A Linear Algebra View of the Wavelet = Transform

From: "Saved by Windows Internet Explorer 8" Subject: A Linear Algebra View of the Wavelet Transform Date: Mon, 29 Nov 2010 01:52:22 +0800 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01CB8F68.10819690" X-MimeOLE: Produced By Microsoft MimeOLE V6.0.6001.18049 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01CB8F68.10819690 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.bearcave.com/misl/misl_tech/wavelets/matrix/index.html A Linear Algebra View of the Wavelet = Transform

A Linear Algebra View of the Wavelet Transform

This web page was written to provide some background explaining the = structure=20 of wavelet algorithms covered on companion web pages. The structure of = wavelet=20 transforms like the Daubechies=20 D4 transform can be more clearly explained in the context of linear = algebra=20 (e.g., matrices).

Wavelet algorithms like the Daubechies D4 transform have special = cases that=20 must be handled in real applications with finite data sets. There are = several=20 methods for addressing the edge problem. One of these is Gram-Schmidt=20 orthogonalization, which is a matrix technique from linear algebra.

Linear algebra provides a tool for illuminating some wavelet = algorithms and=20 for developing wavelet and scaling function coefficients for the edges = of a=20 finite signal. In practice matrices are not used to calculate the = wavelet=20 transform. The matrix form of the wavelet transform is both = computationally=20 inefficient and impractical in its memory consumption. A single wavelet=20 transform step using a matrix algorithm involves the multiplication of = the=20 signal vector by a transform matrix, which is an ON² = operation=20 (where N is the data size for each transform step). In contrast, each = step of=20 the standard transform has a computational complexity of ON.

The Haar Wavelet Transform

The forward transform

Each step in the forward Haar transform calculates a set of wavelet=20 coefficients and a set of averages. If a data set s₀, = s₁,=20 ... s_N-1 contains N elements, there will be N/2 averages and = N/2=20 coefficient values. The averages are stored in the lower half of the N = element=20 array and the coefficients are stored in the upper half. The averages = become the=20 input for the next step in the wavelet calculation, where for iteration = i+1,=20 N_i+1 =3D N_i/2. The recursive iterations continue = until a=20 single average and a single coefficient are calculated. This replaces = the=20 original data set of N elements with an average, followed by a set of=20 coefficients whose size is an increasing power of two (e.g., = 2⁰,=20 2¹, 2² ... N/2 ).

The Haar equations to calculate an average (a_i) and a = wavelet=20 coefficient (c_i) from an odd and even element in the data set = are=20 shown below:

In wavelet terminology the Haar average is calculated by the scaling=20 function. The coefficient is calculated by the wavelet function.

The inverse transform

The data input to the forward transform can be perfectly = reconstructed using=20 the following equations:

Haar forward transform via matrix multiply

In the linear algebra view of the forward Haar transform, the first = average=20 is calculated by the inner product of the signal [s₀, = s₁,=20 ... s_N-1] and the vector, of the same size, [0.5, 0.5, 0, 0, = ..., 0].=20 This is the scaling vector. The first coefficient is calculated by the = inner=20 product of the signal and the vector [0.5, -0.5, 0, 0, ..., 0]. This is = the=20 wavelet vector.

The next average and coefficient are calculated by shifting the = scaling and=20 wavelet vectors by two and calculating the inner products.

In the wavelet literature scaling and wavelet values are sometimes=20 represented by h_i and g_i respectively. In the case = of the=20 Haar transform the scaling and wavelet values would be

 =
scaling function coefficients
  h₀ =3D  0.5
  h₁ =3D  0.5

 wavelet function coefficients
  g₀ =3D  0.5
  g₁ =3D -0.5

The scaling and wavelet values for the Haar transform are shown below = in=20 matrix form.

=20

The first step of the forward Haar transform for an eight element = signal is=20 shown below. Here signal is multiplied by the forward transform matrix. =

=20

The arrow represents a split operation that reorders the result so = that the=20 average values are in the first half of the vector and the coefficients = are in=20 the second half. To complete the forward Haar transform there are two = more=20 steps. The next step would multiple the a_i values by a 4x4 = transform=20 matrix, generating two new averages and two new coefficients which would = replace=20 the averages in the first step. The last step would multiply these new = averages=20 by a 2x2 matrix generating the final avarage and the final coefficient. =

The inverse transform

Like the forward Haar transform, a step in the inverse Haar transform = can be=20 described in linear algebra terms. The matrix operation to reverse the = first=20 step of the Haar tranform for an eight element signal is shown below. =

=20

In this case the arrow represents a merge operation that interleaves = the=20 averages and the coefficients.

Java Software

Java software that implements the forward and inverse Haar transform = using=20 matrices can be downloaded here. This software includes a modest linear = algebra class=20 (e.g., matrix multiply and various vector operations).

A note on terminology: basis and basis functions

One of the struggles I've had with the wavelet literature is the = terminology.=20 As with any field of specialty, mathematics assumes a shared basic set = of=20 knowledge, which includes areas like linear algebra. The wavelet = literature in=20 some cases seems to be written for those with a graduate level = background in=20 mathematics. This certainly does not describe my background. I am not a=20 mathematician and the linear algebra I know is self taught.

The wavelet literature sometimes refers to wavelet basis functions. = Given my=20 shallow background in linear algebra, I feel a bit unsure defining these = terms,=20 but I'll walk out on a limb (if I'm wrong, please send me e-mail with the correct = explanation,=20 preferably couched in the kind of simple terms used here).

The 8x8 matrix above has a row and column basis of 8 (which is = sometimes=20 represented as = ). A=20 wavelet basis function refers to the number of coefficients in the = scaling and=20 wavelet function. The Haar transform has an R² basis = and the=20 Daubechies D4 has an R⁴ basis.

Wavelet packets attempt to find the "best basis". The details of = wavelet=20 packets are beyond this web page, but in this case "basis" refers to the = region=20 of the original signal over which the scaling or wavelet function is = applied.=20

Looked at in terms of the matrix operations discussed on this web = page, the=20 scaling and wavelet functions apply to matrix rows (row basis). With = each step=20 of the wavelet calculation the basis changes (e.g., in this example, = from 8, to=20 4, to 2). Conceptually the scaling and wavelet functions span larger and = larger=20 sections of the signal as the basis decreases. In the case of the Haar=20 transform, as the basis gets smaller the averages and differences = represent the=20 average or average change over larger and larger portions of the signal. =

The Daubechies D4 Wavelet Tranform

As noted above, the Haar scaling and wavelet functions are calculated = using=20 two coefficients, h₀, h₁ and g₀, = g₁,=20 respectively. As the name suggests, the scaling and wavelet functions of = the=20 Daubechies D4 wavelet transform are calculated using four coefficients,=20 h₀, h₁, h₂, h₃ and = g₀,=20 g₁, g₂, g₃. The derivation for the = coefficient=20 values can be found in Ingrid Daubechies work and in section 5.5 of = Wavelets=20 and Filter Banks by Gilbert Strang and Truong Nguyen, = Wellesley-Cambridge=20 Press, 1997.

The scaling function coefficient values are:

The wavelet function coefficient values are:

g₀ =3D h₃
g₁ =3D=20 -h₂
g₂ =3D h₁
g₃ =3D=20 -h₀

The scaling values, a_i, and the wavelet values, = c_i are=20 calculated by taking the inner product of the h_j and = g_j=20 coefficients and the signal. The equations for the scaling and wavelet = inner=20 products are shown below. The second verson of each equation shows the = indexing=20 of the signal.

As with the Haar transform discussed above, the Daubechies scaling = and=20 wavelet function coefficients shift from right to left by two places in = each=20 iteration of a wavelet transform step.

The Daubechies transform has no special cases when applied to an = infinite=20 signal. In the finite world outside of mathematics there would be a = signal with=20 N elements (ranging from s[0] to s[N-1]). When the scaling and wavelet = functions=20 are shifted so that the value of i in the equations above is N-3, = two=20 coefficients will stick out beyond the end of the signal (e.g., the = inner=20 product will be calculated with s[N] and s[N+1]). Methods for dealing = with this=20 are discussed on my Daubechies=20 wavelet web page.

Although in practice we don't have infinite signals (or transform = matrices)=20 ignoring the special cases allows us to look at the Daubechies transform = as a=20 matrix, showing the structure of the coefficients.

Daubechies forward transform matrix

=20

The inverse Daubechies D4 transform matrix is the transpose of the = forward=20 transform matrix:

Daubechies inverse transform matrix

=20

Haar vs. Daubechies transforms

The wavelet literature covers a wide variety of wavelet algorithms, = which are=20 drawn from an infinite set of wavelet algorithms. When I first started = reading=20 about wavelets one of the first questions I had was "which algorithm = should I=20 use". The choice of the wavelet algorithm depends on the application. =

The result of the wavelet transform produces a "down sampled" = smoothed=20 version of the signal (calculated by the wavelet scaling function) and a = "down=20 sampled" version of the signal that reflects change between signal = elements. The=20 smoothing function is sometimes referred to as a low pass filter. The = wavelet=20 function is sometimes referred to as a high pass filter.

If we compare the Haar forward transform matrix to the Daubecies D4 = transform=20 matrix, we can see that there is no overlap between successive pairs of = scaling=20 and wavelet functions, as there is with the Daubechies transform.

The Haar high pass filter (wavelet function) produces a result that = reflects=20 the difference between an even element and an odd element. The = difference=20 between an odd element and its even successor will not be reflected in = the=20 coefficient band calculated by a single step of Haar high pass filter = (although=20 this change will be picked up by later steps). In contrast, there is = overlap=20 between successive Daubechies high pass filters, so change between any = two=20 elements will be reflected in the result.

The Daubechies D4 wavelet transform is more "accurate", since change = in the=20 input data set is reflected in the high pass filter results at each = transform=20 step. The cost if using the Daubechies algorithm is higher computation = overhead=20 (twice the number of operations, compared to Haar) and a more = complicated=20 algorithm (the algorithm must properly handle the edge condition where = i=3D0).=20 Whether the higher accuracy of the Daubechies algorithm is worth the = cost is=20 application dependent.

References

Wavelets and Filter Banks by Gilbert Strang and Truong = Nguyen,=20 Wellesley-Cambridge Press, 1997

Strang and Nguyen make heavy use of linear algebra in their = coverage of=20 wavelets and wavelet filters. As noted above, this text also include = the=20 derivation of the Daubechies wavelet functions. This book is most = easily read=20 by a reader with a background in digital signal processing and linear = algebra.=20
Ripples in Mathematics: the Discrete Wavelet Transform by = Arne Jense=20 and Anders la Cour-Harbo, Springer, 2001

So far this is the best book I've been able to find on wavelets, = from an=20 implementation point of view. Ripples provides less = mathematical depth=20 than Strange and Nguyen, but the material is more accessible. =

Ian Kaplan, January 2002
Revised:

=20

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