Prime Coset Sum: A Systematic Method for Designing Multi-D Wavelet Filter Banks with Fast Algorithms

Prime Coset Sum: A Systematic Method for Designing Multi-D Wavelet Filter Banks with Fast Algorithms

Abstract

As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D lowpass filters, with one of whom being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation, 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D lowpass filters, and furthermore, 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.

1 Preliminaries

1.1 Introduction

Wavelet representation has been one of the most popular data representations in the last two decades. Wavelet filter banks, which can lead to wavelet systems in under some well-understood constraints, has been widely used in Signal Processing applications. In order to obtain wavelet representation for multi-dimensional (multi-D) data, one needs multi-D wavelets. Tensor product is the most common method for constructing multi-D wavelets, and the resulting wavelets are typically referred to as the separable wavelets. However, the separable wavelets constitute only a small portion of multi-D wavelets, and they have some unavoidable limitations. One of the limitations of tensor-product-based wavelets is that the resulting multi-D filters have dense supports. It is well known that the fast algorithms associated with tensor-product-based wavelets have a complexity constant (cf. Section 3.2 for the definition of complexity constant) that increases linearly with the spatial dimension . While this complexity may be satisfactory for many signal processing applications, it can pose a problem for many other signal processing applications, including the case when we deal with large volume data such as medical images in [12], Geographic Information Systems images in [38] and seismic data in [41]. Moreover, it is known that tensor-product-based discrete wavelet transform is memory consuming and cannot directly obtain the target subband signals, due to its dependent subband decomposition process [16]. There have been many researches on improving the implementation of the existing tensor-product-based wavelets [28, 26, 29, 8], as well as on constructing new non-tensor-based multi-D wavelets [22, 7, 23, 1, 27, 32, 9, 14, 20, 2, 3, 21, 24, 40, 35, 34, 5, 6, 37, 11, 15, 31, 43, 44]. However, most of these new constructions work only for low dimensions or have additional constraints on the lowpass filters. Furthermore, most of them are not associated with fast algorithms, preventing them from being widely used in practice.

Recently, the authors introduced a new method called coset sum for constructing non-tensor-based multi-D wavelets in [19]. There it was shown that the resulting wavelets are associated with fast algorithms whose complexity constant does not increase as the spatial dimension increases. It was also shown there that many features of tensor product that makes it attractive in wavelet construction still hold true for coset sum.

However, similar to the tensor product method, coset sum also assumes the dyadic dilation. We recall that the matrix is called a dilation matrix if it is an integer matrix whose spectrum lies outside the closed unit disc. It determines the exact way of how downsampling and upsampling are performed in wavelets or wavelet filter banks. The dilation is called scalar if the dilation matrix is a scalar multiple of the identity matrix , i.e., with an integer. In particular, it is called dyadic if . In this paper, we say that the dilation is prime if for a prime number . Wavelets with dyadic dilation are referred to as dyadic wavelets. Dyadic wavelets are the standard and traditional types of wavelets, however they are not suitable for all applications (see, for example, [30, 42, 13]).

In this paper, we show that we can generalize the coset sum in the sense that multi-D wavelet filter banks with fast algorithms can be constructed for any prime dilation . We also show that the complexity constant for our fast algorithms with prime dilation is independent of the spatial dimension.

The organization of this paper is as follows. The rest of Section 1 is a brief review of some relevant concepts including the coset sum method. In Section 2 we discuss a possible generalization of the coset sum, which we call prime coset sum, together with its properties. In Section 3 we present a new method to construct multi-D wavelet filter banks based on the prime coset sum refinement masks and show that they are associated with fast algorithms. Section 4 is a summary of our results. Some technical proofs and details in this paper are placed in Appendix.

1.2 Notation and Basic Concepts

Let be a dilation matrix and let . In the multiresolution analysis [25] setting, the (compactly supported) scaling or refinable function (with dilation ) satisfies the following refinement relation:

(1)

where is the associated finitely supported filter with dilation .

A mask associated with a finitely supported filter is a Laurent trigonometric polynomial defined as

for any . That is, is the Fourier transform of the filter , up to a normalization. Throughout this paper, we use to denote this Fourier transform of .

By taking the Fourier transform of (1), the refinement relation can be recast as

where is the mask associated with , and the superscript is used to denote the conjugate transpose of a matrix, and hence is the same as , the transpose of , in this case.

A mask with is typically referred to as a wavelet mask. In this paper, we use the normalization of the mask so that a mask with is referred to as a refinement mask. This is equivalent to , which is our normalization for a filter to be lowpass. A refinement mask is called interpolatory if, for any ,

where is a complete set of representatives of the distinct cosets of containing . For example, for the scalar dilation with , the set can be used for . We note that is interpolatory if and only if its corresponding filter satisfies

(2)

The order of zeros of at is called the accuracy number of . Throughout this paper, we assume that all refinement masks have at least accuracy number one. The order of zeros of at the origin is called the number of vanishing moments of . Thus a mask is a wavelet mask if and only if it has at least one vanishing moment. The order of zeros of at the origin is called the flatness number of . Thus a mask is a refinement mask if and only if it has at least flatness number one. Throughout this paper, we use the accuracy number, the number of vanishing moments, and the flatness number both for a mask and for the filter associated with it.

Two refinement masks and are called biorthogonal if

for any . Here and below, the overline is used to denote the complex conjugate. For the corresponding filters and of and , respectively, the biorthogonality condition becomes

For a pair of biorthogonal refinement masks and and wavelet masks and , , we refer to and as the combined biorthogonal masks if they satisfy the following condition: for every ,

(3)

It is well known that the combined biorthogonal masks can give rise to a biorthogonal wavelet system in (see, for example, [33]).

A filter bank is a finite set of filters. We consider only the filter banks that are non-redundant with the perfect reconstruction property [36]. A (non-redundant) filter bank consists of analysis bank and synthesis bank, which are collections of filters linked by downsampling and upsampling operators, respectively, associated with the dilation matrix . The analysis bank splits the input signal into signals typically called subband signals using a parallel set of bandpass filters. The synthesis bank reconstructs the original data from subband signals. We are interested in the wavelet filter bank for which each of analysis and synthesis banks has exactly one lowpass filter and the rest of them are all highpass filters. We recall that a filter is highpass if the associated mask is a wavelet mask, i.e. . The filters associated with the combined biorthogonal masks constitute a wavelet filter bank. Furthermore, it is well known that the minimum of accuracy numbers of lowpass filters in a given wavelet filter bank provides a lower bound for the number of vanishing moments of the highpass filters in the given wavelet filter bank [4].

1.3 Multi-D Wavelet Construction Methods: Tensor Product and Coset Sum

When is large, in general, it is not easy to find the combined biorthogonal masks and . However, if the dilation is dyadic (i.e. and ) and the spatial dimension satisfies , then the well-known tensor product and more recent coset sum can be used. Below we provide a brief review of these methods.

We recall that the -D tensor product mask from (possibly distinct) -D masks is defined as, for ,

Then starting from -D combined biorthogonal masks and with dyadic dilation, one can construct -D combined biorthogonal masks with dyadic dilation by setting the -D biorthogonal refinement masks as

and the -D wavelet masks , , , as

It is well known that the above tensor product method has many advantages: 1) it preserves the interpolatory property and the accuracy number of -D refinement masks; 2) it also preserves the biorthogonality between two refinement masks; and 3) the resulting separable wavelets are associated with fast algorithms (cf. Section 3.2). However, as discussed in Section 1.1, the limitations of the separable wavelets constructed from the tensor product are widely known.

Aa an alternative to the tensor product, a new method called coset sum for constructing -D dyadic refinement masks from -D dyadic refinement masks is recently proposed [19]. The coset sum refinement mask for a -D dyadic refinement mask is defined as

The following results about coset sum refinement masks and coset sum wavelet systems have been proved in [19].

Result 1

Let be the coset sum, and let and be univariate dyadic refinement masks.

  1. is interpolatory if and only if is interpolatory.

  2. Suppose that one of and is interpolatory. Then and are biorthogonal if and only if and are biorthogonal.

  3. Suppose that is interpolatory. Then and have the same accuracy number.

Result 2

Suppose that and are 1-D biorthogonal dyadic refinement masks, and that is interpolatory. Define -D biorthogonal refinement masks as

and -D wavelet masks , , as

(4)

Then there exist wavelet masks , , such that and are -D combined biorthogonal masks with dyadic dilation.

As we can see above, the coset sum and the tensor product method share many useful properties. In addition, the coset sum wavelets can overcome some of the limitations of the separable wavelets. For example, attributed to the smaller supports (number of nonzero entries) of the resulting multi-D filters, as well as the special structure of the filters, the coset sum can be associated with fast algorithms whose complexity constant does not increase with the spatial dimension. Therefore, in higher dimension, coset sum fast algorithms can be much faster than the tensor product fast algorithms. For more details about the coset sum including its comparison with the tensor product, we refer to [19].

2 Prime Coset Sum

Since coset sum has many useful properties including fast algorithms, which can be much faster than the existing tensor product fast algorithms, in this section, we try to extend the coset sum method to non-dyadic scalar dilations. The following simple lemma plays an important role in our generalization of coset sum.

Lemma 1

Let be a fixed spatial dimension. Let be a prime number, and let and be the complete set of representatives of the distinct cosets of and , respectively, containing . Then for every , we have

Remark 1

A special case of Lemma 1 for is used for the coset sum (cf. (19) in [19]).

Remark 2

In general, Lemma 1 does not hold true if is not a prime number. For example, when and , we can take and . Then, it is easy to see that if or , then the cardinality of the set is 1 (in fact, in both cases), whereas if , then and hence its cardinality is . As we will see below, in our proof of the lemma, we used crucially the fact that is a finite field for a prime number , which does not hold true anymore if is not a prime number.

Proof 1 (Proof of Lemma 1)

First of all, we claim that, without lose of generality, we may assume and . This is because for any other and , there is a one-to-one correspondence between the elements of and , and between the elements of and . To be more specific, for any other and , and for any and , there exist unique and such that

and vice versa. Therefore, . Hence the cardinality of the set is the same as the cardinality of the set .

Now for any , and , we let , and let and , , be the -th component of and . Then both and lie in the set . Since , at least one of ’s is not 0. Without loss of generality, we may assume . Furthermore, if and only if .

For any , and any , , let satisfy

Since is a finite field for a prime number , there exists a unique multiplicative inverse of such that . Then there exists a unique satisfies

Thus

Since there are different choices for , for any , we have

With Lemma 1 in hand, we define a particular generalization of coset sum for the prime dilation , where is a prime number. Let and be defined as in Lemma 1. For example, and can be used.

Motivated by the definition of the original coset sum (cf. Section 1.3), we consider a generalized coset sum of the form

where , and and are constants that will be determined soon. To pin down the constants and , we impose two conditions that we consider natural on the map . Firstly, we require to map a 1-D refinement mask with dilation to an -D refinement mask with dilation . That is, we want whenever . From this we get the equation

(5)

Secondly, we require the accuracy number of to be at least one whenever the accuracy number of the 1-D refinement mask is at least one. That is, we want, for any ,

where the last equality is due to Lemma 1. This gives the equation

(6)

By solving and that satisfy (5) and (6) simultaneously, we reach the following definition of a generalized coset sum for prime dilations.

Definition 1

Let be a prime number. We define the prime coset sum that maps a 1-D refinement mask with dilation to an -D refinement mask with dilation as follows: for any ,

where .

Remark 3

We refer to the refinement mask obtained by as the prime coset sum refinement mask. We notice that the prime coset sum with reduces to the original coset sum for dyadic dilation, i.e. (cf. Section 1.3 for the choice of and [19] for more general choice of ).

Let be the 1-D lowpass filter associated with the 1-D refinement mask . Let be the -D lowpass filter associated with the -D refinement mask . We refer to such a filter as the prime coset sum lowpass filter. For any nonzero , we define a set as . Then the -D prime coset sum lowpass filter can be written in terms of the 1-D lowpass filter as follows:

(7)

Now we give a simple example to show the construction of multi-D prime coset sum lowpass filters.

Example 1 (Centered -D Haar lowpass filter with dilation )

Consider the centered -D Haar lowpass filter with dilation :

Let us take . Then it is easy to check that the -D prime coset sum lowpass filter constructed from the -D centered Haar is

Figure 1 shows the -D filter and the resulting -D filter .

1  1  1      1 1 1 1 1 1 1 1 1

Figure 1: Construction of centered -D Haar lowpass filter with dilation using prime coset sum (cf. Example 1) 1

Some of the properties of the original coset sum (cf. Section 1.3) still hold true for the generalized prime coset sum.

Lemma 2

Let be the prime coset sum, and be a univariate refinement mask with dilation . If is interpolatory, then is interpolatory.

Proof 2

See Appendix .1.

Lemma 3

Let be the prime coset sum, be a univariate refinement mask with dilation , and let and be positive integers. Suppose that has accuracy and flatness. Then has at least accuracy.

Proof 3

See Appendix .2. Similar arguments to the ones given in [19] are used in our proof.

Remark 4

If is interpolatory, then . Hence, the above lemma says that, when is interpolatory, the accuracy number of is at least as much as the accuracy number of . For the case of the original coset sum with dyadic dilation, the accuracy number of is exactly the same as the accuracy number of when is interpolatory (cf. Result 1(c)). We do not yet know whether this result would hold true for the prime coset sum in general.

Lemma 4

Let be the prime coset sum, and be a univariate refinement mask with dilation . Then the flatness number of is at least the flatness number of .

We omit the proof of Lemma 4 as it is a simple variant of our proof of Lemma 3.

Unlike the original coset sum with dyadic dilation (cf. Result 1(b)), in general, the prime coset sum does not preserve the biorthogonality of 1-D refinement masks when , even if one of them is interpolatory. Let us look at two examples to this end. Both of them are related with the Haar refinement masks with dilation .

Example 2 (Centered -D Haar refinement mask with dilation )

Let us consider the centered 1-D Haar refinement mask as in Example 1:

Then the above mask has dilation and it is associated with the refinable function . If we define both and to be this centered -D Haar refinement mask with dilation , then they are interpolatory and biorthogonal with one accuracy.

Let us now take . Then, it is easy to see that transforming and to -D using the prime coset sum with produces two -D refinement masks and (cf. Figure 1) that are not only interpolatory with one accuracy, but also biorthogonal.

Example 3 (Non-centered -D Haar refinement mask with dilation )

Now let us consider the non-centered -D Haar refinement mask with dilation :

that is associated with the refinable function , where is the characteristic function on . Let both and be the above non-centered -D Haar refinement mask with dilation . Then it is easy to see that and are interpolatory and biorthogonal, and they have one accuracy.

We use this time. By transforming and to -D masks using the prime coset sum with , we see that and are still interpolatory and they still have one accuracy, but that they are no longer biorthogonal.

3 Multi-D Wavelet Filter Banks with Fast Algorithms

3.1 Theory

Suppose that and are 1-D biorthogonal refinement masks with dilation , and that is interpolatory. Since the -D prime coset sum refinement masks and are not necessarily biorthogonal (cf. Example 3 in Section 2), it is not trivial to construct wavelet filter banks from and directly. We propose to use a recent method developed by the first author [17]. This method can construct wavelet filter banks from two refinement masks that are not necessarily biorthogonal, as long as one of them is interpolatory. Noting that is interpolatory (cf. Lemma 2), we apply this method to and to construct wavelet filter banks. As we will see later (cf. Section 3.2), similar to the coset sum case, the resulting wavelet filter banks using this method can be associated with fast algorithms, that are faster than the tensor product fast algorithms.

Since the method in [17] works for any dilation matrix , below we present it for the general dilation matrix with . Let and be the complete set of representatives of the distinct cosets of and , respectively, containing . The following result is from [17] written in terms of our notation.

Result 3

Suppose and are two -D lowpass filters with dilation , and is interpolatory. Then the two -D refinement masks defined as

for every , and the -D wavelet masks defined as

and

for every , and , form the combined biorthogonal masks (cf. (3)).

Proof 4

Result 3 is proved in [17], but under slightly different settings. For completeness, we provide an alternative proof that does not rely on the results of [17]. Our proof is placed in Appendix .4.

Remark 5

In fact, the results in [17] say that, if we assume that, in addition to the assumptions of Result 3, has accuracy, has accuracy, and flatness, then has at least accuracy. In such a case, and , , have at least vanishing moments (cf. Section 1.2).

For the rest of this section, we assume that the dilation is prime, i.e. , and that the sets and are associated with the prime dilation, i.e., and are the complete set of representatives of the distinct cosets of and , respectively, containing . In particular, we have in this case.

Before presenting our main theorem, let us first define a map

with , where is a complete set of representatives of the distinct cosets of that contains 0. For example, the set can be used for . Let . Then there exists the unique multiplicative inverse of (cf. Remark 2 in Section 2). After computing the multiplication in the usual sense, we define to be the element in so that

By the above conditions, is uniquely well defined as an element in since is in but not in . For example, if , , and , then and .

Now we are ready to present our result.

Theorem 1

Suppose that and are two 1-D lowpass filters with dilation , and that is interpolatory. Let and be the 1-D refinement masks associated with and , and let be the prime coset sum. Define -D biorthogonal refinement masks as

for every , and -D wavelet masks as

(8)

and

(9)

for , and for every , where , and , .2 Then and form -D combined biorthogonal masks.

Remark 6

In the dyadic setting, i.e., when , one can take and . Then, since is the only element in and for all , the -D wavelet masks in (8) become

where the second identity is from the definition of and the third identity is from the fact that is interpolatory. The above wavelet masks are the same as the wavelet masks in the coset sum wavelet system (cf. (4) in Result 2) up to a normalization factor. In fact, the exact forms of for coset sum wavelet system are also provided in [19], and similar calculation shows that they are the same as in (9) up to a normalization factor when . Hence we conclude that Theorem 1 reduces to the known result of the original coset sum case when .

Remark 7

We refer to the wavelet filter bank associated with the combined biorthogonal masks constructed in Theorem 1 as the prime coset sum wavelet filter bank. There are many potentially useful properties of the prime coset sum wavelet filter banks. One important property is that it can be implemented by fast algorithms (cf. Section 3.2).

Remark 8

In addition to the assumptions of Theorem 1, if we assume that has accuracy, has accuracy, and flatness, then by Lemma 3 and Lemma 4, has at least accuracy, has at least accuracy, and at least flatness. Combining these with Remark 5, we conclude that has at least accuracy, and and , , have at least vanishing moments.

In order to prove Theorem 1, we use the following lemma which connects the polyphase decomposition of the 1-D lowpass filter and the polyphase decomposition of the -D prime coset sum lowpass filter obtained from . Polyphase decomposition is a common method in Signal Processing and we give a brief review in Appendix .3.

Lemma 5

Let be a 1-D lowpass filter with dilation , and let be the -D lowpass filter obtained from by applying the prime coset sum . Let the sets and , and the map be defined as before. Then for any ,

Proof 5

First it is easy to see that (cf. (15) in Appendix .3)

Using this identity and the definition of prime coset sum, we get

(10)

Next we use another identity that can be quickly derived (cf. (48) in [17]):

(11)

By using (10), (11), and the fact that , for any , , and , we obtain

Then we use the following simple identity (cf. (20)):

to get