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Experiment with the fractional spline wavelet transform in 2D or 3D and perform simple processing in the transform domain (denoising, filtering, or user-specified). See the user manual.

Performs a non-redundant, separable fractional wavelet transform in 2D. The wavelet basis is specified within the family of fractional splines, which are the only wavelets to date that are tunable in a continuous fashion. You can adjust and visualize the basis functions and apply the wavelet transform to your images.

Open an image

The Wavelet Transform plug-in for ImageJ computes the 2D wavelet transform of an image.

• Open an image or image sequence using the standard menu of ImageJ: File->Open...
• Click on the image frame to put it in focus and then start the plug-in by selecting: Plugins->Fractional Spline Wavelet->Wavelet Transform
• Once the plug-in is started, all operations are done on the selected image or image sequence

Functioning of the Wavelet Transform plug-in

• The plug-in operates in two modes:
  1. Analysis: computes the wavelet transform of an image
  2. Synthesis: reconstructs the image from its wavelet transform
• Any processing using ImageJ can be done on the wavelet coefficients before reconstruction since the plug-in window is not modal.

Step 1 - Discrete Wavelet Transform: Analysis side

The complete family of (bi)-orthogonal spline wavelets with fractional degree (α) and arbitrary shift (τ) is available in this plug-in

• Filter: orthonormal (same analysis and synthesis filters), biorthogonal (B-spline analysis or synthesis and dual-spline synthesis or analysis);
• Iterations: the number or scales;
• Degree: the fractional degree of the spline. The Haar transform is obtained with degree=0.0 and shift=0.50;
• Shift: the shift of the spline function (0.0 results into symmetrical filters).

For more information on these transform parameters, refer to wavelet properties.

The scaling function or the wavelet are displayed in the central part of the plug-in window. In this way, the user can visualize the effects of changing the transform parameters.

When the image dimensions are not powers of two, an "Image size control" window allows you to crop or extend the original image dimensions. If the size is not a power of two, the algorithm still works but the processing will be very slow.

Optionally - Processing

You can use any ImageJ manipulations to process the wavelet coefficients

Step 2 - Inverse Discrete Wavelet Transform: Synthesis side

The final step reconstructs the (processed) image or image sequence using the same settings as the analysis side

Open an image or image sequence

The Wavelet Module plug-in for ImageJ works both with images 2D and with image stacks.

• Open an image or image sequence using the standard menu of ImageJ: File->Open...
• Click on the image frame to put it in focus and then start the plug-in by selecting: Plugins->Fractional Spline Wavelet->Wavelet Module
• Once, the plug-in is started and all operations will be done on the image or image sequence that was last in focus

Functioning of the Wavelet Module 3D plug-in

• The plug-in operates in three steps:
  1. Analysis: computes the forward discrete wavelet transform for orthogonal and bi-orthogonal spline wavelets
  2. Processing: performs an operation in the wavelet domain
  3. Synthesis: computes the inverse discrete wavelet transform
  
• For each step, one can choose to produce an intermediate output by enabling the checkbox "Output"
• The steps can be done all at once by clicking "Run all steps", or "Run step : xxx"

When the image dimensions are not powers of two, an "Image size control" window allows you to crop or extend the original image dimensions. If the size is not a power of two, the algorithm still works but the processing will be very slow.

Step 1 - Discrete Wavelet Transform: Analysis side

The complete family of (bi)-orthogonal spline wavelets with fractional degree (α) and arbitrary shift (τ) is available in this plug-in

• Filter: orthonormal (filters for analysis and synthesis are equal), biorthogonal (B-spline at the analysis or the synthesis side, the dual the other way around);
• Iterations: can be selected independently for each of the three dimensions; if one selects 0, no transform in this dimension will be done;
• Degree: the fractional degree of the spline can be selected, for Haar wavelets select degree 0.0 and shift 0.50;
• Shift: the shift of the spline function (0.0 results into symmetrical filters).

For more information on these transform parameters, refer to wavelet properties.

To explore the family of wavelets and scaling functions available, we refer to the Wavelet Transform plug-in.

Step 2 - Processing

To illustrate the potential of wavelet transform domain processing, we have included several versions of popular algorithms:

• Bypass: no processing
• Hard threshold: puts coefficients whose absolute value is below the "Threshold" to zero
• Soft threshold: same as "Hard threshold", but compensates the offset at the threshold position
• Enhancement: enhances the lowest subband by "Factor"
• Rescale: rescales the coefficients of each subband to make them visually easier to interpret

Step 3 - Inverse Discrete Wavelet Transform: Synthesis side

The final step reconstructs the (processed) image or image sequence using the same settings as the analysis side

Adding functionality to the Wavelet Module

You can program your own processing algorithm (that will be executed when you select "Your processing here" in the dialog box) by modifying and compiling the source file CoefProcessing.java. More details can be found in this file.