4.3.8 Wavelet

Summary

Wavelet transforms are useful for analyzing signals which exhibit sudden changes of phase and frequency, local maxima and minima, or related parameters. Wavelet transforms have become a popular tool in time-frequency analysis, especially for analysis of non-stationary signals. OriginPro provides wavelet transform tools for both continuous and discrete transforms.

What You Will Learn

This tutorial will show you how to:

  1. Perform one-level discrete wavelet decomposition and reconstruct a signal from approximation coefficients and detail coefficients.
  2. Apply multi-level discrete wavelet decomposition.
  3. Perform continuous wavelet transform.
  4. Remove noise from signals by using wavelet transform.
  5. Perform 2D wavelet decomposition and reconstruction on matrix data.
  6. Convert an image to matrix data.
  7. Merge graph windows into one graph.

1D Wavelet Transform

Decomposition

  1. Start with a new workbook, and select menu Data: Import from File: Single ASCII... to import the data <Origin Installation Directory>\Samples\Signal Processing\Chirp Signal.dat into the worksheet.
    SP Tutorial Wavelet 1.png
  2. Highlight column B, and select menu item Analysis: Signal Processing: Wavelet: Decompose... to open the Decompose: dwt dialog.
  3. In the dialog, choose DB6 for Wavelet Type & Order, and Zero-padded for Extension Mode.
    SP Tutorial Wavelet 2.png
  4. Click OK to close the dialog box and output the approximation coefficients and the detail coefficients.
  5. Highlight columns B, C, and D (columns C and D should contain your approximation coefficients and your detail coefficients, respectively). From the menu, select Plot: Multi-panel: Stack... to open the Stack: plotstack dialog box.
  6. In this dialog, clear the Link Layers check box in the Options node. Check the Auto Preview check box at the bottom to see the preview graph in the right panel.
    SP Tutorial Wavelet 3.png
  7. Click OK to close the dialog box and create a stack graph.
    SP Tutorial Wavelet 4.png

Note: To perform level 2 (3, 4, etc.) discrete wavelet decomposition, repeat steps 2 to 4 on the approximation coefficients (column C, in our example). Multi-scale wavelet decomposition (available in OriginPro) is demonstrated below.

Reconstruction

Reconstruction is the inverse operation of decomposition and this example will recover the signal from the result generated above.

  1. Highlight columns C and D (your approximation coefficients and detail coefficients, from above).
  2. Choose menu item Analysis: Signal Processing: Wavelet: Reconstruction... to bring up the Reconstruction: idwt dialog box.
  3. To reconstruct the signal, we need to specify the same Wavelet Type, and Boundary, so set them to DB6 and Zero-padded, respectively.
    SP Tutorial Wavelet 5.png
  4. Click OK and the recovered signal is output to column E of the worksheet.
  5. Press CTRL and select columns B and E. From the menu, choose Plot: Line: Line to make a graph with these two columns of data.
    SP Tutorial Wavelet 6.png
  6. We can see from the resulting graph that the original signal and the recovered signal overlay each other.

Multi-Scale Wavelet Decomposition

  1. Start with a new workbook, and then import the same data used in the Decomposition section above.
  2. Highlight column B and select menu item Analysis: Signal Processing: Wavelet: Multi-Scale DWT... to open the Multi-Scale DWT: mdwt dialog box.
  3. In the dialog, change Extension Mode to Zero-padded, and Decomposition Level to 3. Click the arrow button beside Multiple Coeff Data and select [<input>]<input> from the menu.
    SP Tutorial Wavelet 7.png
  4. Click OK to obtain 3rd level discrete wavelet decomposition results. The coefficients are stored in the same worksheet as the input signal.
  5. Select all columns in the worksheet, then choose menu Plot: Multi-panel: Stack... to open the Stack: plotstack dialog box. Click the arrow button to the right of Dialog Theme and select System Default. Check the Auto Preview check box at the bottom of the dialog box to preview the graph.
    SP Tutorial Wavelet 8.png
  6. Click OK. The resulting graph is shown below.
    SP Tutorial Wavelet 9.png

Continuous Wavelet Transform

  1. Start with a new workbook with two empty columns (columns A and B) in a single worksheet.
  2. Select column A and then right-click and choose Set Column Values... from the shortcut menu. This opens the Set Values dialog box. In the text box, enter data(0, 1, 1/1023) and click Apply. This fills column A with data.
    SP Tutorial Wavelet 10.png
  3. Click the >> button to select column B. Enter sin(1.5/(0.25-col(A)))+cos(0.2/(0.2-col(A))) and click OK.
    SP Tutorial Wavelet 11.png
  4. We can make a plot to see how the data looks. With column B selected, choose Plot: Line: Line from the menu.
    SP Tutorial Wavelet 13.png
  5. With the worksheet active, add a new column to the worksheet by selecting Column: Add New Columns... from the menu. In the dialog box, accept the default value of 1 and click OK.
    SP Tutorial Wavelet 12.png
  6. Column C is added to the worksheet. Select it, right-click and choose Set Column Values from the shortcut menu. In the text box, enter data(1, 512) and click OK.
    SP Tutorial Wavelet 14.png
  7. Select both columns B and C, then from the menu, select Analysis: Signal Processing: Wavelet: Continuous Wavelet... to open Continuous Wavelet: cwt dialog box.
  8. In the dialog, set Wavelet Type to MexHat, and check Coefficient Matrix.
    SP Tutorial Wavelet 15.png
  9. Click OK button to exit dialog.
  10. The coefficients are output to a new worksheet and a matrix and a contour plot is added. Double-click on the plot to open the Plot Details dialog box. Select the Colormap tab and click on the Fill... heading. The Fill dialog box opens with color contols. Click Load Palette, and choose Pumpkin Patch from the Select Palette menu. Click OK to return to the Plot Details dialog box.
    SP Tutorial Wavelet 16.png
  11. Click OK to change the color of the contour plot. It should now look like the graph below.
    SP Tutorial Wavelet 17.png

Noise Removal by Wavelet Transform

  1. Start with a new workbook and from the menu, select Data: Import from File: Single ASCII.... Import the file <Origin Installation Directory>\Samples\Signal Processing\Signal with Shot Noise.dat .
    SP Tutorial Wavelet 18.png
  2. We can see from the Sparklines in the image above that there is shot noise in the signal. Highlight column B and select menu Analysis: Signal Processing: Wavelet: Denoise... to bring up the Denoise: wtdenoise dialog box.
  3. Check Auto Preview to preview your result. Set Wavelet Type to DB9, Extension Mode to Zero-padded, Threshold Type to sqtwolog, and Thresholding Level to 7.
    SP Tutorial Wavelet 19.png
  4. Click OK to remove the shot noise from the signal. Results are output to worksheet columns C and D.
    SP Tutorial Wavelet 20.png
  5. Highlight all columns in the worksheet, then choose menu item Plot: Line: Line to make a graph with the original signal and the cleaned-up signal. We see from the graph below that the shot noise has been removed.
    SP Tutorial Wavelet 21.png

2D Wavelet Transform

2D Wavelet Decomposition

  1. Start with a new matrix book. From the menu, choose Data: Import from File: Image to Matrix... and import the image, <Origin Installation Directory>\Samples\Image Processing and Analysis\Car.bmp.
    SP Tutorial Wavelet 22.png
  2. To begin, we need to convert the image into data. To do this, select menu item Image: Conversion: Convert to Data. The Convert to Data: img2m opens. Set Type to byte(1).
    SP Tutorial Wavelet 23.png
  3. Click OK to obtain the converted matrix data.
  4. With the converted matrix active, select menu Analysis: Signal Processing: Wavelet: 2D Decompose... and open the 2D Decompose: dwt2 dialog box.
    SP Tutorial Wavelet 24.png
  5. Accept the default settings and click OK. This performs 2D wavelet decomposition on the matrix data. A matrix book with four matrix sheets (CA, CH, CV, and CD) is generated. CA, CH, CV, and CD contain approximation coefficients, horizontal detail coefficients, vertical detail coefficients, and diagonal detail coefficients respectively. You can change the view mode to image mode by choosing View: Image Mode from the menu.
    SP Tutorial Wavelet 25.png
  6. With the CA matrix sheet active, make an image plot by selecting Plot: Image Plot from the menu. Repeat for CH, CV, and CD matrix sheets.
  7. Double click on graph of them to open the Plot Details dialog, in the left panel, activate Layer1, and then go to the Size/Speed tab in the right panel, and uncheck Matrix data, maximum points per dimension. Click OK. select the axes related objects (including axis, label) in the graph and press the DELETE key to delete them.