData`.
`call`	Arguments of the call to `markPoisson`.

Joern Schulz jschulz78@web.de.

Schulz, Joern, Diploma Thesis: Analyse von PET Daten unter Einsatz adaptiver Glaettungsverfahren, Humboldt-Universitaet zu Berlin, Institut fuer Mathematik, 2006.

Gentle, J.E., Elements of Computational Statistics, Springer-Verlag New York Berlin Heidelberg, 2002.

radon, iradon, iradonIT

## Not run: 
P <- phantom()
rP <- radon(P)
mP1 <- markPoisson(P)
mP2 <- markPoisson(P, nSample = 1000000)
viewData(list(P, mP1$Data, mP2$Data, rP$rData, mP1$rData, mP2$rData),
         list("Phantom", "nSample = 200000", "nSample = 1000000",
              "Radon Transfom of Phantom",  "nSample = 200000", 
              "nSample = 1000000"))
cat("Number of generated accepted events for mP2:",sum(mP2$Data),"\n")
rm(mP1,mP2,P,rP)

## End(Not run)