k-Wave

1. spg2716
   

   Member
   Joined: Oct '18
   Posts: 2

   Dear Bradley Sir,

   I have some conceptual doubt with respect to,
   1. What is meant by inverse crime?
   2. Just by changing the grid size how can we avoid the inverse crime ?
   3. Please suggest reference or textbook to help me understand the matrix A.

   The forward model of PAT is represented as
   Ax = b, A = system matrix, x = initial pressure and b = measurement.

   Based on understanding gained - The system matrix is obtained by solving the governing photoacoustic wave propagation equation (either using Green's function approach or pseudospectral methods or finite difference). In kWave, the command- "kspaceFirstOrder2D" helps us create the A matrix and x to obtain b.
   (This becomes our Forward model)

   My intention is compute the system matrix and apply the SVE/SVD method to understand the behavior of it.

   In the following code , while doing reconstruction (Inverse model/process),
   you change the kgrid size in order to avoid the inverse crime but for the rest for the Time reversal example the same kgrid is used (does that mean inverse crime is committed and ignored).

   Reference Code:- (i) Attenuation Compensation Using Time Reversal Example
   (ii) 2D Time Reversal Reconstruction For A Line Sensor Example

   Please do correct me in case I am wrong

   Thank you for your time and consideration.

   Regards,
   Soma

   Posted 3 years ago #

2. bencox
   

   Developer
   Joined: Mar '10
   Posts: 285

   Hi Soma,

   The 'inverse crime' refers to the situation when exactly the same set-up and model are used for both generating the data and performing the inversion. (See
   https://arxiv.org/abs/math-ph/0401050.) In any practical/experimental situation there will be noise in the data and uncertainty in other parameters such as the detector positions, and how sensitive the inversion is to these is crucial to how well is will perform in an application. The worry is that these aspects are not captured by a solution that commits the 'inverse crime'. On the other hand, performing an inversion that commits the inverse crime can be a useful check: if the inversion doesn't even work in this case there may be something wrong with the code, or perhaps the problem under investigation is very ill-posed. Common ways to ameliorate the inverse crime are to add noise to the simulated data, and use a different model for simulating the data and in the inversion (or at least using a different computational grid for the inversion).

   On your other question regarding the system matrix. You could construct it using k-Wave or some other model, or an analytical formula (Green's function), but it can get very large very easily. Each column will consist of a stack of time series at all the detector points generated by a point source at one grid point. So you could build it column-by-column by running lots of simulations, each with one point source at one grid point. (This will result in a full matrix, and it may be better to find a way to exploit some sparsity to reduce the memory required.) We don't usually construct the matrix explicitly, but if you do then it allows you to use all the usual tools of linear algebra to solve it.

   Hope that helps.

   Best wishes,
   Ben

   Posted 3 years ago #

3. spg2716
   

   Member
   Joined: Oct '18
   Posts: 2

   Thank you very much Sir.

   Posted 3 years ago #

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