CWT Denoising

CWT Denoising is a specialized continuous wavelet transform (CWT) filtration procedure that sets a signal-strength threshold in the time-scale (wavelet) domain, retaining coefficients above the threshold and zeroing those below. This option is used exclusively to remove noise. Because the wavelet's scale adapts locally to peak width, it is often the most effective denoiser when a record contains peaks or features of very different widths in the same trace, a case where a single Fourier cutoff or a single eigendecomposition order can under-serve either the broad or the narrow features. For records whose features share a similar width or whose noise is simple, Fourier Denoising or Eigendecomposition Denoising can be equally effective and faster.

Uniformly spaced data are needed so that no additional noise is introduced. This procedure does not automatically generate uniform data for the transform. If needed, the Spline Estimation or Local Regression Spline Estimation option can be used to create an interpolated data set with constant X-spacing.

The option presents the input and denoised output time domain data together, so the effect of each setting can be judged directly on the graph.

Wavelet Family

The transform can be computed with any of four mother wavelets:

· Morlet - a modulated Gaussian; the most commonly used wavelet in the signal-processing literature, offering good frequency resolution at a given time resolution.

· Paul - an asymmetric, better time-localized alternative to the Morlet.

· Gaussian Derivative - the derivative-of-Gaussian family; order 2 is the classic "Mexican hat" wavelet.

· Morse - a two-parameter family (order and symmetry) that spans a wide range of time-frequency trade-offs, including close approximations to the Morlet at high order.

Each family exposes one or two parameters that set its order or shape (for the Morse wavelet, both an order and a second shape parameter are adjustable). A Complex option computes the transform with the analytic (complex-valued) form of the wavelet rather than the real-valued form; the complex form is generally the more robust choice and is recommended unless you have a specific reason to inspect the real-valued coefficients directly.

Threshold

The main task in this procedure is to set a threshold, expressed in normalized dB, below which the wavelet coefficients are treated as noise and zeroed. The threshold is applied to the whole time-scale surface as a coherent region, not coefficient by coefficent; this is a requirement of the transform, not a stylistic choice, since zeroing scattered individual coefficients degrades the reconstruction rather than improving it.

An AI (automatic) threshold option estimates the threshold directly from a handful of blind statistics of the incoming record itself (its noise level, its noise color, its length, and the correlation between neighboring points), with no need to know the noise level or the signal-to-noise ratio in advance. This automatic estimate tracks the true optimal threshold closely across a very wide range of noise levels and noise colors, including noise that has already been shaped by an earlier instrument filter, and is the recommended starting point for most records. You can also enter the threshold manually, or fine-tune the AI estimate, if the automatic result under- or over-smooths a particular record.

For records that are already very clean, the procedure may find that no denoising threshold usefully improves the record, since at very high signal-to-noise the reconstruction's own numerical floor can compete with the small remaining noise. In this situation leaving the record unfiltered is the correct choice, and the AI option is designed to recognize it rather than force a threshold onto clean data.

A Mode setting selects how the threshold is applied to form the reconstruction:

· Band (soft) - the default; coefficients are faded out smoothly across a narrow transition band at the threshold rather than zeroed abruptly, avoiding an artificial ripple at the cutoff.

· Band (hard) - coefficients are zeroed abruptly at the threshold.

· Contour - the reconstruction follows a fixed-depth contour shell below the surface's peak power, rather than an absolute dB value.

· Ridges - reconstructs only along the ridge lines (the scale at which each feature's power is locally maximal), which is a strong denoiser for records containing distinct, separable peaks.

· Components - like Ridges, but keeps each detected ridge as its own separate curve (up to 24 components), in addition to the union of all of them. This lets you inspect or export the reconstruction contributed by one individual peak or feature at a time, the wavelet-domain equivalent of reconstructing a single retained component in Fourier Denoising or a single eigenmode in Eigendecomposition Denoising.

Reconstruction

The retained coefficients above the threshold are reconstructed into the denoised output curve by summing the real part of the surviving coefficients across scale, then applying a two-parameter affine rescale fitted to the observed data. This makes the reconstruction numerically exact when the whole surface is retained, and well-behaved as the threshold removes progressively more of the surface. Each surviving ridge or component can be reconstructed this same way on its own, letting individual peaks or features be isolated, compared, or exported independently of the full denoised curve.

Estimated Noise Reduction

TableCurve Studio offers a robust noise estimation procedure that may be of some value for low-frequency signals. A cubic polynomial interpolation is made for each point using the two points to the left and the two to the right (excluding the current point). The difference between the interpolated and data values is used to generate a measure of the white noise present in the data. This assumes that the data can be locally characterized by a smooth cubic interpolant. Also, the signal component(s) should exist only in the lower quarter of the Nyquist range. If a high frequency signal component is present, these estimates of noise will be invalid.

The In value reports the estimated white noise in the incoming data, the Out value the estimated white noise for the filtered signal. The percent is given as the amount of estimated noise remaining after filtration.

Power Reduction

In this section the In value reports the TISA (time-integral squared amplitude) power in the incoming data. The Out value is the TISA power for the filtered signal. The percent is given as the amount of power remaining after filtration. For most S/N ratios, the reduction in power should be minimal. These values can alert you when signal components are being discarded.

Correlation Coefficient

The r-squared correlation coefficient should also remain high. An r squared of 1 is a perfect correlation while a value of 0 means the filtered and unfiltered signals are uncorrelated. Low r squared values are also indicative of signal components being lost in the thresholding.

List

The List Data option lists the index, time, and output signal in a three column table. The listing uses the TableCurve Studio text viewer facility.

Copy

The Copy Data to Clipboard option copies the time and output signal values to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors).

Save

The Save Data to Disk option writes the time and output values to a supported file format, including ASCII and Excel (xlsx/xls).

Production Facility

The Automation facility allows unattended processing of large numbers of data sets. The data sets can come from a text/CSV file or an Excel workbook. The graphs and reports can be exported to an RTF file or streamed to Microsoft Word, while the processed data can be exported to an Excel (xlsx) workbook.

The Reset button restores the data to its state when first entering the procedure.

Residuals

The Residuals button is used to graphically display the difference between the original and filtered data.

Numeric Summary

The Numeric Summary offers a full CWT report, including the wavelet and threshold settings used.

MS Word/RTF Export

The MS Word/RTF File Export option is used to save the current graphs and reports to either an MS Word file or a portable RTF (Rich Text Format) file. The graphs are inserted into the file as Windows metafiles.

Automatic Update

The List, Residuals, and Numeric Summary windows are automatically updated when any change is made to the algorithm's settings.

Updating the Data Table

When exiting this procedure with the OK button, an option will be presented to update TableCurve Studio's main data table with the denoised data.

If Background Thread Fitting is active, the fitting will be initiated as soon as the filtering is accepted and the data are rescanned.