7

OTHER WAVELET TOPICS

This chapter contains a variety of topics of a more advanced nature that relate to wavelets. Since these topics are more advanced, an overview will be presented with details left to the various references.

7.1 COMPUTATIONAL COMPLEXITY

7.1.1 Wavelet Algorithm

In this section we briefly discuss the number of operations required to compute the wavelet decomposition of a signal (the computational complexity of the wavelet decomposition algorithm).

To take a concrete setting, suppose f is a continuous signal defined on the unit interval 0 ≤ t ≤ 1. Let

be a discrete sampling of ƒ. We wish to count the number of multiplications in the decomposition algorithm in terms of the number N = 2ⁿ.

The decomposition algorithm stated in Eq. (5.12) or Eq. (5.17) computes the a^j-i and b^j–i, j = n,..., 1, using the formulas

Note that there are half as many a^j-i coefficients as there are a^j coefficients. So the index l on the left runs from 0 to 2^j–

This result is often summarized by stating that the wavelet decomposition algorithm requires O(N) multiplication operations where N is the number of data at the top level. Here, O(N) stands for a number that is proportional to N. The proportionality constant in this case is 2(L + 1) (which is 6 in the case of Haar or 10 in the case of Daubechies).

By comparison, the number of multiplicative operations for the fast Fourier transform is 0(N logN) (see Section 3.1.3). However, this comparison is unfair since the fast Fourier transform decomposes a signal into all its frequency components between 0 and N/2. By contrast, the wavelet decomposition algorithm decomposes a signal into its frequency components which lie in ranges bounded by powers of 2. For example,

is the decomposition of f given in Theorem 5.11. This should be regarded as the component of f which involves frequencies in the range 2^7–1 to 2^j. Therefore, the wavelet decomposition algorithm does not decompose the signal into the more detailed range of frequencies that are offered by the discrete Fourier transform.

7.1.2 Wavelet Packets

Wavelets can be used to obtain a decomposition of a signal into a finer gradation of frequency components using wavelet packets. We briefly describe the technique for doing this, which requires O (N log N) operations, and therefore it is no more efficient than the fast Fourier transform. A more complete discussion may be found in Meyer (1993, Chapter 7).

We illustrate the idea of wavelet packets in the specific case when n = 3. A signal f3 ∈ V₃ can be decomposed via Theorem 5.11 as

We can put this decomposition into a more hierarchical structure

The arrows marked “H” are the projection operators onto the wavelet components. The arrows marked “L” stand for the projection to the next lower level (e.g., the projection from V₂ to V₁). The H stands for “high-pass filter” since the wavelet part of a particular level represents the component with higher frequency (e.g., the W₁ component of V₂ = W₁ ⊕ V₁ represents higher frequencies that are not in the V₁ component). Likewise, the L stands for “low-pass filter” since it projects to the next lower level.

The ƒ3 component contains the frequency range from 1 to 8. The w₂ component contains frequency components from 5 to 8. The analogous frequency ranges for the w₁, w₀, and w₀ components are 3 to 4, 2, and 1, respectively. The idea behind wavelet packets is to apply the high-pass and low-pass filters to each of w₂, w₁, and w₀. The high-pass filter and low-pass filters applied to w₂ create new components W₂₁ and w₂₀, respectively, which are orthogonal to one another. The process is iterated until a complete decomposition into all frequencies is obtained.

In the case of ƒ3, a complete decomposition into all components is illustrated in the following diagram.

To count the number of operations required, imagine that this diagram were extended to handle n levels (n = 3 is illustrated). Each component in the kth

level of the diagram requires O(2^N–k) computations. Since there are 2^k components in the kh level, the total number of computations for this level is 0(2”) (independent of k). Since there are n levels (not counting the first, which only contains the original signal), the total number of operations is 0(n2ⁿ) or Nlog₂ N, where N = 2ⁿ. This is the same number of operations required by the fast Fourier transform.

7.2 WAVELETS IN HIGHER DIMENSIONS

The signal processing of two-dimensional video images or of digitized three-dimensional scans of solid objects can be done using wavelets, as generalized to multidimensions. Wavelets in higher dimensions can be obtained in several ways (Daubechies, 1992). We will briefly describe a two-dimensional multiresolution framework commonly used in various image processing tasks.

Suppose that ϕand ψ are the scaling function and wavelet for a multiresolution analysis and that they are real-valued and compactly supported. The Haar system or the ones corresponding to Daubechies wavelets are examples. As shown in previous sections, the functions

are orthonormal bases for the spaces V_j and W_j, respectively. Moreover, each ψ_jk is orthogonal to all the ϕjk- Finally, the ϕand ψ satisfy the scaling and wavelet relations below.

Because ϕ is real and has compact support, the P_k’s are real and are nonzero for only a finite number of k’s. The “inverse” of the scaling relation, which expresses ϕ (2x) in terms of (x – k) and ψ(x – k), is Eq. (5.10) with j = 1:

In the two-dimensional case, we will need one scaling function but three wavelet functions: horizontal, vertical, and diagonal. To help in describing these, let x = (x₁, x₂), where x₁ and x₂ are in R, and let k = {k₁, k₂), where k₁ and k₂ are integers; we will write this as k ∈ Z². The scaling function is

and we also set

The horizontal, vertical, and diagonal wavelets are, respectively (Mallat, 1998),

We define 1, 2, 3, analogously to Φ_j,k It is easy to show (exercise 6) that Φ satisfies the scaling relation,

(7.1)

and that the Ψ^γ’s satisfy wavelet relations,

(7.2)

Analogous to Eq. (5.10) in the one-variable setting, the inverse scaling relation in the two-dimensional case is

(7.3)

The space of finite-energy, two-dimensional “signals” is the Hubert space L²(R²), which comprises all functions f(x) = f(x₁,X₂) such that

where || f ||_L2_(S2) is the norm. The inner product for L²(R²) is

Its role in image processing is analogous to L²(R) in one-dimensional signal processing.

The scaling functions and wavelets are orthonormal in L²(R²) for fixed j,

The wavelets, but not the scaling functions, are also orthogonal at different levels—that is, different j’s. Except for there being three wavelets instead of one, these orthogonality relations are analogous to those for the one-dimensional case.

We can use the scaling function Φ to introduce a two-dimensional multireso-lution analysis. Define the approximation space V_j to be the span of the Φ_j,k’s in L²(R²). In a way similar to that used for denning the V_j’s, we use the three wavelets to define the wavelet spaces , and For example, is the span of { k ∈ Z}, and similarly for the other spaces. These three wavelet spaces are all orthogonal to each other and to V,. The wavelet spaces, but not the approximation spaces, are also orthogonal for different levels j, j’. The resulting multiresolution analysis is described in this result:

Proposition 7.1 The approximation spaces V_j, j ∈ Z, together with the scaling function Φ, satisfy all of the properties of a multiresolution analysis, with L²(R) replaced by L²(R²). In addition, the following decomposition formula holds:

(7.4)

Note. Equation (7.4) is the multidimensional analogue of V)+i = V) ⊕ Wj in the one-variable setting (see Chapter 5).

Proof. The defining properties of a multiresolution analysis are given in Definition 5.1. The scaling relation (7.1) implies Φ(x – 1) is a linear combination of the Φ(2x – k)’s. Consequently, Φ(x – 1) is in V₁. Since every function in V₀ is in the span of the Φ(x – l)’s, we have that V₀ C V₁. A similar argument using Φ_j,1 and Φ_j+1,k gives us that the approximation spaces are nested:

(7.5)

Of course, V₀ was constructed from the basis Φ(x – 1), so that V₀ is generated by shifts of the scaling function. The remaining properties needed for a multires olution analysis—namely, density, separation, and scaling—are left as exercises.

The decomposition (7.4) follows from Eqs. (7.1), (7.2), and (7.3).

Our convention for the coefficients for expansions in the bases for these spaces is this. Let f ∈ L²(R²). If f_j,- ∈ V_j is the projection of f onto V_j and is the projection of f onto , then the orthonormality of the bases implies that

As we did earlier, we introduce “a” and “b” notation:

With this and a little algebra, it is easy to see that the expansions above become

(7.6)

Wavelet analysis relies on the decomposition of Vy+i in Eq. (7.4). It implies that the projection of f onto V₇₊₁ can be written as

Expressing f_j+1 in terms of the basis {Φ_j+i,k}k∈Z² for V_j+1 and equating it to the f_j, expansions from Eq. (7.6), which come from the bases {Φ_j,k}k∈z^{2 U we} obtain decomposition and reconstruction formulas for the coefficients at the levels j and j + 1:

(7.7)

(7.8)

These equations are the multidimensional analogs of Eqs. (5.12) and (5.13) in one-variable.

Signal processing with two-dimensional wavelets follows the same overall steps as in one dimension: sampling, decomposition, processing (e.g., compression, denoising), and reconstruction. Let’s illustrate these steps for an image f(x¹,x₂) on the unit square, where {{x₁,x₂); 0 ≤x₁,x₂ ≤ 1}. Suppose that the sampled image has n² pixels, where n = 2^j+1. The samples are then just f(2^–J–1k₁,2^–J–1k₂). Here k = (k₁k₂), 0 ≤ k₁ k₂ ≤ n – l.

The decomposition step requires initializing the highest-level approximation coefficients, the Suppose Φ a scaling function with compact support, so that Φ(2^j+1x – k) has support within a few pixels of the point 2^–j–1 k = {2^–J–1 2^–j–lk₂). Then

(7.9)

When either or both of the components k₁ or k₂ are outside of the interval [0, n – 1], we assume that f(2^j–1k) = 0. This allows us to define for all k. The rest of the decomposition step amounts to iterating Eqs. (7.7) to obtain the approximation and wavelet coefficients down to some desired level, denoted by j₀ < J. The wavelet coefficients can then be processed accordingly.

For example, if filtering is the goal, then the wavelet coefficients corresponding to the unwanted frequencies can be set to zero. If compression is the goal, then the wavelet coefficients whose absolute values are below some desired threshold can be set equal to zero (and thus it is not necessary to store or transmit these zeros). It is these modified wavelet coefficients that are used in the reconstruction step by iterative use of Eq. (7.8) starting with j = jo and proceeding to the highest level j = J. As indicated in Eq. (7.9), the value of the modified image at x = 2^–j–1k is approximated by the (now modified) value of

For signal processing, it is essential to understand what the approximation and wavelet coefficients mean. Let’s look at a simple example where we use the Haar scaling function and wavelet for ϕ and ψ These have p₀ = p₁ = 1 and q₀ = 1, q1 = – 1. The corresponding p_k’s and P£’s in Eqs. (7.1) and (7.2), which are all 0 unless k is (0, 0), (1, 0), (0, 1), or (1,1), are best displayed in a matrix form in which the entries correspond to k = (0, 0), k = (1, 0), and so on:

We can calculate the level J approximation coefficients initialized by taking Eq. (7.9) for level J + 1 approximation coefficients in Eqs. (7.7). Combining this with the pk above, we see that

where x₁ = 2^–J and y_1,0 = 2^–j–1(1, 0), and so on. The point here is that is the average of f evaluated at the corners of the square of length 2^–j–1 on each side, with 2^–jl as its lower left-hand corner. It is thus a smoothed or “blurred” version of the image using 1/4 of the original number of pixels, with each new pixel being 2^–J on a side rather than 2^–j–1 on a side. A similar analysis, along with a little algebra, shows that the wavelet coefficient has the form

which is the difference of the horizontal pixel averages on the square. Such wavelet coefficients will be able to pick up changes among horizontal lines. Coefficients for the form

pick up differences among vertical lines. Finally, the wavelet coefficients of the form

pick up changes among diagonal lines. The behavior of these wavelet coefficients is the origin of the designation of a wavelet as horizontal, vertical, or diagonal.

The coefficient formulas that we’ve just discussed are specific to the Haar mul-tiresolution analysis. Even so, the general interpretation of coefficients remains for many other one-dimensional multiresolution analyses. We point out that when other ϕ and ψ are used, the above formulas must be modified. Moreover, other technical difficulties with filter lengths and edges arise. Still, processing a two-dimensional signal or image is done with these interpretations for the coefficients kept in mind.

EXERCISES ON 2D WAVELETS

1. Let f ∈ L²(R²). For the two-dimensional Haar multiresolution analysis, with scaling function Φ, show that the scaling coefficient for the projection of f onto the space V_j is the exactly the average of f over the support of Φ(2^jx – k). This allows us to interpret as a pixel value associated with the square having length 2^–j on a side and lower left-hand corner 2^–j {k₁, k₂).

2. The square S below is divided into congruent squares, each having sides of length 2^–J–l. The lower left-hand corner L of S has coordinates 2^–j {k₁, k₂).

(a) Let p₁,...,p₄ be the pixel values of f ∈ L²(R²) associated with S₁,..., S₄. (See exercise 1.) Show that ps = is the average of these four pixel values.

(b) Express the wavelet coefficients bfc , X = 1, 2, 3 in terms the pixel values pi,..., P4. Briefly discuss why these expressions illustrate the interpretation of the wavelet coefficients as vertical, horizontal, and diagonal.

3. Let f(x¹,x₂) = 5 – x₁ – 3x₂ on the square [0,1] × [0,1], and 0 elsewhere. Use exercise 1 to calculate the 16 pixel values for the projection of f onto level j = 2. Use exercise 2 to calculate pixel values for j = í and j = 0. Plot the resulting images. Notice the “blurring” that occurs as j decreases.

4. Provide filter diagrams for the two-dimensional decomposition and reconstruction formulas.

5. Complete the proof of Proposition 7.1.

6. Show that (7.1) and (7.2) hold.

7. This exercise is really a small project. You will need Matlab’s Wavelet Toolbox to carry it out. Find two gray-scale images on the web. Color images can be used, but doing so complicates matters. The idea is to hide one image inside of the other. Using two-dimensional wavelets, try to hide in such a way that the hidden image cannot be seen just by looking at it, but it can be recovered from a wavelet analysis of the combined image. Try several different wavelets to see which one works best.

7.3 RELATING DECOMPOSITION AND RECONSTRUCTION

In this section we analyze the decomposition and reconstruction algorithms and we show that the latter involves the I² adjoint of the former. As before, we assume that ɸand ψ are the scaling and wavelet functions associated to a multiresolution analysis. We rewrite the decomposition and reconstruction algorithms using the orthonormal bases ϕ_jk(x) = 2^j/2ϕ(2^Jx – k) ∈ V_j) and ϕ_jk(x) = 2^j/2ϕ(2^jx – k) ∈ Vj. To do this, it is more convenient to use coefficients := (f,ϕ and : = (f,ψ) which are related directly to the orthonormal bases for V_j and W_j. The ’s and ’s introduced in Section 5.1.4 relate to the ’s and ’s via.

The decomposition algorithm (5.9) becomes the following:

Theorem 7.2 (Decomposition). A function where j = n is the top level, can be decomposed as

with

where

In terms of convolution and the down-sampling operator D

The reconstruction algorithm (5.11) can be restated as follows.

Theorem 7.3 {Reconstruction). Suppose

and let

Then the coefficients c_l^j can be computed from the sequences c^j–1 and d^–1 by

In terms of convolution and the up-sampling operation U, we obtain

where P* = ±p_k and Q* = -*._Pl__k(-if.

Except for the factor of which results from using the orthonormal bases ϕ_jl and ψ_ji, these theorems are identical to the algorithms in Eqs. (5.12) and (5.13) [or Eqs. (5.17) and (5.22)].

Recall that l² is the space of all sequences x = (x_j); j ∈ Z with ∑_j |x_j|² < ∞. In practical applications, there are only a finite number of nonzero X_j and so the condition ∑_j |x_j|² < ∞ is automatically satisfied. The key operators involved with decomposition algorithm are the down-sampling operator D : l² → l² and the operators T_P : l² → l² and T_Q : l² → l²

where P and Q are the sequences defined in Theorem 7.2. Likewise, the key operators in the reconstruction algorithm are the up-sampling operator U and the operators T_P* : l² → l² and Tq* : l² → l².

where P* and Q* are the sequences defined in Theorem 7.3.

Our goal now is to show that the operators T_P and T_P* are adjoints of one another and likewise for the operators involving Q. Recall that if T : V → V is a linear operator on an inner product space V, then its adjoint T* is defined by

Lemma 7.4

• The adjoint of T_P is T_P*.
• The adjoint of Tq is Tg*.
• The adjoint of D (the down-sampling operator) is U (the up-sampling operator).

Proof. The first two properties follow from Theorem 3.14, which states that the adjoint of the operator associated with the convolution with a sequence f_n is the convolution operator associated with the sequence Thus, the first property follows from the fact that Likewise, and so the second property follows from Theorem 3.14.

For the third property, let x and y be sequences in I². Using the definition of D (see Definition 5.13),

On the other hand, only the even entries of Uy are nonzero and (Uy)2_n = y_n (see Definition 5.15). Therefore

By comparing these two sets of equations, we conclude {Dx, y) = (x, Uy} as desired. ¦

Using the lemma and Theorems 7.2 and 7.3, the decomposition and reconstruction algorithms can be restated (essentially) as being adjoints of each other.

Theorem 7.5 Decomposition. Let and Suppose

where D is the down-sampling operator and Tp (respectively Tq) is the operator that convolves with the sequence P (respectively Q). A function ƒ, =∑i a¡(pji(x) can be decomposed as

with

where

(7.10)

Conversely, the sequence a⁷ can be reconstructed from a^{7 x} and V ^x by

(7.11)

Proof. The decomposition formulas follow from Theorem 7.2. For the reconstruction formula, Theorem 7.3 implies

Equations (7.10) and (7.11) can be combined as

which is another way of stating that the reconstruction process is the adjoint of decomposition and that the reconstruction process inverts decomposition [because (Tq °To + T* ° T) sends the sequence a⁷ to itself]. Operators with this property have a special name.

Definition 7.6 A pair of filters, Tp and Tq (i.e., convolution operators with the sequences P and Q, respectively) is called a quadrature mirror filter if the associated maps T₀ = D°T_P and T = D°Tq satisfy

(7.12)

for all sequences x = (...,x i,x₀,x,...) ∈ l²(Z).

An equivalent formulation of Eq. (7.12) is

For if equation (7.12) holds, then

The I² norm of a signal x = (...,x-,xq,x,...), (i.e., ∑_n^xn)> is proportional to the energy of the signal. So a physical interpretation of Eq. (7.12) is that a quadrature mirror niter preserves the energy of a signal.

7.3.1 Transfer Function Interpretation

Our goal in this section is to transcribe the denning property of a quadrature mirror niter

(7.13)

into a condition on the transfer functions of the niters Tp and Tq.

First, we recall the definition of a transfer function given just after Theorem 3.13. A sequence x = (..., x_–1, x₀, x₁,...) ∈ l²(Z) has a discrete Fourier transform ‘x which is a function on the interval – π ≤θ≤π defined by

If Tp is a convolution operator with the sequence P_n—that is, (T_px)_n = ∑_k P_n–kx_k—then the transfer function of T_p is the function _p on the interval —it ≤6 ≤ 7t defined by

As shown in Theorem 3.13,

(7.14)

Another key property that we will use is the computation of the transfer function of the transpose of a convolution operator, that is,

(7.15)

which is established in Theorem 3.14.

To transcribe Eq. (7.13) into one involving transfer functions, we take the discrete Fourier transform (the hat of) both sides of this equation. In what follows, we will use the notation T{x){9) = x~(6) for expressions x that are long. We start with the (Tq o Tq)(x) term on the left side,

(7.16)

(7.17)

(7.18)

where the last two equalities use Eqs. (7.14) and (7.15), respectively. Now if y is the sequence y_n, n ∈ Z, then from the last section we have (Dy)_n = y2_n. So,

Therefore,

which can be re-described as

(the terms when n is odd cancel since then e –^inπ = –1). Therefore

Applying this equation to the sequence y_n = T_p(x)_n yields

Inserting this into the end of Eq. (7.18), we obtain

Similarly,

Adding these two equations and setting it equal to x~(6) [as required by Eq. (7.12)] yields

Let G (6) be the term inside the first parentheses on the left and let H(6) be the term on the inside of the second parentheses. The preceding equation now reads

which must hold for all sequences x. Let us first apply this equation when x(λ) = e^iλ, which occurs when x is the sequence given by x₀ = 1 and x_n = 0 for nonzero n. We obtain

Now apply the equation when x(λ) = e^ie, which occurs when x is the sequence given by x_i = 1 and x_n = 0 for all other n. After dividing by e’^e, we obtain

Adding these two equations, we obtain G (6) = 1. Subtracting these two equations yields H(6) = 0. Inserting the expressions for G and H, we obtain the following theorem.

Theorem 7.7 The filters Tp and Tq form a quadrature mirror filter if and only if

or, equivalently, the matrix

is unitary.

When the quadrature mirror filters come from a scaling function and wavelet, then (the transpose of) the unitary matrix in the preceding theorem becomes

with p(θ) = P(e–^iθ) and q(θ) = Q{e–^iθ) as defined in Section 6.1. This is the same matrix as that given after Theorem 5.21, and we saw that this matrix was unitary from the techniques given in that section.

7.4 WAVELET TRANSFORM

In Chapter 2, which discussed the Fourier transform, we discussed the Fourier inversion formula. In this section we develop an analogous wavelet transform and its associated inversion formula. Let us first review the case of the Fourier transform. For a function, ƒ, in L²(R), its Fourier transform is given by

Another notation for the Fourier transform is f (A). The inverse Fourier transform of a function g ∈ L²(R) is given by

The content of the Fourier inversion Theorem 2.1 is that ^–1 as defined previously, really is the inverse operator of T, that is,

As discussed in Chapter 2, {f){k) roughly measures the component of f which oscillates with frequency k. The inversion formula when written as

expresses the fact that f can be written as a weighted sum (or integral) of its various frequency components.

7.4.1 Definition of the Wavelet Transform

The wavelet transform, along with its associated inversion formula, also decomposes a function into a weighted sum of its various frequency components. However, this time the weights involve a given wavelet, rather than the exponential term e^iλx.

To introduce the wavelet transform, we assume that a wavelet function, ψ(x), is given that satisfies the following two requirements.

1. ψ(x) is continuous and has exponential decay (i.e., ψ(x) ≤ Me–^c|^x| for some constants C and M).

2. The integral of ψ is zero, (i.e., ).

An example of a suitable wavelet function is ψ(x) = xe^–x2 whose graph appears in Figure 7.1. Another example is the wavelet function of Daubechies constructed in the previous chapter. The Daubechies wavelet is only nonzero on a finite interval, so the first requirement is automatically satisfied (by taking the constant M large enough). Note that no assumptions are made concerning orthogonality of the wavelet with its translates. The Daubechies wavelets are constructed so that its translates are orthogonal; but the preceding example, //(x) = xe~^x , is not orthogonal to its translates.

In the derivations that follow, we will actually assume that ¡r{x) equals zero outside /A,which is a stronger condition than the first condition just given. However, every derivation given can be modified to include wavelets with exponential decay.

We are now ready to state the definition of the wavelet transform.

Definition 7.8 Given a wavelet ψ satisfying the two requirements just given, the wavelet transform of a function f ∈ L²(R) is a function Wf : R² → R given by

From the preceding definition, it is not clear how to define the wavelet transform at a = 0. However, the change of variables y = (x — b)/a converts the wavelet transform into the following:

From this representation, clearly W/(a, b) = 0 when a = 0. As a gets small, the graph of

becomes tall and skinny as illustrated in the graphs of ψ_1,0 and ψ_1/2,0 with ψ(x) = xe^–x₂ given in Figures 7.1 and 7 .2, respectively.

Therefore the frequency of ψ_a,b increases as a gets small. Also note that if most of the support of ψ (i.e., the nonzero part of the graph of ψ) is located near the origin (as with the preceding example), then most of the support of ψ_a,b will be located near x = b. So W_f(a, b) measures the frequency component of f that vibrates with frequency proportional to 1/a near the point x = b.

7.4.2 Inversion Formula for the Wavelet Transform

The inversion formula for the wavelet transform is given in the following theorem.

Theorem 7.9 Suppose ψ is a continuous wavelet satisfying the following:

• ψ has exponential decay at infinity.
• 

Then for any function f e L²(R), the following inversion formula holds:

where

As with the Fourier inversion theorem, the preceding wavelet inversion theorem essentially states that a function f can be decomposed as a weighted sum (or integral) of its frequency components, as measured by W_f(a, b). The difference is that with the wavelet inversion theorem, two parameters, a and b, are involved since the wavelet transform involves a measure of the frequency (using the parameter a) of f near the point x =b.

Before we start the proof, we should state why Cf is finite (since the integrand defining it appears to become infinite at X = 0). First, we split the integral that defines Cf into two pieces: one involving an integral over |A.| > 1 and a second involving the integral over |A.| < 1. For the first integral, we note that ψ has exponential decay and so ψ is a member of L²(R). Therefore its Fourier transform also belongs to L²(R) and so

For the second integral, first note that the exponential decay assumption on ψ implies ψ is differentiable. From a first-order (Taylor) expansion about X = 0, we have

where O(λ) stands for terms that are bounded by C|λ| for some constant C. We also have by the second assumption on ψ stated in the theorem. Therefore, the second integral can be estimated by

Since both integral pieces of C_ψ are finite, we conclude that C_ψ is finite as well.

Proof of the Wavelet Transform Theorem. Let F(x) be the quantity given on the right side of the main statement of the theorem, that is,

We must show that F(x) is equal to f (x). We shall accomplish this using the following two steps.

• Step 1. We first show

where F_b{·} stands for the Fourier transform of the quantity inside the brackets {,} with respect to the variable b .This step follows by applying Plancherel’s Theorem (Theorem 2.12), which states that F(u) .This theorem is applied to the e-integral occurring in the definition of F(x) and where v(b) = ψ((x – b)/a) and u(b) = W_f(a, b) (with, x and a constant). In order to apply the Plancherel Theorem, both of these functions must belong to L²(R) .If f and ψ have finite support, then the b-support of W_f(a,b) will also be finite (as you can check) and so Wf(a,b) and are L² functions in b.

• Step 2. The next step is to evaluate the two Fourier b-transforms occurring in the integrand of Step 1. We will show

We shall establish Step 2 later. Assuming this step for the moment, we proceed to the conclusion. Combining Steps 1 and 2, we obtain

(7.19)

(7.20)

where the last equality lollows by interchanging the order oí the y and a-integrals. To calculate the a-integral on the right, we make a change of variables u = ay (and so du = y da) provided that y ≠ 0 to obtain

(7.21)

(7.22)

If y = 0, then by assumption (0) =0 and so the rightmost integral in Eq. (7.20) is zero. Therefore the rightmost integral in Eq. (7.20) is equal to Cψ/2π for all values of y except y = 0. Since the y-integral in Eq. (7.20) is not affected by the value of the integrand at the single point y = 0, we can substitute Eq. (7.22) into Eq. (7.20) to obtain

where the last equality follows from the Fourier inversion theorem. This finishes the proof, modulo the derivation of Step 2, which we present next.

Proof of Step 2. The first equality in Step 2 follows from the sixth property of Theorem 2.6. In more detail, we use the definition of the Fourier transform to obtain

After the change of variables v = (x — b)/a, this becomes

(7.23)

(7.24)

(7.25)

Taking conjugates of both sides of this equation establishes the first equality in Step 2.

To establish the second equality, we use the definition of the wavelet transform

The next step is to bring the Fourier transform operator F_b under the integral sign. This can be accomplished by viewing the integral as a sum and using the linearity of the Fourier transform operator. More precisely, we approximate the integral on the right by the following Riemann sum

This approximation becomes more accurate as the partition gets finer (i.e., as Δx → 0). The Fourier transform operator, F_b is linear and can be brought inside the sum to obtain

Now, we let the partition get finer and obtain (in the limit as Δx → 0)

Inserting Eq. (7.25) into this equation, we obtain

as claimed in the second equation of Step 2. This completes the derivation in Step 2 and the proof of the theorem.