0.1 Signal dictionaries and representations (Page 2/2)

Concise signal models Page 2 / 2

Wavelets

Closely related to the Fourier transform, wavelets provide a framework for localized harmonic analysis of asignal [link] . Elements of the discrete wavelet dictionary are local, oscillatory functions concentrated approximately ondyadic supports and appear at a discrete collection of scales, locations, and (if the signal dimension $D > 1$ ) orientations.

Scale

In wavelet analysis and other settings, we will frequently refer to a particular scale of analysis for a signal. Consider, for example, continuous-time functions $f$ defined over the domain $D = {[0, 1]}^{D}$ . A dyadic hypercube $X_{j} \subseteq {[0, 1]}^{D}$ at scale $j \in N$ is a domain that satisfies

X_{j} = [β_{1} 2^{- j}, (β_{1} + 1) 2^{- j}] \times \dots \times [β_{D} 2^{- j}, (β_{D} + 1) 2^{- j}]

with

β_{1}, β_{2}, \dots, β_{D} \in {0, 1, \dots, 2^{j} - 1}

. We call

X_{j}

a dyadic interval when

D = 1

or a dyadic square when

D = 2

(see [link] ). Note that

X_{j}

has sidelength

2^{- j}

Dyadic partitioning of the unit square at scales $j = 0, 1, 2$ . The partitioning induces a coarse-to-fine parent/child relationshipthat can be modeled using a tree structure.

For discrete-time functions the notion of scale is similar. We can imagine, for example, a “voxelization” of the domain ${[0, 1]}^{D}$ (“pixelization” when $D = 2$ ), where each voxel has sidelength $2^{- B}$ , $B \in N$ , and it takes $2^{B D}$ voxels to fill ${[0, 1]}^{D}$ . The relevant scales of analysis for such a signal would simply be $j = 0, 1, \dots, B$ , and each dyadic hypercube $X_{j}$ would refer to a collection of voxels.

Wavelet fundamentals

The wavelet transform offers a multiscale decomposition of a function into a nested sequence of scaling spaces $V_{0} \subset V_{1} \subset \dots \subset V_{j} \subset \dots$ . Each scaling space $V_{j}$ is spanned by a discrete collection of dyadic translations of alowpass scaling function $ϕ_{j}$ , and the difference between adjacent scaling spaces $V_{j}$ and $V_{j + 1}$ is spanned by a discrete collection of dyadic translations of a bandpass wavelet function $ψ_{j}$ . [link] shows an example of this multiscale organization in the case of the Haar wavelet dictionary. Each wavelet function at scale $j$ is concentrated approximately on some dyadic hypercube $X_{j}$ , and between scales, both the wavelets and scaling functions are “self-similar,” differing only by rescaling and dyadicdilation. When $D > 1$ , the difference spaces are partitioned into $2^{D} - 1$ distinct orientations (when $D = 2$ these correspond to vertical, horizontal, and diagonal directions). The wavelet transform can be truncated atany scale $j$ . We then let the basis $Ψ$ consist of all scaling functions at scale $j$ plus all wavelets at scales $j$ and finer.

Multiscale wavelet representations on the interval $[0, 1]$ . (a) Haar scaling functions spanning $V_{j}$ with $j = 2$ . (b) Haar wavelet functions spanning the difference space between $V_{j}$ and $V_{j + 1}$ . (c) Haar scaling functions spanning $V_{j + 1}$ . (d) Two example functions belonging to the spaces (left) $V_{j}$ and (right) $V_{j + 1}$ .

Wavelets are essentially bandpass functions that detect abrupt changes in a signal. The scale of a wavelet, which controls itssupport both in time and in frequency, also controls its sensitivity to changes in the signal. This is made more precise byconsidering the wavelet analysis of smooth signals. Wavelet are often characterized by their number of vanishing moments ; a wavelet basis function is said to have $H$ vanishing moments if it is orthogonal to (its inner product is zero against)any $H$ -degree polynomial. Sparse (Nonlinear) models discusses further the wavelet analysis of smooth and piecewise smooth signals.

The dyadic organization of the wavelet transform lends itself to a multiscale, tree-structured organization of the waveletcoefficients. Each “parent” function, concentrated on a dyadic hypercube $X_{j}$ of sidelength $2^{- j}$ , has $2^{D}$ “children” whose supports are concentrated on the dyadic subdivisions of $X_{j}$ . This relationship can be represented in a top-down tree structure, as demonstrated in [link] . Because the parent and children share a location, they will presumably measure related phenomena about thesignal, and so in general, any patterns in their wavelet coefficients tend to be reflected in the connectivity of the treestructure. [link] and [link] show an example of the wavelet transform applied to the Cameraman test image; since the dimension $D = 2$ , each scale is partitioned into vertical, horizontal, and diagonal wavelet analysis, and each parent coefficient has $2^{D} = 4$ children.

Cameraman test image (size $256 \times 256$ ) for use in wavelet decomposition and approximation examples.

Wavelet analysis of the Cameraman test image. (a) One-level wavelet transform, where the $N$ -pixel image is transformed into four sets of $N / 4$ coefficients each. The top left quadrant represents the scaling coefficients at the next coarser scale (relative to the scale of pixelization). The remaining quadrants represent the wavelet coefficients from the difference spaces, partitioned into the vertical, horizontal, and diagonal subbands. (b) Three-level wavelet transform, where the wavelet decomposition has been iterated twice more on the scaling coefficients. The multiple scales of wavelet coefficients exhibit a parent-child dependency. The largest coefficients tend to concentrate at the coarsest scales and around high-frequency features such as edges in the image.

In addition to their ease of modeling, wavelets are computationally attractive for signal processing; using a filterbank, the wavelet transform of an $N$ -voxel signal can be computed in just $O (N)$ operations.

Other dictionaries

A wide variety of other dictionaries have been proposed in signal processing and harmonic analysis. As one example, complex-valuedwavelet transforms have proven useful for image analysis and modeling [link] , [link] , [link] , [link] , [link] , [link] , [link] , thanks to a phase component that captures location information ateach scale. Just a few of the other harmonic dictionaries popular in image processing include wavelet packets [link] , Gabor atoms [link] , curvelets [link] , [link] , and contourlets [link] , [link] , all of which involve various space-frequency partitions. We mention additional dictionaries in Compression .

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Source: OpenStax, Concise signal models. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10635/1.4

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