pylops.CausalIntegration¶
-
class
pylops.
CausalIntegration
(N, dims=None, dir=-1, sampling=1, halfcurrent=True, dtype='float64', kind='full', removefirst=False)[source]¶ Causal integration.
Apply causal integration to a multi-dimensional array along
dir
axis.Parameters: - N :
int
Number of samples in model.
- dims :
list
, optional Number of samples for each dimension (
None
if only one dimension is available)- dir :
int
, optional Direction along which smoothing is applied.
- sampling :
float
, optional Sampling step
dx
.- halfcurrent :
bool
, optional Add half of current value (
True
) or the entire value (False
). This will be deprecated in v2.0.0, use instead kind=half to obtain the same behaviour.- dtype :
str
, optional Type of elements in input array.
- kind :
str
, optional Integration kind (
full
,half
, ortrapezoidal
).- removefirst :
bool
, optional Remove first sample (
True
) or not (False
).
Notes
The CausalIntegration operator applies a causal integration to any chosen direction of a multi-dimensional array.
For simplicity, given a one dimensional array, the causal integration is:
\[y(t) = \int\limits_{-\infty}^t x(\tau) \,\mathrm{d}\tau\]which can be discretised as :
\[y[i] = \sum_{j=0}^i x[j] \,\Delta t\]or
\[y[i] = \left(\sum_{j=0}^{i-1} x[j] + 0.5x[i]\right) \,\Delta t\]or
\[y[i] = \left(\sum_{j=1}^{i-1} x[j] + 0.5x[0] + 0.5x[i]\right) \,\Delta t\]where \(\Delta t\) is the
sampling
interval, and assuming the signal is zero before sample \(j=0\). In our implementation, the choice to add \(x[i]\) or \(0.5x[i]\) is made by selectingkind=full
orkind=half
, respectively. The choice to add \(0.5x[i]\) and \(0.5x[0]\) instead of made by selecting thekind=trapezoidal
.Note that the causal integral of a signal will depend, up to a constant, on causal start of the signal. For example if \(x(\tau) = t^2\) the resulting indefinite integration is:
\[y(t) = \int \tau^2 \,\mathrm{d}\tau = \frac{t^3}{3} + C\]However, if we apply a first derivative to \(y\) always obtain:
\[x(t) = \frac{\mathrm{d}y}{\mathrm{d}t} = t^2\]no matter the choice of \(C\).
Attributes: Methods
__init__
(N[, dims, dir, sampling, …])Initialize this LinearOperator. adjoint
()Hermitian adjoint. apply_columns
(cols)Apply subset of columns of operator cond
([uselobpcg])Condition number of linear operator. conj
()Complex conjugate operator div
(y[, niter, densesolver])Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\). dot
(x)Matrix-matrix or matrix-vector multiplication. eigs
([neigs, symmetric, niter, uselobpcg])Most significant eigenvalues of linear operator. matmat
(X)Matrix-matrix multiplication. matvec
(x)Matrix-vector multiplication. rmatmat
(X)Matrix-matrix multiplication. rmatvec
(x)Adjoint matrix-vector multiplication. todense
([backend])Return dense matrix. toimag
([forw, adj])Imag operator toreal
([forw, adj])Real operator tosparse
()Return sparse matrix. trace
([neval, method, backend])Trace of linear operator. transpose
()Transpose this linear operator. - N :