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Sections G: General purpose processes

Forward model inversion

Inversion methods

Code OSIPI name Alternative names Notation Description Reference
G.MI1.001 Analytical inversion -- -- This method is used when the solution of the model inversion is well-defined and the model parameters of interest can be calculated analytically.
Input:
Forward model (M.GF1.001),
Static model parameters (Q.AI1.001),
[Data (Q.GE1.002), Data grid (Q.GE1.001)]
Output:
[Model parameters (Q.OP1.001)]		 --
G.MI1.002 Optimization Model fitting -- Inversion of a forward model by iteratively adjusting the set of model parameters in order to minimize a similarity measure between the data and the model.
Input:
Optimizer (select from optimizers)
Output:
[Estimated model parameters (Q.OP1.003)]		 --
G.MI1.003 Deconvolution -- -- Method which can be used when a model is given as a convolution with known h(x) and f(x) to determine g(x).
Input:
Deconvolution method (select from deconvolution methods)
Output:
[Data (Q.GE1.002), Data grid (Q.GE1.001)] = [g(x), x]		 --
G.MI1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Optimization

Optimizers

Code OSIPI name Alternative names Notation Description Reference
G.OP1.001 Levenberg-Marquardt -- LM An algorithm that interpolates between the Gauss-Newton algorithm and the method of gradient descent.
Input:
Cost function (select from cost functions),
Initial model parameters (Q.OP1.006)
Optional:
Model parameter lower bounds (Q.OP1.007),
Model parameter upper bounds (Q.OP1.008),
Data weights (Q.OP1.009),
Maximum number of iterations (Q.OP1.010),
Convergence threshold (Q.OP1.011)
Output:
Estimated model parameters (Q.OP1.003),
Cost value minimum (Q.OP1.005)		 --
G.OP1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Cost functions

Code OSIPI name Alternative names Notation Description Reference
G.OP2.001 Non-linear least squares -- NLLS , where f is a forward model describing the data, x is the data grid, y(x) is the measured data and is the L2-norm. The forward model is non-linear in the model parameters.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
Forward model (M.GF1.001),
(Q.OP1.001),
(Q.OP1.002)
Output:
Cost value (Q.OP1.004)
--
G.OP2.002 Linear least squares -- LLS , where x is the data grid, y(x) is the measured data and is the L2-norm.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
(Q.OP1.001),
A (Q.OP1.012)
Output:
Cost value (Q.OP1.004)
--
G.OP2.003 Standard-Form Tikhonov -- SFT , where x is the data grid, y(x) is the measured data , is the L2-norm and L is the identity matrix (same dimensions as A).
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
(Q.OP1.001),
A (Q.OP1.012),
(Q.OP1.013)
Output:
Cost value (Q.OP1.004)
--
G.OP2.004 Generalized cross validation -- GCV , where x is the data grid, y(x) is the measured data, is the L2-norm, I is the identity matrix of the same dimensions a A, is the solution of the matrix equation obtained from the SVD for a certain regularization parameter λ and is defined by the relationship .
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
(Q.OP1.001),
A (Q.OP1.012),
(Q.OP1.013)
Output:
Cost value (Q.OP1.004)
--
G.OP2.005 L-curve -- LC TO DO
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
(Q.OP1.001),
A (Q.OP1.012),
(Q.OP1.013)
Output:
Cost value (Q.OP1.004)
--
G.OP2.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Regularization parameter

Code OSIPI name Alternative names Notation Description Reference
G.OP3.001 Fixed -- -- A fixed value of λ , rather than a determined value is assumed.
Input:
λfixed (Q.OP1.015)
Output:
λ(Q.OP1.013)		 --
G.OP3.002 Generalized Cross Validation -- GCV λ is determined by minimizing the generalized cross validation cost function with respect to λ.
Input:
Optimizer (select from optimizers) with a GCV cost function (G.OP2.004) and Φ(Q.OP1.001) = λ(Q.OP1.013)
Output:
λ(Q.OP1.013)		 --
G.OP3.003 L-Curve criterion -- LCC λ is determined by minimizing the L-curve cost function with respect to λ.
Input:
Optimizer (select from optimizers) with a L-curve cost function (G.OP2.005) and Φ(Q.OP1.001) = λ(Q.OP1.013).
Output:
λ(Q.OP1.013)		 --
G.OP3.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Deconvolution

Code OSIPI name Alternative names Notation Description Reference
G.DE1.001 Discretization method -- -- Method to transfer continuous models, functions and equations into discrete counterparts. Select from Discretization methods. --
G.DE1.002 Regularization method -- -- Method to control an excessively fluctuating function such that the coefficients do not take extreme values. This is done by adding an additional penalty term in the cost function. Select a regularized cost function from Cost functions. --
G.DE1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Deconvolution methods

Code OSIPI name Alternative names Notation Description Reference
G.DE2.001 Singular Value Decomposition -- SVD Algebraic deconvolution of with f(x) and h(x) sampled at discrete points [f(x), x] and [g(x), x]. The convolution equation is discretized according to a given discretization method and the resulting matrix equation is solved as a regularized least-squares problem with a given regularization method.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)] = [f(x), x],
[Data (Q.GE1.002), Data grid (Q.GE1.001)] = [g(x), x],
Discretization method (select from discretization methods ),
Regularization method
Output:
[Data (Q.GE1.002), Data grid (Q.GE1.001)] = [g(x), x]
--
G.DE2.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Discretization methods

In this group, the following notation is assumed for all functions f: fn= f(xn).

Code OSIPI name Alternative names Notation Description Reference
G.DI1.001 Numerical convolution (first order) -- -- Convolutions are calculated as
--
G.DI1.002 Block-circulant -- -- Convolutions are calculated as .
This requires hi to be pre-padded with N zeros such that hi < 0 = 0.		 Wu 2004
G.DI1.003 Volterra linear -- -- Convolutions are calculated as
Sourbron 2003
G.DI1.004 Singular -- -- Convolutions are calculated as
.
It is assumed that gi = 0 for i < 0 and there is an index k such that hn = 0* for n <- k.		 Ostergaard 1996
G.DI1.005 Hybrid -- -- Convolutions are calculated as
.		 Willats 2006
G.DI1.006 Exponential convolution -- -- Convolutions are calculated as
with
.		 Flouri 2016
G.DI1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Curve descriptive processes

General processes applied to a given data set, e.g. processes to derive descriptive quantities are defined in this group.

Code OSIPI name Alternative names Notation Description Reference
G.CD1.001 Calculate value at data grid point -- Calcf(xi) This process returns the data value f(xi) at the data grid point xi.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
i (Q.GE1.003)
Output:
f(xi) (Q.CD1.001)		 --
G.CD1.002 Calculate maximum of data -- Calcfmax This process returns the maximum data value fmax .
Input:
Data (Q.GE1.002)
Output:
fmax (Q.CD1.002)		 --
G.CD1.003 Calculate data grid point of maximum data value -- Calcxmax This process returns the data grid point at which the maximum of a given data set occurs.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)]
Output:
xmax (Q.CD1.003)		 --
G.CD1.004 Calculate minimum of data -- Calcfmin This process returns the minimum data value fmin .
Input:
Data (Q.GE1.002)
Output:
fmin (Q.CD1.004)		 --
G.CD1.005 Calculate data grid point of minimum data value -- Calcxmin This process returns the data grid point at which the minimum of a given data set occurs.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)]
Output:
xmin (Q.CD1.005)		 --
G.CD1.006 Calculate value of final data point -- Calcffin This process returns the value of the data at the final data grid point.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)]
Output:
ffin (Q.CD1.006)		 --
G.CD1.007 Calculate final data grid point -- Calcxfin This process returns the last data grid point of a given data grid.
Input:
Data grid (Q.GE1.001)
Output:
xfin (Q.CD1.007)		 --
G.CD1.008 Calculate maximum deviation from baseline -- This process returns the maximum absolute deviation of a given data set and baseline.
Input:
Data (Q.GE1.002),
Baseline value (Q.BL1.001)
Output:
(Q.CD1.008)		 --
G.CD1.009 Derivative at data grid point -- This process returns the value of the derivative of a given data set at the data grid point xi.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
i (Q.GE1.003)
Output:
(Q.CD1.009)
--
G.CD1.010 Calculate time to peak -- CalcTTP This process returns the time to peak for a given bolus arrival time and data grid point of maximum value.
Input:
xmax (Q.CD1.003),
BAT (Q.BA1.001)
Output:
TTP (Q.CD1.010)		 --
G.CD1.011 Calculate wash-in slope -- CalcWIS This process returns the wash-in-slope for a given baseline, maximum value and time to peak of a data set.
Input:
fmax (Q.CD1.002),
fBL (Q.BL1.001),
TTP (Q.CD1.010)
Output :
WIS (Q.CD1.011)		 --
G.CD1.012 Calculate wash-out slope -- CalcWOS This process returns the wash-out-slope for a given maximum value, final data value and the data grid points of the maximum and final data value of a data set.
Input:
fmax (Q.CD1.002),
ffin (Q.CD1.006),
xmax (Q.CD1.003),
xfin (Q.CD1.007)
Output:
WOS (Q.CD1.012)		 --
G.CD1.013 Calculate area under curve -- This process returns the integral of data on a data grid in between a range of data grid points xstart and xend.
Input:
[Data (Q.GE1.002), Data grid (Q.GE1.001)],
[xstart (Q.GE1.013), xend(Q.GE1.014)]
Output:
AUCxstart,xend (Q.CD1.013)		 --
G.CD1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Segmentation

Processes related to segmentation are listed in this section.

Code OSIPI name Alternative names Notation Description Reference
G.SE1.001 Create binary mask -- -- This process creates a binary segmentation mask on a given data set using a specified segmentation method.
Input:
Data (Q.GE1.002),
Segmentation method (select from segmentation methods)
Output:
Binary mask (Q.SE1.001)		 --
G.SE1.002 Apply binary mask -- -- This process masks a given data set with a given mask.
Input:
Data (Q.GE1.002),
Binary mask (Q.SE1.001)
Output:
Data (Q.GE1.002)		 --
G.SE1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Segmentation methods

Code OSIPI name Alternative names Notation Description Reference
G.SE2.001 Freehand -- -- Manual freehand drawing of contours.
Input:
Data (Q.GE1.002)
Output:
Binary mask (Q.SE1.001)		 --
G.SE2.002 Threshold -- -- This method selects all input data with values in a specified range between lower and upper threshold.
Input:
Data (Q.GE1.002),
Lower threshold (Q.GE1.010),
Upper threshold (Q.GE1.011)
Output:
Binary mask (Q.SE1.001)		 --
G.SE2.003 Region growing -- -- This method grows a region from selected seeds with values between the lower and upper value threshold in the neighborhood of the seeds.
Input:
Data (Q.GE1.002),
Seeds (Q.SE1.004),
Lower threshold (Q.GE1.010),
Upper threshold (Q.GE1.011)
Output:
Binary mask (Q.SE1.001)		 --
G.SE2.004 k-means clustering -- -- This method partitions the input data in a number of clusters using the K-means clustering algorithm and selects the cluster with the ith index as binary mask.
Input:
Data (Q.GE1.002),
Number of k-Means clusters (Q.SE1.005),
i (Q.GE1.003)
Output:
Binary mask (Q.SE1.001)		 --
G.SE2.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --

Uncertainty estimation

This section is currently work in progress

Code OSIPI name Alternative names Notation Description Reference
-- -- -- -- -- --

Averaging

Code OSIPI name Alternative names Notation Description Reference
G.AV1.001 Calculate Average -- CalcAv This process returns the average of input data according to a specified averaging method.
Input:
Data (Q.GE1.002), Averaging method (select from uncertainty estimation and statistics processes e.g. (G.US1.001) )
Output:
Data (Q.GE1.002)		 --
G.AV1.999 Method not listed -- -- This is a custom free-text item, which can be used if a method of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. --