Section 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 \(h(x) = f(x) \ast g(x)\) 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 | \(\lVert f(\phi, \theta; x) - y(x) \rVert_{2}^{2}\) , where \(f\) is a forward model describing the data, \(x\) is the data grid, \(y(x)\) is the measured data and \(\lVert \rVert_2\) 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), \(\phi\) (Q.OP1.001), \(\theta\) (Q.OP1.002) Output: Cost value (Q.OP1.004) |
-- |
| G.OP2.002 | Linear least squares | -- | LLS | \(\lVert A\phi - y(x) \rVert_2^2\) , where \(x\) is the data grid, \(y(x)\) is the measured data and \(\lVert \rVert_2\) is the L2-norm. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(\phi\) (Q.OP1.001), A (Q.OP1.012) Output: Cost value (Q.OP1.004) |
-- |
| G.OP2.003 | Standard-Form Tikhonov | -- | SFT | \(\lVert A\phi - y(x) \rVert_2^2 + \lambda^2 \lVert Ix \rVert_2\) , where \(x\) is the data grid, \(y(x)\) is the measured data , \(\lVert \rVert_2\) is the L2-norm and \(L\) is the identity matrix (same dimensions as \(A\)). Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(\phi\) (Q.OP1.001), A (Q.OP1.012), \(\lambda\) (Q.OP1.013) Output: Cost value (Q.OP1.004) |
-- |
| G.OP2.004 | Generalized cross validation | -- | GCV | \(\frac{\lVert A\phi_\lambda - y(x) \rVert_2^2}{trace\left( I-AA_\lambda^\zeta \right)^2}\), where \(x\) is the data grid, \(y(x)\) is the measured data, \(\lVert \rVert_2\) is the L2-norm, \(I\) is the identity matrix of the same dimensions as \(A\), \(\phi_\lambda\) is the solution of the matrix equation obtained from the SVD for a certain regularization parameter \(\lambda\) and \(A^\zeta_\lambda\) is defined by the relationship \(\phi_\lambda=A_{\lambda}^{\zeta}y(x)\). Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(\phi\) (Q.OP1.001), A (Q.OP1.012), \(\lambda\) (Q.OP1.013) Output: Cost value (Q.OP1.004) |
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| G.OP2.005 | L-curve | -- | LC | \(\frac{\hat{\rho^\prime}\hat{\eta^{\prime\prime}} - \hat{\rho^{\prime\prime}}\hat{\eta^\prime}}{\left( (\hat{\eta^\prime})^2 + (\hat{\rho^\prime})^2 \right)^{3/2}}\), where \(\phi_\lambda\) is the solution of the matrix equation obtained from the SVD for a certain regularization parameter \(\lambda\), \(\rho(\lambda) = \lVert A\phi_\lambda - y(x) \rVert_2\) and \(\eta(\lambda) = \lVert \phi_\lambda \rVert_2\), \(\hat{\rho} = ln(\rho)\), \(\hat{\eta} = ln(\eta)\) and \(\prime\) and \(\prime\prime\) are the derivatives with respect to \(\lambda\). Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(\phi\) (Q.OP1.001), A (Q.OP1.012), \(\lambda\) (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 \(\lambda\) , rather than a determined value is assumed. Input: \(\lambda_{fixed}\) (Q.OP1.015) Output: \(\lambda\) (Q.OP1.013) |
-- |
| G.OP3.002 | Generalized Cross Validation | -- | GCV | \(\lambda\) is determined by minimizing the generalized cross validation cost function with respect to \(\lambda\). Input: Optimizer (select from optimizers) with a GCV cost function (G.OP2.004) and \(\phi\) (Q.OP1.001) = \(\lambda\)(Q.OP1.013) Output: \(\lambda\)(Q.OP1.013) |
-- |
| G.OP3.003 | L-Curve criterion | -- | LCC | \(\lambda\) is determined by minimizing the L-curve cost function with respect to \(\lambda\). Input: Optimizer (select from optimizers) with a L-curve cost function (G.OP2.005) and \(\phi\) (Q.OP1.001) = \(\lambda\)(Q.OP1.013). Output: \(\lambda\)(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 \(h(x) = f(x) \ast g(x)\) 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 \(f\): \(f_n = f(x_n)\), \(g_i^- = (2g_i + g_{i-1})/6\), \(g_i^+ = (2g_i + g_{i+1})/6\) and \(g_i^\pm = g_i^+ + g_i^-\).
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 | -- | Calc \(f(x_i)\) | This process returns the data value \(f(x_i)\) at the data grid point \(x_i\). Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], i (Q.GE1.003) Output: \(f(x_i)\) (Q.CD1.001) |
-- |
| G.CD1.002 | Calculate maximum of data | -- | Calc \(f_{max}\) | This process returns the maximum data value \(f_{max}\) . Input: Data (Q.GE1.002) Output: \(f_{max}\) (Q.CD1.002) |
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| G.CD1.003 | Calculate data grid point of maximum data value | -- | Calc \(x_{max}\) | 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: \(x_{max}\) (Q.CD1.003) |
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| G.CD1.004 | Calculate minimum of data | -- | Calc \(f_{min}\) | This process returns the minimum data value \(f_{min}\) . Input: Data (Q.GE1.002) Output: \(f_{min}\) (Q.CD1.004) |
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| G.CD1.005 | Calculate data grid point of minimum data value | -- | Calc \(x_{min}\) | 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: \(x_{min}\) (Q.CD1.005) |
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| G.CD1.006 | Calculate value of final data point | -- | Calc \(f_{fin}\) | 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: \(f_{fin}\) (Q.CD1.006) |
-- |
| G.CD1.007 | Calculate final data grid point | -- | Calc \(x_{fin}\) | This process returns the last data grid point of a given data grid. Input: Data grid (Q.GE1.001) Output: \(x_{fin}\) (Q.CD1.007) |
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| G.CD1.008 | Calculate maximum deviation from baseline | -- | Calc \(\Delta f_{BL,max}\) | 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: \(\Delta f_{BL,max}\) (Q.CD1.008) |
-- |
| G.CD1.009 | Derivative at data grid point | -- | Calc \(\frac{df(x_i)}{dx}\) | 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: \(\frac{df(x_i)}{dx}\) (Q.CD1.009) |
-- |
| G.CD1.010 | Calculate time to peak | -- | Calc \(TTP\) | This process returns the time to peak for a given bolus arrival time and data grid point of maximum value. Input: \(x_{max}\) (Q.CD1.003), \(BAT\) (Q.BA1.001) Output: \(TTP\) (Q.CD1.010) |
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| G.CD1.011 | Calculate wash-in slope | -- | Calc \(WIS\) | This process returns the wash-in-slope for a given baseline, maximum value and time to peak of a data set. Input: \(f_{max}\) (Q.CD1.002), \(f_BL\) (Q.BL1.001), \(TTP\) (Q.CD1.010) Output : \(WIS\) (Q.CD1.011) |
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| G.CD1.012 | Calculate wash-out slope | -- | Calc \(WOS\) | 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: \(f_{max}\) (Q.CD1.002), \(f_{fin}\) (Q.CD1.006), \(x_{max}\) (Q.CD1.003), \(x_{fin}\) (Q.CD1.007) Output: \(WOS\) (Q.CD1.012) |
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| G.CD1.013 | Calculate area under curve | -- | Calc \(AUC_{x_{start}, x_{end}}\) | This process returns the integral of data on a data grid in between a range of data grid points \(x_{start}\) and \(x_{end}\). Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], [\(x_{start}\) (Q.GE1.013), \(x_{end}\) (Q.GE1.014)] Output: \(AUC_{x_{start}, x_{end}}\) (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
Uncertainty estimation and statistics processes
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| G.US1.001 | Calculate arithmetic mean (sample) | -- | Calc \(\bar{x}\) | This process returns the arithmetic mean from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(n\) (Q.GE1.015) Output \(\bar{x}\) (Q.US1.001) |
-- |
| G.US1.002 | Calculate geometric mean (sample) | -- | Calc \(\bar{x}_{geom}\) | This process returns the geometric mean from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(n\) (Q.GE1.015) Output \(\bar{x}_{geom}\) (Q.US1.002) |
-- |
| G.US1.003 | Calculate variance (sample) | -- | Calc \(s^2\) | This process returns the variance from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(n\) (Q.GE1.015) Output \(s^2\) (Q.US1.003) |
-- |
| G.US1.004 | Calculate standard deviation (sample) | -- | Calc \(s\) | This process returns the standard deviation from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(n\) (Q.GE1.015) Output \(s\) (Q.US1.004) |
-- |
| G.US1.005 | Calculate median value (sample) | -- | Calc \(x_{median}\) | This process returns the median from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], Output \(x_{median}\) (Q.US1.005) |
-- |
| G.US1.006 | Calculate coefficient of variation (sample) | -- | Calc \(CV\) | This process returns the coefficient of variation from a given data sample. Input: \(s\) (Q.US1.004), \(\bar{x}\) (Q.US1.001) Output CV (Q.US1.006) |
-- |
| G.US1.007 | Calculate standard error | -- | Calc \(SEM\) | This process returns the standard error of the mean value (sample) from a given data sample. Input: \(\bar{x}\) (Q.US1.001), \(n\) (Q.GE1.015) Output SEM (Q.US1.008) |
-- |
| G.US1.008 | Calculate range | -- | Calc \(r\) | This process returns the range from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)] Output \(r\) (Q.US1.013) |
-- |
| G.US1.009 | Calculate interquartile range | -- | Calc \(IQR\) | This process returns the interquartile range from a given data sample. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)] Output \(IQR\) (Q.US1.014) |
-- |
| G.US1.010 | Calculate \(\gamma\)-quantile | -- | Calc \(f_\gamma\) | This process returns the gamma-quantile from a given data sample and a given gamma-quantile value. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(\gamma\) (Q.US1.015) Output \(f_\gamma\) (Q.US1.016) |
-- |
| G.US1.011 | Calculate percentile range | -- | Calc \(PR\) | This process returns the percentile range from a given data sample and two given gamma-quantile values. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], \(\gamma_1\) (Q.US1.015) \(\gamma_2\) (Q.US1.015) Output \(PR\) (Q.US1.017) |
-- |
| G.US1.012 | Calculate confidence interval | -- | Calc \(CI\) | This process returns the confidence interval from a given data sample for a given confidence interval probability. Input: [Data (Q.GE1.002), Data grid (Q.GE1.001)], SEM (Q.US1.008) \(df\) (Q.US1.025) \(\alpha\) (Q.US1.018) Output \(CI\) (Q.US1.019) |
-- |
| G.US1.013 | Calculate signal to noise ratio | -- | Calc \(SNR\) | This process returns the signal to noise ratio. Input: \(s\) (Q.US1.004), \(\bar{x}\) (Q.US1.001) Output \(SNR\) (Q.US1.020) |
-- |
| G.US1.014 | Calculate contrast to noise ratio | -- | Calc \(SNR\) | This process returns the contrast to noise ratio. | -- |
| -- | -- | -- | -- | The lexicon needs to be updated on google docs! | -- |
| G.US.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 the doi of a literature reference and request the item to be added to the lexicon for future usage. | -- |
Averaging¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| G.AV1.001 | Calculate Average | -- | CalcAverage | 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. Calc \(\bar{x}\) (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. | -- |