MimWaveletTransform() uses wavelets to decompose or recompose the specified source (typically image data). To remove noise, MimWaveletDenoise() also uses wavelets.

Wavelets refer to mathematical algorithms that process data in both the space and frequency domain, offering a compromise between each. By using wavelets, MimWaveletTransform() results can therefore represent space and frequency. This makes MimWaveletTransform() especially effective at filtering images. It is typically useful for signal coding, signal denoising, data compression, and pattern recognition.

Transformations conventionally represent either space or frequency data. For example, calling MimTransform() with an image can produce a precise frequency result, but it lacks spatial information. Drawing such a result looks like a signal, with no resemblance to the original source image. Drawing a wavelet transformation result resembles the original source image, since some spatial information is kept. The following illustration shows a MimWaveletTransform() and a MimTransform() (FFT) result of a source image representing several lead stripes. You can tell which is the wavelet transformation, since it resembles the source.

Numerous types of wavelet results are available. The one above represents a high-frequency wavelet transformation (vertical direction). For more information about wavelet results, see the Results and drawings subsection of this section.

Note that the uncertainty principal in mathematics proves that representing data with the same precision in both space and frequency impossible. Wavelets are recognized as offering an effective balance between these two domains.

Both MimWaveletTransform() and MimWaveletDenoise() require a wavelet image processing context, which you must allocate by calling MimAlloc() with M_WAVELET_TRANSFORM_CONTEXT or M_WAVELET_TRANSFORM_CUSTOM_CONTEXT. With M_WAVELET_TRANSFORM_CONTEXT, you have access to predefined wavelet filters. With M_WAVELET_TRANSFORM_CUSTOM_CONTEXT, you must provide your own wavelet filters. Information about wavelets applies to both contexts unless otherwise specified.

The wavelet context stores the filter and mode used to perform MimWaveletTransform() and MimWaveletDenoise().

To inquire about what is stored in a wavelet context, call MimInquire(). For example, you can inquire whether the context's filter uses complex numbers (M_TRANSFORMATION_DOMAIN), as well as retrieve the identifier of the internal buffer that contains the actual filter values (M_FILTER_..._ID).

Before calling a wavelet operation, you should also allocate a wavelet result buffer, using MimAllocResult() with M_WAVELET_TRANSFORM_RESULT. Remember to free all image processing buffers with MimFree().

To set a predefined wavelet filter (only for M_WAVELET_TRANSFORM_CONTEXT), call MimControl() with M_WAVELET_TYPE. The default is M_HAAR. This typically applies when transforming or denoising images with distinct intensity transitions. Other filter types are categorized as Daubechies (generally appropriate for intensity discontinuities) or Symmetry (a mathematically more symmetric version of Daubechies), and the number of precision points (vanishing moments) the filter employs (for example, M_DAUBECHIES_3). More vanishing moments typically result in the representation of more complex features, and longer processing time.

A further distinction between filters is whether they use complex or real coefficients. The mathematical domain of complex filters (for example, M_DAUBECHIES_3_COMPLEX) consists of wavelet coefficients that have a real part (real numbers) and an imaginary part (imaginary numbers). The mathematical domain of filters that are not complex consist of wavelet coefficients that have real numbers only. Results that include imaginary numbers are generally longer to calculate, but have more information (representing the signal's amplitude and phase).

To set custom wavelet filters (only for M_WAVELET_TRANSFORM_CUSTOM_CONTEXT), call
MimWaveletSetFilter()
and specify the filter values. The custom wavelet context must
contain your filter values before calling MimWaveletTransform()
or MimWaveletDenoise().
The custom filters that you specify must be separable 2D wavelets
that are factorized in their 1D form. For more information, see
"Stéphane Mallat. * A Wavelet Tour of Signal Processing*
. USA: Academic Press, 2008. ".

When calling the wavelet function (MimWaveletTransform() or MimWaveletDenoise()), you must set the number of levels (iterations) with which to apply the filter, up to an internally established maximum. This is the level at which calculations at subsequent levels do not result in relevant data. The maximum level depends on the wavelet function, the source, and the wavelet image processing context. If the specified level exceeds the maximum level, MIL uses the maximum level. You can determine the actual number of levels used to produce wavelet results by calling MimGetResult() with M_NUMBER_OF_LEVELS.

For more information on wavelet filters, see
"Christopher Torrence and Gilbert P. Compo. * A Practical Guide
to Wavelet Analysis* . USA: AMS, 1998. ".

MIL uses filters in dyadic (default) or undecimated mode. In dyadic mode, processing involves sampling the wavelet coefficients by a factor of 2 at each level. For example, when drawing dyadic results, they are at different sizes, at different levels. Undecimated processing does not sample. The filter values themselves are adjusted. When drawing undecimated results, they are at the same size regardless of level.

Dyadic generally applies to signal coding and data compression. Undecimated applies more to signal denoising and pattern recognition. To modify the mode, set the M_TRANSFORMATION_MODE control to M_DYADIC or M_UNDECIMATED.

Performing MimWaveletDenoise() with M_DYADIC is typically faster than M_UNDECIMATED, though the quality of denoising is usually higher with M_UNDECIMATED. In general, M_DYADIC uses less resources (processing time and memory) than M_UNDECIMATED.

For more information about dyadic and undecimated
wavelet modes, see "Stéphane Mallat. * A Wavelet Tour of Signal
Processing* . USA: Academic Press, 2008. ".

MimWaveletTransform() can perform a forward (M_FORWARD) or reverse (M_REVERSE) wavelet transformation on the specified source. Results are written in the specified destination. A forward transformation decomposes the source, which can be an image or wavelet result; the destination must be a wavelet result. Reverse transformations recompose the source, which must be a wavelet result; the destination can be an image or wavelet result.

The basic methodology for performing wavelet transformations typically requires calling MimWaveletTransform() with an image and M_FORWARD, and then using the result with your own specialized operations, typically related to signal coding, signal denoising, data compression, and pattern recognition. The last step is to transform those results using M_REVERSE and produce the modified image. Since you must use M_REVERSE with a wavelet result (not an image), you must have at some previous point used M_FORWARD with an image to produce a result.

To retrieve general types of wavelet results, call MimGetResult(). For example, to determine the actual number of transformation levels used, get the M_NUMBER_OF_LEVELS result.

To retrieve information about a specific wavelet result, use MimGetResultSingle(). To identify the individual result, you must specify the resulting wavelet coefficient and transformation level with M_DIAGONAL_LEVEL(Level), M_HORIZONTAL_LEVEL(Level), or M_VERTICAL_LEVEL(Level).

These results represent the image separated into its high-frequency components at a specific level and oriented along a specific direction. For example, if you specify M_DIAGONAL_LEVEL(3) with the M_WAVELET_COEFFICIENTS_IMAGE_ID result, you will retrieve the identifier of the internal image buffer that holds the high-frequency results oriented along the diagonal plane at the third level of transformation.

You can also call MimGetResultSingle() with M_APPROXIMATION to retrieve results about the approximation (the low frequency rendition) of the wavelet transformation at the last calculated level.

To perform drawing operations with wavelet results, call MimDraw(). The entire content of the wavelet result is drawn.

For dyadic modes (MimControl() with M_TRANSFORMATION_MODE set to M_DYADIC), drawings are in the top-right (vertical coefficient), bottom-right (diagonal coefficient), and bottom-left (horizontal coefficient) corners of the display. This drawing pattern repeats for each level calculated (MimGetResult() with M_NUMBER_OF_LEVELS). Since dyadic transformations sample wavelet coefficients by a factor of 2 per level, drawings are resized at each level. MIL also draws the approximation (the low frequency rendition) of the wavelet transformation at the last level, in the top-left corner of the display.

The following is an example of drawing dyadic results (three level transformation).

For undecimated modes (M_UNDECIMATED), drawings are in one row, per level. Each row is split into three columns, representing the horizontal (left column), diagonal (middle column), and vertical (right column) wavelet coefficients for that level. Since undecimated transformations are not sampled, drawings are all the same size, regardless of level. MIL also draws the approximation (the low frequency rendition) of the wavelet transformation at the last level, in the first column of the first row. The middle and right columns in this row are blank.

The following is an example of drawing undecimated results (three level transformation).

Depending on the specifics of your application, such as the source and destination image, context type, transformation mode, and filter type, calculations can require MIL to internally add padding data to the image's border. MimDraw() allows you to draw without (M_DRAW_WAVELET) or with (M_DRAW_WAVELET_WITH_PADDING) this padding. You can also retrieve related results with our without padding. For example, to get the width required to draw wavelet results, you can call MimGetResult() with M_WAVELET_DRAW_SIZE_X or M_WAVELET_DRAW_SIZE_X_WITH_PADDING.

The following example performs a wavelet transformation and then displays the resulting wavelet transforms.

MimWaveletDenoise() uses wavelet denoising techniques to remove noise from the specified source, which must be an image or wavelet result.

If the source is an image, MimWaveletDenoise() produces the actual denoised image in the specified destination. Since the destination is an image (not a result), you cannot perform result type operations with it, such as drawing the calculated wavelet coefficients using MimDraw().

If the source is a result, it must come from MimWaveletTransform(). The specified destination must also be a result, allocated using MimAllocResult() with M_WAVELET_TRANSFORM_RESULT.

As previously discussed, the denoising process uses the filter type and mode indicated in the specified wavelet context. Specifically, MIL decomposes the image according to the specified number of levels, performs the denoising, and recomposes the image using the same number of levels. The maximum level depends on the source image and the wavelet image processing context.

When calling MimWaveletDenoise(), you must also specify a specific wavelet shrinkage process: M_BAYES_SHRINK, M_NEIGH_SHRINK, or M_SURE_SHRINK.

M_BAYES_SHRINK is good at minimizing Gaussian noise. For minimizing Mean Square Errors (MSE), M_SURE_SHRINK is better. When noise is difficult to characterize, try NeighShrink. This setting mimics the general behavior of wavelet coefficients by taking neighborhood statistics and dismissing outliers. Though NeighShrink is less precise than the others, it can yield the best results under uncertain conditions. NeighShrink is usually the fastest setting.

If you are having difficulty denoising with MimWaveletDenoise(), try using some of MIL's other denoising techniques, such as spatial filtering (in particular for removing salt-and-pepper noise). For more information, see the Denoise using spatial filtering and area open and close operations section of Chapter 3: Fundamental image processing.

The following illustration shows a member of the Periphrastic family of near passenger birds (that is, a Toucan). The first image has no noise, while the second has white Gaussian noise.

The first two images below show the removal of noise using median ranking and smoothing. The third image shows the removal of noise using MimWaveletDenoise() with M_BAYES_SHRINK.

To remove Gaussian noise, a BayesShrink wavelet process proves most effective.

For more information about wavelet denoising
techniques, see "Rohit Sihag, Rakesh Sharma, and Varun Setia.
* Wavelet Thresholding for Image DE-noising* . USA: IJCA
Proceedings on International Conference on VLSI, Communications and
Instrumentation (ICVCI) (14):20–24, 2011. pp. 20-24." and "David L.
Donoho and Iain M. Johnstone.

The following example performs a wavelet denoising, and compares it with other conventional denoising operations, such as rank and spatial filtering.