33institutetext: National Engineering Research Center for Big Data Technology
and System,
Services Computing Technology and System Lab,
Cluster and Grid Computing Lab,
School of Computer Science and Technology,
Huazhong University of Science and Technology,
430074, Wuhan, China
<EMAIL_ADDRESS>22institutetext: Computer Vision Institute,
School of Computer Science and Software Engineering,
Shenzhen University
<EMAIL_ADDRESS><EMAIL_ADDRESS>33institutetext: School of Hydropower
and Information Technology,
Huazhong University of Science and Technology,
430074, Wuhan, China
<EMAIL_ADDRESS>
# Context Aware 3D UNet for Brain Tumor Segmentation

Parvez Ahmad 11 0000-0003-1409-3175 Saqib Qamar 22 0000-0002-5980-5976 Linlin
Shen 22 Adnan Saeed 33 0000-0002-0784-5861

###### Abstract

Deep convolutional neural network (CNN) achieves remarkable performance for
medical image analysis. UNet is the primary source in the performance of $3$D
CNN architectures for medical imaging tasks, including brain tumor
segmentation. The skip connection in the UNet architecture concatenates
features from both encoder and decoder paths to extract multi-contextual
information from image data. The multi-scaled features play an essential role
in brain tumor segmentation. However, the limited use of features can degrade
the performance of the UNet approach for segmentation. In this paper, we
propose a modified UNet architecture for brain tumor segmentation. In the
proposed architecture, we used densely connected blocks in both encoder and
decoder paths to extract multi-contextual information from the concept of
feature reusability. In addition, residual-inception blocks (RIB) are used to
extract the local and global information by merging features of different
kernel sizes. We validate the proposed architecture on the multi-modal brain
tumor segmentation challenge (BRATS) 2020 testing dataset. The dice (DSC)
scores of the whole tumor (WT), tumor core (TC), and enhancing tumor (ET) are
$89.12\%$, $84.74\%$, and $79.12\%$, respectively.

###### Keywords:

CNN UNet Contexual information Dense connections Residual inception blocks
Brain tumor segmentation.

##  1 Introduction

Brain tumor is the growth of irregular cells in the central nervous system
that can be life-threatening. Primary and secondary are two types of brain
tumors. Primary brain tumors originate from brain cells, whereas secondary
tumors metastasize into the brain from other organs. Gliomas are primary brain
tumors. Gliomas can be further sub-divided into high-grade glioblastoma (HGG)
and low-grade glioblastoma (LGG). In the diagnosis and treatment planning of
glioblastoma, brain tumor segmentation results can derive quantitative
measurements. While radiologists have manually analyzed magnetic resonance
imaging (MRI) modalities to derive information quantitatively, however
segmenting 3D modalities is a time-consuming task with deviations and errors.
This difficulty is further increases if organs have variation in terms of
shape, size, and location. Conversely, Convolutional Neural Networks (CNNs)
can apply to the MRI images to develop automatic segmentation methods. Deep
CNNs have achieved remarkable performances for brain tumor segmentation [6],
[12], [8], [15], [23]. A $3$D UNet is a popular variation of UNet architecture
for automatic brain tumor segmentation [9], [13], [22]. The multi-scale
contextual information of the encoder-decoder paths is effective for the
accurate brain tumor segmentation task. Researchers have presented variant
forms of the $3$D UNet to extract the enhanced contextual information from MRI
[14], [16]. Network’s depth is a common factor to improve the performances
among approaches. Residual networks [10] and dense connections [11] are
effective to acquire the possible depth in the architecture. Our proposed
method used dense connections and the residual-inception blocks [24] to
extract the meaningful contextual information from brain MRIs. We used densely
connected blocks in both encoder-decoder paths to obtain more abstract
features. In the previous approach [1], we have used a low number of densely
connected blocks as compared to the proposed approach. In the meantime,
residual-inception block (RIB) is used to extract local and global information
by merging features of different kernel sizes. Our proposed model gives
scalable $3D$ UNet architecture for brain tumor segmentation in the view of
these combinations. The key contributions of this study are as follows:

  * •

We proposed a novel densely connected $3D$ encoder-decoder architecture to
extract context features at each level of the network.

  * •

We used residual-inception block (RIB) to extract local and global information
by merging features of different kernel sizes.

  * •

Our network achieves state-of-the-art performance as compared to other recent
methods.

##  2 Proposed Method

Fig. 1 is shown our proposed architecture for brain tumor segmentation. In our
previous work [1], we have proposed the combined benefits of the residual and
dense connections by using Atrous Spatial Pyramid Pooling (ASPP) [7]. However,
insufficient contextual information in each block of the encoder-decoder paths
limits the previous model’s performance. Moreover, an insufficient number of
higher layers degrades the scores of the model. We designed the novel densely
connected $3$D encoder-decoder architecture for brain tumor segmentation to
address these issues. We used dense connections in our proposed architecture
while enhancing the maximum features’ size to $32$ in the final output layer.
Therefore, the number of features is twice as compared to the previous
architecture. The output features at the levels of the encoder path are $32$,
$64$, $128$, $256$, and $512$. The proposed work can be divided into (i) dense
blocks, which are building blocks of the encoder-decoder paths, and (ii)
residual-inception blocks, which are used to the first dense block of the
encoder path and along with the upsampling layers of the decoder path.

Figure 1: Proposed densely-connected $3$D UNet architecture. Each dense block
(green) has three convolution layers. The first block of the encoder path has
features of RIB to address the sizes of tumors. A similar approach is employed
after each upsampling process (red) in the decoder part.

###  2.1 Dense Blocks

Dense connections [11] have been exceptional in delivering high accuracy both
in the medical [6, 17, 8] and non-medical domains [11]. Dense connections have
a feature reuse property, in which output feature maps of all previous layers
are the inputs to the subsequent layers. Thus, the feature reusability
property of dense connection reduces the network’s parameters and improve
segmentation accuracy. In addition, dense connections enable multi-path flow
for gradients between layers during training by back-propagation and hence
does implicit deep-supervision. Therefore, inspiring by the dense networks, we
use them for each dense block. Each block of the encoder-decoder paths has
three convolution layers. The first dense block of the encoder path is shown
in Fig. 2. Here, the output feature maps of a residual-inception block (RIB)
and the first convolution layer are concatenated. The concatenated features
are then passed to the first convolution layer of a dense block. In addition,
we doubled the output feature maps of the first dense block. Subsequently, the
output feature maps of the remaining dense blocks in the encoder path are
doubled to improve the contexts. We also use a growth-rate value of $2$, which
aids in reducing the training parameters and enables the proposed model to fit
into the GPU memory.

Figure 2: Proposed densely connected structure of the first block of the
encoder path. The first convolution layer and residual-inception block (RIB)
(gold) generate $16$ concatenated output feature maps to the first convolution
layer ($3\times 3\times 3$ CONV-Instance Normalization-ReLU) of a dense block.
In a dense block, a dropout rate of $0.2$ is used after the first convolution
layer. Simultaneously, all the preceding output feature maps are the inputs to
the next layers of a dense block, followed by a concatenation operation. Here,
the output feature maps are doubled (from $16$ to $32$). Finally, the
resulting output feature maps are passed to the next dense blocks.

###  2.2 Residual-Inception Blocks

A residual-inception block (RIB) is designed with three parallel dilated
convolution layers (of rates $1$, $3$, and $5$). It is employed along with the
first dense block of the encoder path. Subsequently, RIB is employed after
each upsampling operation in the decoder path. The residual connections are
used in the RIB to prevent the vanishing gradient problem, while the inception
blocks provide multi-scale contexts to the existing residual networks. In this
way, the combined RIB architecture improves the size of the features. Fig. 3
is shown a RIB architecture, which is proposed to address the problem of
different tumors through multiple sizes of the receptive field. Multi-scale
contexual information of a RIB can reduce the number of false-positives and
prevent the failed segmentation problem. Moreover, instead of concatenation
operations, we used the addition operations for skip-connections, which are
more computationally and memory-efficient. Furthermore, we also keep the dense
connections to improve the feature’s strength by concatenating feature maps of
addition operations and the decoder part’s dense blocks. Finally, the softmax
layer is employed for the outcomes.

Figure 3: Proposed RIB structure of the $3$D UNet. The output feature maps of
the previous layer are the inputs to the three parallel dilated layers. Output
maps of every parallel layer are enhanced with dense connections (dashed red
lines) to minimize the problem of multiple sizes of the tumors.

##  3 Experimental Results

###  3.1 Dataset

The BRATS aims to bring the research communities together, along with their
brilliant ideas for different tasks. Especially for the segmentation task,
public benchmark datasets are provided by the organizers. In BRATS challenges
[2], [3], [4], [5], [20] organizers provide various independent datasets for
training, validation, and testing. In this paper, we use BRATS $2018$, BRATS
$2019$, and BRATS $2020$ datasets to train and evaluate our proposed work.
Details of each year’s BRATS dataset are shown in Table 1. Here, we can notice
each training dataset’s further classification into high-grade glioblastoma
(HGG) and low-grade glioblastoma (LGG). Furthermore, we have only access to
the training and validation datasets of BRATS $2018$ and BRATS $2019$. Four
different types of modalities, i.e., native (T1), post-contrast T1-weighted
(T1ce), T2-weighted (T2), and Fluid Attenuated Inversion Recovery (FLAIR), are
related to each patient in the training, validation, and testing datasets. For
each training patient, the annotated labels have the values of $1$ for the
necrosis and non-enhancing tumor (NCR/NET), $2$ for peritumoral edema (ED),
$4$ for enhancing tumor (ET), and $0$ for the background. The segmentation
accuracy is measured by several metrics, where the predicted labels are
evaluated by merging three regions, namely whole tumor (Whole Tumor or Whole:
label $1$, $2$ and $4$), tumor core (Tumor Core or Core: label $1$ and $4$),
and enhancing tumor (Enhancing Tumor or Enhancing: label $4$). The organizers
performed necessary pre-processing steps for simplicity. However, the truth
label is not provided for the patients of the validation and testing datasets.

Table 1: Details of BRATS $2018$, $2019$ and $2020$ datasets.

Dataset | Type | Patients | HGG | LGG
---|---|---|---|---
BRATS 2018 | Training | 285 | 210 | 75
| Validation | 66 |  |
BRATS 2019 | Training | 335 | 259 | 76
| Validation | 125 |  |
BRATS 2020 | Training | 369 | 293 | 76
| Validation | 125 |  |
| Testing | 166 |  |

###  3.2 Implementation Details

Since $19$ institutes are involved in data collection, these institutes used
multiple scanners and imaging protocols to acquire the brain scans. Thus,
normalization would be necessary to establish a similar range of intensity for
all the patients and their various modalities to avoid the network’s initial
biases. Here, normalization of entire data may degrade the segmentation
accuracy. Therefore, we normalize each MRI of each patient independently. We
extract patches of size $128\times 128\times 128$ from $4$ MRI modalities to
feed them into the network. We used five-fold cross-validation, in which each
time our network is trained $300$ epochs. The batch size is $1$. Adam is the
optimizer with the initial learning rate $7\times 10{{}^{-5}}$, which is
dropped by $50\%$ if validation loss not improved within $30$ epochs.
Moreover, we used augmentation techniques during the training by randomly
rotating the images within a range of $[$$,$$]$ and random mirror flips (on
the x-axis) with a probability of $0.5$.

During the designing of our network, we have tuned several hyperparameters,
such as the number of layers for the dense and residual-inception blocks, the
initial number of channels for training, the growth rate’s value, and the
number of epochs, etc. Hence, to avoid any further hyperparameter tunning and
inspired by a non-weighted loss function’s potential, we employed the
previously proposed multi-class dice loss function [21]. Thus, the earlier
mentioned dice loss function can be easily adapted with our proposed model and
summarized as

$Loss=-\frac{2}{D}\sum_{d\in
D}\frac{\sum_{j}P_{(j,d)}T_{(j,d)}}{{\sum_{j}P_{(j,d)}}+{\sum_{j}T_{(j,d)}}}\linebreak$
(1)

where $P_{(j,d)}$ and $T_{(j,d)}$ are the prediction obtained by softmax
activation and ground truth at voxel $j$ for class $d$, respectively. $D$ is
the total number of classes.

###  3.3 Qualitative Analysis

Fig. 4 and Fig. 5 are shown the segmentation results of our proposed
architecture. Fig. 4 shows the $T1ce$ axial slice of $HGG$ patient from the
training dataset, while Fig. 5 depicts the $T1ce$ sagittal and coronal slices
of four different $HGG$ patients. Fig. 4(a) and Fig. 4(b) depicted the truth
and segmented labels overlaid on $T1ce$ axial slices. Fig. 5(a) and Fig. 5(c)
shows the truth labels overlaid on $T1ce$ sagittal and coronal slices, while
the overlaying of the segmented labels on $T1ce$ sagittal and coronal slices
is shown in Fig. 5(b) and Fig. 5(d), respectively. Based on the visualized
slices, our proposed model can accurately segment the truth labels of axial,
sagittal and coronal slices.

(a) Axial Truth

(b) Axial Segmented

Figure 4: Segmentation results. $(a)$ and $(b)$ represent the truth and
segmented labels overlaid on $T1ce$ axial slices. Different colors represent
different parts of the tumor: red for TC, green for WT, and yellow for ET.

(a) Sagittal Truth

(b) Prediction

(c) Coronal Truth

(d) Prediction

Figure 5: Segmentation results. $(a)$ and $(c)$ represent the overlaying of
truth labels on $T1ce$ sagittal and coronal slices. Simultaneously, $b$ and
$d$ shows the segmented labels overlaid on $T1ce$ sagittal and coronal slices.
Different colors represent different parts of the tumor: red for TC, green for
WT, and yellow for ET.

###  3.4 Quantitative Analysis

We now evaluate our proposed work on BRATS datasets of $2018$, $2019$, and
$2020$. In this paper, our main contribution is to propose a variant form of
$3$D UNet, which can improve context information. Hence, a simple training
procedure is performed to check the potential of the proposed model. For deep
learning models, cross-validation is a powerful strategy, which is useful with
a limited dataset in reducing the variance. Therefore, we perform a five-fold
cross-validation procedure on the BRATS $2020$ training dataset. After
training, five models are used to evaluate the BRATS $2020$ validation
dataset. Furthermore, a simple post-processing step is performed to remove
false-positive voxels from the training and validation predicted datasets. At
a threshold value of $0.5$, all enhancing tumor regions with less than 500
voxels are replaced with the necrosis [13]. Finally, an average operation is
performed on five predicted training and validation sets for the final
submission. The scores of the BRATS $2020$ training, validation, and testing
datasets are shown in Table 2 (see top three rows).

While the proposed model has obtained encouraging scores, however, GPU memory
consumption is high. Therefore, we will modify our proposed work to fit into
the lowest available GPU memory in the future. Nevertheless, in this paper, we
evaluate the proposed model on the BRATS $2019$ and $2018$ datasets.
Therefore, a five-fold cross-validation strategy is also performed on the
BRATS $2019$ and BRATS $2018$ training datasets. However, the results are
based only on the best models (training models of the BRATS $2019$ and BRATS
$2018$ based on the highest mean dice scores are respectively used to evaluate
the $125$ cases of the BRATS $2019$ validation and $66$ patients of the BRATS
$2018$ validation datasets). The scores of the BRATS $2019$ and BRATS $2018$
datasets are shown in Table 2 (see the fourth row for BRATS $2019$ and the
fifth row for BRATS $2018$). Furthermore, the Hausdorff metric scores are not
included in Table 2 because the Hausdorff metric is highly susceptible to the
outlier. Hence, it is not a reliable metric for medical image segmentation
[25].

Table 3 and Table 4 show the comparisons between the proposed model and the
state-of-the-art methods in the MICCAI BRATS $2018$ and $2019$ validation
datasets, respectively. Table 4 shows the mean DSC value of our previous and
proposed works. Context-aware $3D$ UNet approach gains state-of-the-art
performances for brain tumor segmentation. We used different dilation rates
($1$, $3$ and $5$) in the residual-inception blocks to address the problem of
losing information due to sparsed kernels [26]. The DSC values for WT and TC
are best, and ET is lower for both BRATS $2018$ and $2019$ validation
datasets. ET value can be improved by using some post-processing strategies
(McKinley [Filtered Output] [19]). We are currently studying the influence of
dilated and non-dilated convolution layers on the DSC value of the ET. We will
try to improve the mean DSC score of the ET by augmentation techniques based
on the classes [27].

Table 2: The average scores of different metrics. For BRATS $2018$ and $2019$,
only validation scores are presented. The BRATS organizers validate all the
given scores.

Dataset | Metrics | Whole | Core | Enhancing
---|---|---|---|---
BRATS 2020 Training | DSC | 93.680 | 91.829 | 81.677
| Sensitivity | 94.052 | 92.189 | 82.839
| Specificity | 99.934 | 99.962 | 99.975
BRATS 2020 Validation | DSC | 90.678 | 84.248 | 75.635
| Sensitivity | 90.390 | 80.455 | 75.300
| Specificity | 99.929 | 99.975 | 99.975
BRATS 2020 Testing | DSC | 89.120 | 84.674 | 79.100
| Sensitivity | 89.983 | 85.551 | 84.287
| Specificity | 99.929 | 99.969 | 99.961
BRATS 2019 Validation | DSC | 90.217 | 83.435 | 72.289
| Sensitivity | 91.570 | 82.175 | 79.863
| Specificity | 99.370 | 99.744 | 99.829
BRATS 2018 Validation | DSC | 91.173 | 84.108 | 77.000
| Sensitivity | 91.830 | 82.059 | 84.367
| Specificity | 99.520 | 99.830 | 99.762

Table 3: Performance evaluation of different methods on the BRATS $2018$
validation dataset. For comparison, only DSC scores are shown. All scores are
evaluated online.

Methods | Enhancing | Whole | Core
---|---|---|---
Isensee [baseline] [13] | 79.590 | 90.800 | 84.320
McKinley [Base + U + BE] [18] | 79.600 | 90.300 | 84.700
Myronenko [Single Model] [22] | 81.450 | 90.420 | 85.960
Proposed [Single Model] | 77.000 | 91.173 | 84.108

Table 4: Performance evaluation of different methods on the BRATS $2019$
validation dataset. For comparison, only DSC scores are shown. All scores are
evaluated online.

Methods | Enhancing | Whole | Core
---|---|---|---
Previous Work [ours [1]] | 62.301 | 85.184 | 75.762
Zhao [BL+warmup+fuse] [14] | 73.700 | 90.800 | 82.300
McKinley [Raw Output] [19] | 75.000 | 91.000 | 81.000
McKinley [Filtered Output] [19] | 77.000 | 91.000 | 81.000
Proposed [Single Model] | 72.289 | 90.217 | 83.435

##  4 Discussion and Conclusion

We have proposed a unique $3D$ UNet model for brain tumor segmentation. The
proposed architecture consists of two sub-modules: (i) dense connections at
each level of the encoder-decoder paths, (ii) RIB to extract local and global
contextual information by merging feature maps of different kernel’s rate.
This study addressed a lack of essential information by using context features
at each level of encoder-decoder paths. Therefore, the mean average DSC scores
of the TC and the WT are improved. In the meantime, the ET score does not
improve due to various reasons: (i) zero value of the label ET in the $LGG$
dataset of the BRATS training and the validation datasets, respectively (ii)
all the presented scores are based on unbiased corrected brain MRI volumes.
(iii) the given model has $84$ layers, and we are trying to build more depth
to address it. At the same time, the proposed work has obtained competitive
scores. However, the number of channels, which doubled at the end of each
dense block in the encoder path, requires huge GPU memory. Therefore, the
proposed model should be modified to fit into the low GPU memory. In the
future, we will try to develop light $3D$ CNN architectures and investigate
the augmentation techniques based on the classes to minimize the class
imbalance problem. In summary, our proposed model has the potential to address
the issue of other medical imaging tasks.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China
under Grant No. 91959108.

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Characterization of the pseudo-scaling functions on Vilenkin group

Prasadini Mahapatra

###### Abstract

The study of wavelets, originated from it’s applications in diverse fields,
was combined together by it’s Mathematical properties. Initially, all the
wavelets and it’s variants were explored in the real space $\mathbb{R}^{n}$.
But now, these are being studied in different abstract settings. The present
paper also contributes to this extension. Vilenkin groups, introduced by F. Ya
Vilenkin, form a class of locally compact abelian groups. In this paper,
Parseval frame multiwavelets associated to multiresolution analysis (MRA) are
characterized in $L^{2}(G)$, where $G$ is the Vilenkin group. Further, we
introduce the pseudo-scaling function along with a class of generalized low
pass filters and study their properties in Vilenkin group.

##  1 Introduction

During the last two decades, several authors studied more generalizations and
extensions of wavelets. Multiresolution analysis (MRA) is very fundamental
tool to establishment of scaling function, which appeared in very different
contexts. This paper is related to one such generalization of wavelets.

Walsh functions were introduced by J. Walsh In 1923, J. Walsh introduced that
the linear combination of Haar functions is known as Walsh functions. At
first, N. J. Fine and N. Ya Vilenkin independently determined that Walsh
system is the group of characters of the Cantor dyadic group. A large class of
locally compact abelian groups, called Vilenkin groups is introduced by
Vilenkin. Cantor dyadic group is a particular case. Refinable equation gives
refinable function which generates MRA and hence wavelets, if the mask
satisfies certain conditions. Necessary and sufficient conditions were given
over the mask of scaling function $\phi$ in terms of modified Cohen’s
condition and blocked sets such that $\phi$ generates an MRA.

Wavelets and multiwavelets related work on Vilenkin group has been done in
several paper. In case of Vilenkin group if the associated prime $p$ is
greater than 2, then MRA generates a multiwavelet set having $p-1$ functions.

In section 2, It has been introduced that the pseudo scaling functions
associated with the filters and generalized filters. Then we characterize the
generalized low pass filters, which motivate us to construct the subclass of
MRA Parseval frame multiwavelets. Furthermore, we have studied that the
associated class of pseudo-scaling functions which are not necessarily
obtained from a multiresolution analysis.

###  1.1 Preliminaries

Vilenkin group $G$ is defined as the group of sequences

$x=(x_{j})=(...,0,0,x_{k},x_{k+1},x_{k+2},...),$

where $x_{j}\in\\{0,1,...,p-1\\}$, $p$ is prime, for $j\in\mathbb{Z}$ and
$x_{j}=0$, for $j<k=k(x)$. The group operation on $G$, denoted by $\oplus$, is
defined as coordinatewise addition modulo $p$:

$(z_{j})=(x_{j})\oplus(y_{j})\Leftrightarrow
z_{j}=x_{j}+y_{j}(\text{mod}\>p),\text{ for }j\in\mathbb{Z}.$

$\theta$ denotes the identity element (zero) of $G$. Let

$U_{l}=\\{(x_{j})\in G:x_{j}=0\>\text{for}\>j\leq l\\},\qquad l\in\mathbb{Z},$

be a system of neighbourhoods of zero in $G$. In case of topological groups if
we know neighbourhood system $\\{U_{l}\\}_{l\in\mathbb{Z}}$ of zero, then we
can determine neighbourhood system of every point $x=(x_{j})\in G$ given by
$\\{U_{l}\oplus x\\}_{l\in\mathbb{Z}}$, which in turn generates a topology on
$G$.

Let $U=U_{0}$ and $\ominus$ denotes the inverse operation of $\oplus$. The
Lebesgue spaces $L^{q}(G),\>1\leq q\leq\infty$, are defined with respect to
the Haar measure $\mu$ on Borel subsets of $G$ normalized by $\mu(U)=1$.

The group dual to $G$ is denoted by $G^{*}$ and consists of all sequences of
the form

$\omega=(\omega_{j})=(...,0,0,\omega_{k},\omega_{k+1},\omega_{k+2},...),$

where $\omega_{j}\in\\{0,1,...,p-1\\}$, for $j\in\mathbb{Z}$ and
$\omega_{j}=0$, for $j<k=k(\omega)$. The operations of addition and
subtraction, the neighbourhoods $\\{U_{l}^{*}\\}$ and the Haar measure
$\mu^{*}$ for $G^{*}$ are defined as above for $G$. Each character on $G$ is
defined as

$\chi(x,\omega)=\exp\bigg{(}\frac{2\pi
i}{p}\sum_{j\in\mathbb{Z}}{x_{j}w_{1-j}}\bigg{)},\quad x\in G,$

for some $\omega\in G^{*}$.

Let $H=\\{(x_{j})\in G\>|\>x_{j}=0,\;\text{ for }j>0\\}$ be a discrete
subgroup in $G$ and $A$ be an automorphism on $G$ defined by
$(Ax)_{j}=x_{j+1}$, for $x=(x_{j})\in G$. From the definition of annihilator
and above definition of character $\chi$, it follows that the annihilator
$H^{\perp}$ of the subgroup $H$ consists of all sequences $(\omega_{j})\in
G^{*}$ which satisfy $\omega_{j}=0$ for $j>0$.

Let $\lambda:G\longrightarrow\mathbb{R}_{+}$ be defined by

$\lambda(x)=\sum_{j\in\mathbb{Z}}{x_{j}p^{-j}},\qquad x=(x_{j})\in G.$

It is obvious that the image of $H$ under $\lambda$ is the set of non-negative
integers $\mathbb{Z}_{+}$. For every $\alpha\in\mathbb{Z}_{+}$, let
$h_{[\alpha]}$ denote the element of $H$ such that
$\lambda(h_{[\alpha]})=\alpha$. For $G^{*}$, the map
$\lambda^{*}:G^{*}\longrightarrow\mathbb{R}_{+}$, the automorphism
$B\in\text{Aut }G^{*}$, the subgroup $U^{*}$ and the elements
$\omega_{[\alpha]}$ of $H^{\perp}$ are defined similar to $\lambda$, $A$, $U$
and $h_{[\alpha]}$, respectively.

The generalised Walsh functions for $G$ are defined by

$W_{\alpha}(x)=\chi(x,\omega_{[\alpha]}),\qquad\alpha\in\mathbb{Z}_{+},x\in
G.$

These functions form an orthogonal set for $L^{2}(U)$, that is,

$\int_{U}{W_{\alpha}(x)\overline{W_{\beta}(x)}d\mu(x)}=\delta_{\alpha,\beta},\qquad\alpha,\beta\in\mathbb{Z}_{+},$

where $\delta_{\alpha,\beta}$ is the Kronecker delta. The system
${W_{\alpha}}$ is complete in $L^{2}(U)$. The corresponding system for $G^{*}$
is defined by

$W_{\alpha}^{*}(\omega)=\chi(h_{[\alpha]},\omega),\qquad\alpha\in\mathbb{Z}_{+},\omega\in
G^{*}.$

The system $\\{W_{\alpha}^{*}\\}$ is an orthonormal basis of $L^{2}(U^{*})$.

For positive integers $n$ and $\alpha$,

$U_{n,\alpha}=A^{-n}(h_{[\alpha]})\oplus A^{-n}(U).$

###  1.2 Wavelets on Vilenkin group

In [farkov3], Farkov considered the Strang-fix condition, partition of unit
property and the stability of scaling functions on Vilenkin group. Necessary
and sufficient conditions are given for scaling functions to generate an MRA
in the $L^{2}$ space on Vilenkin groups by using modified Cohen’s condition
and blocked sets.

####  1.2.1 Refinable function and mask

######  Definition 1.1.

Let $L_{c}^{2}(G)$ be the set of all compactly supported functions in
$L^{2}(G)$. A function $\phi\in L_{c}^{2}(G)$ is said to be a refinable
function, if it satisfies an equation of the type

$\phi(x)=p\sum_{\alpha=0}^{p^{n}-1}{a_{\alpha}\phi(Ax\ominus h_{[\alpha]})}.$
(1)

The above functional equation is called the refinement equation. The
generalized Walsh polynomial

$m(\omega)=\sum_{\alpha=0}^{p^{n}-1}{a_{\alpha}\overline{W_{\alpha}^{*}{(\omega)}}}$
(2)

is called the mask of the refinement equation (or the mask of its solution
$\phi$).

######  Theorem 1.2.

Let $\phi\in L_{c}^{2}{(G)}$ be a solution of the refinement equation, and let
$\widehat{\phi}(\theta)=1$. Then,

$\sum_{\alpha=0}^{p^{n}-1}{a_{\alpha}}=1,\qquad\text{supp }\phi\subset
U_{1-n},$

and

$\widehat{\phi}(\omega)=\prod_{j=1}^{\infty}{m(B^{-j}\omega)}.$

Moreover, the following properties are true:
$1$. $\widehat{\phi}(h^{*})=0$, for all $h^{*}\in
H^{\perp}\setminus{\\{\theta\\}}$ (the modified Strang-Fix condition),
$2$. $\sum_{h\in H}{\phi(x\oplus h)}=1$, for almost every $x\in G$ (the
partition of unit property).

######  Definition 1.3.

A set $M\subset U^{*}$ is said to be blocked (for the mask $m$) if it
coincides with some union of the sets $U_{n-1,s}^{*}$, $0\leq s\leq
p^{n-1}-1$, does not contain the set $U_{n-1,0}^{*}$, and satisfies the
condition

$T_{p}{M}\subset M\cup\\{\omega\in U^{*}:m(\omega)=0\\},$

where

$T_{p}M=\bigcup_{l=0}^{p-1}\\{B^{-l}\omega_{[l]}+B^{-1}(\omega):\omega\in
M\\}.$

####  1.2.2 Multiresolution analysis

######  Definition 1.4.

A collection of closed subspaces $V_{j}\subset L^{2}{(G)},j\in\mathbb{Z}$, is
called a Multiresolution analysis (MRA) in $L^{2}{(G)}$ if the following hold:
(i) $V_{j}\subset V_{j+1}$, for all $j\in\mathbb{Z}$
(ii) $\overline{\cup_{j\in\mathbb{Z}}V_{j}}=L^{2}{(G)}$ and
$\cap_{j\in\mathbb{Z}}V_{j}=\\{0\\}$
(iii) $f(\cdot)\in V_{j}\Leftrightarrow f(A\cdot)\in V_{j+1}$, for all
$j\in\mathbb{Z}$
(iv) $f(\cdot)\in V_{0}\Rightarrow f(\cdot\ominus h)\in V_{0}$, for all $h\in
H$
(v) there is a function $\phi\in L^{2}{(G)}$ such that the system
$\\{\phi(\cdot\ominus h)|h\in H\\}$ is an orthonormal basis of $V_{0}$.

The function $\phi$ in condition (v) is called a scaling function of the MRA
$(V_{j})_{j\in\mathbb{Z}}$.

For $\phi\in L^{2}{(G)}$,

$\phi_{j,h}(x)=p^{j/2}{\phi(A^{j}{x}\ominus h)},\qquad j\in\mathbb{Z},\;\;h\in
H$

and the system $\\{\phi_{j,h}:h\in H\\}$ forms an orthonormal basis of
$V_{j}$, for every $j\in\mathbb{Z}$.

######  Theorem 1.5.

A function $\phi\in L^{2}(G^{*})$ is a scaling function for an MRA of
$L^{2}(G^{*})$ if and only if

  1. 1.

$\sum_{h\in H^{\perp}}|\hat{\phi}(\omega\oplus h)|^{2}=1$, for a.e $\omega\in
G^{*}$

  2. 2.

$lim_{j\rightarrow\infty}|\hat{\phi}(B^{-j}\omega)|=1$, for a.e $\omega\in
G^{*}$

  3. 3.

$\hat{\phi}(B\omega)=m(\omega)\hat{\phi}(\omega)$, for a.e $\omega\in G^{*}$.

A function $\phi$ is said to generate an MRA in $L^{2}{(G)}$ if the system
$\\{\phi(\cdot\ominus h)|h\in H\\}$ is orthonormal in $L^{2}{(G)}$ and, the
family of subspaces

$V_{j}=\text{clos}_{L^{2}{(G)}}\text{span}\\{\phi_{j,h}:h\in H\\},\qquad
j\in\mathbb{Z},$

is an MRA in $L^{2}{(G)}$ with scaling function $\phi$. Farkov gave the
following condition under which a compactly supported function $\phi\in
L^{2}(G)$ generates an MRA in $L^{2}(G)$.

######  Theorem 1.6.

Suppose that the refinement equation possesses a solution $\phi$ such that
$\widehat{\phi}(\theta)=1$ and the corresponding mask $m$ satisfies the
conditions

$m(\theta)=1\qquad\Sigma_{l=0}^{p-1}|m(\omega\oplus\delta_{l})|^{2}=1,\;\;\omega\in
G^{*},$

where $\delta_{l}$ is the sequence $\omega=(\omega_{j})$ such that
$\omega_{1}=l$ and $\omega_{j}=0$ for $j\neq 1$. Then the following are
equivalent:
(a) $\phi$ generates an MRA in $L^{2}(G)$.
(b) $m$ satisfies the modified Cohen’s condition.
(c) $m$ has no blocked sets.

Using the above characterization of refinable function and the matrix
extension method Farkov gave a procedure for construction of orthonormal
wavelets $\psi_{1},...,\psi_{p-1}$ such that the functions

$\psi_{l,j,h}(x)=p^{j/2}{\psi_{l}{(A^{j}x\ominus h)}},\qquad 1\leq l\leq
p-1,j\in\mathbb{Z},\;\;h\in H,$

form an orthonormal basis of $L^{2}{(G)}$.

##  2 Characterization the pseudo-scaling functions

In this section, We establish the notion of pseudo scaling function and its
impact on generalized filters.

For multiwavelet $\Psi:=\\{\psi_{1},\psi_{2},...,\psi_{p-1}\\}\subset
L^{2}(G)$, The affine system $\mathcal{A}(\Psi)$ is defined by

$\mathcal{A}(\Psi)=\\{\psi_{j,h}^{l}(x)|\psi_{j,h}^{l}(x)=p^{j/2}\psi^{l}(A^{j}x-h):j\in\mathbb{Z},h\in
H,l=1,2,...,p-1\\}.$ (3)

The following are the two definitions of multiwavelet frame and the
multiwavelet Parseval frame.

######  Definition 2.1.

The affine system $\mathcal{A}\subset L^{2}(G)$ is said to be multiwavelet
frame if the system (3) is a frame for $L^{2}(G)$.

######  Definition 2.2.

The affine system $\mathcal{A}\subset L^{2}(G)$ is said to be multiwavelet
Parseval frame if the system (3) is a Parseval frame for $L^{2}(G)$.

The following theorem is the characterization of Parseval frame for Vilenkin
group, which is one of the particular case of local fields with positive
characteristics.

######  Theorem 2.3.

Suppose $\Psi=\\{\psi_{1},\psi_{2},...,\psi_{p-1}\\}\subset L^{2}(G)$. Then
the affine system $A(\Psi)$ is a Parseval frame for $L^{2}(G)$ if and only if
for a.e. $\omega$, the following holds:

  * 1.

$\sum_{i=1}^{p-1}\sum_{j\in\mathbb{Z}}|\hat{\psi}_{i}(B^{j}\omega)|^{2}=1$

  * 2.

$\sum_{i=1}^{p-1}\sum_{j=0}^{\infty}\hat{\psi}_{i}(B^{j}\omega)\overline{\hat{\psi}_{i}(B^{j}(\omega+\gamma))}=0$,
for $\gamma\in H^{\perp}\setminus BH^{\perp}$.

The proof of this theorem are in [1].

######  Definition 2.4.

Let $M=\\{m_{0},m_{1},...,m_{p-1}\\}\subset L^{\infty}(U)$ be a $H-$ periodic
function, is called a generalized filter if it satisfies the following
equation

$\sum_{i=0}^{p-1}|m_{i}(\omega)|^{2}=1,\text{ a.e }\omega,$ (4)
$m_{0}(\omega)\overline{m_{0}(\omega+\beta)}-(\sum_{i=1}^{p-1}m_{i}(\omega)\overline{m_{i}(\omega+\beta)})=0\text{
a.e }\omega,\text{ where }\beta=B^{-1}\gamma\text{ and }\gamma\text{ as in
Theorem 2.3 }.$ (5)

We define the set of generalized filters is denoted by $\tilde{F}$ and let
$\tilde{F}^{+}=\\{M\in\tilde{F}:m_{0}\geq 0,m_{0}\in M\\}$. Notice that for
$M\in\tilde{F}$,
$M_{|m_{0}|}=\\{|m_{0}|,m_{1},...,m_{p-1}\\}\in\tilde{F}^{+}$.

######  Definition 2.5.

A function $\phi\in L^{2}(G)$ is called a pseudo-scaling function if there
exists a generalized filter $M=\\{m_{0},m_{1},...,m_{p-1}\\}\in\tilde{F}$ such
that

$\hat{\phi}(B\omega)=m_{0}(\omega)\hat{\phi}(\omega),\text{ a.e }\omega.$ (6)

Here we observed that $M$ is not uniquely determined by the pseudo-scaling
function $\phi$. Therefore, we consider the set of all $M\in\tilde{F}$ such
that $M$ satisfies (6) for $\phi$, and is denoted by $\tilde{F}_{\phi}$. In
particular, if $\phi=0$, then $\tilde{F}_{\phi}=\tilde{F}$.

If $M\in\tilde{F}^{+}$, then

$\hat{\phi}_{m_{0}}(\omega)=\prod_{j=1}^{\infty}m_{0}(B^{-j}\omega)\quad\text{
is well defined a.e. },$

since $0\leq m_{0}(\omega)\leq 1$, a.e. $\omega$. Furthermore, we get

$\hat{\phi}_{m_{0}}(B\omega)=m_{0}(\omega)\hat{\phi}_{m_{0}}(\omega),\text{
a.e }\omega.$ (7)

######  Definition 2.6.

Suppose that $M=\\{m_{0},m_{1},...,m_{p-1}\\}\in\tilde{F}_{\phi_{m_{0}}}$. Let
us define

$N_{0}(m_{0})=\\{\omega\in
G^{*}:lim_{j\rightarrow\infty}\hat{\phi}_{m_{0}}(B^{-j}\omega)=0\\}.$ (8)

If $|N_{0}(|m_{0}|)|=0$, then $M$is called a generalized low-pass filter. The
set denoted by $\tilde{F}_{0}$ is the set of all generalized filters
satisfying equation (8).

######  Lemma 2.7.

If $f\in L^{2}(G^{*})$, then $lim_{j\rightarrow\infty}|f(B^{j}\omega)|=0$, for
a.e $\omega\in G^{*}$.

###### Proof.

Let us assume that $f\in L^{2}(G^{*})$ and by applying the monotone
convergence theorem we get

$\displaystyle\int_{G^{*}}\sum_{j\in\mathbb{Z}^{+}}|f(B^{j}\omega)|^{2}d\omega$
$\displaystyle=\sum_{j\in\mathbb{Z}^{+}}\int_{G^{*}}|f(B^{j}\omega)|^{2}d\omega$
$\displaystyle=\sum_{j\in\mathbb{Z}^{+}}p^{-j}\int_{G^{*}}|f(\omega)|^{2}d\omega$
$\displaystyle=\frac{1}{p-1}||f||^{2}<\infty.$

That implies $\sum_{j\in\mathbb{Z}^{+}}|f(B^{j}\omega)|^{2}$ is finite, for
$\omega\in G^{*}$ a.e. Thus, for a.e. $\omega\in G^{*}$,
$lim_{j\rightarrow\infty}|f(B^{j}\omega)|=0$. ∎

######  Definition 2.8.

A Parseval frame multiwavelet(PFMW)
$\Psi=\\{\psi_{1},\psi_{2},...,\psi_{p-1}\\}$ is called an MRA PFMW if there
exists a pseudo-scaling function $\phi$, $M\in\tilde{F}_{\phi}$ and unimodular
functions $s_{i}\in L^{2}(G)$, $1\leq i\leq p-1$ such that

$\hat{\psi}_{i}(B\omega)=W_{\alpha}(\omega)s_{i}(B\omega)\overline{M(\omega)}\hat{\phi}(\omega),\text{
a.e }\omega.$ (9)

The following theorem gives a characterization of the generalized low pass

filter.

######  Theorem 2.9.

Suppose $\Psi=\\{\psi_{1},\psi_{2},...,\psi_{p-1}\\}$ is an MRA PFMW and
$\phi$ is a pseudo-scaling function satisfying (6). If M is defined by (9),
then $M\in\tilde{F}_{0}$.

###### Proof.

Notice that $\Psi$ is an MRA PFMW, by Theorem 2.3, (4), (5) and (9), we have

$\displaystyle 1$
$\displaystyle=\sum_{i=1}^{p-1}\sum_{j\in\mathbb{Z}}|\hat{\psi}_{i}(B^{j}\omega)|^{2}$
$\displaystyle=\sum_{i=1}^{p-1}\sum_{j\in\mathbb{Z}}|W_{\alpha}(\omega)s_{i}(B^{j}\omega)\overline{M(B^{j-1}\omega)}|^{2}|\hat{\phi}(B^{j-1}\omega)|^{2}$
$\displaystyle=\sum_{i=1}^{p-1}\sum_{j\in\mathbb{Z}}|m_{i}(B^{j-1}\omega)|^{2}|\hat{\phi}(B^{j-1}\omega)|^{2}$
$\displaystyle=lim_{n\rightarrow\infty}\sum_{j=-n}^{n}(\sum_{i=1}^{p-1}|m_{i}(B^{j-1}\omega)|^{2})|\hat{\phi}(B^{j-1}\omega)|^{2}$
$\displaystyle=lim_{n\rightarrow\infty}\sum_{j=-n}^{n}(1-|m_{0}(B^{j-1}\omega)|^{2})|\hat{\phi}(B^{j-1}\omega)|^{2}$
$\displaystyle=lim_{n\rightarrow\infty}\\{|\hat{\phi}(B^{-n-1}\omega)|^{2}-|\hat{\phi}(B^{n}\omega)|^{2}\\}.$

By using the Lemma $\eqref{21}$ for $\phi\in L^{2}(G)$, we get
$lim_{n\rightarrow\infty}|\hat{\phi}(B^{n}\omega)|^{2}=0$ for a.e. $\omega$.
Therefore, $lim_{n\rightarrow\infty}|\hat{\phi}(B^{-n-1}\omega)|^{2}=1$ holds
for a.e $\omega$. From (6), we have

$|\hat{\phi}(\omega)|=|\prod_{j=1}^{n}M(B^{-j}\omega)||\hat{\phi}(B^{-j}\omega)|,\text{
a.e. }\omega.$

Using (7), It is obtained that $|\hat{\phi}(\omega)|=\hat{\phi}_{|M|}$ and
$|N_{0}(|M|)|=0$ is clearly satisfied. Thus, by Definition 2.5, we have
$M\in\tilde{F}_{0}$. ∎

######  Lemma 2.10.

Let $\mu$ be a $H-$ periodic, unimodular function. Then there exists a
unimodular function $v$ that satisfies

$\mu(\omega)=v(B\omega)\overline{v(\omega)},\quad\text{ a.e. }\omega.$ (10)

######  Theorem 2.11.

Let $M\in\tilde{F}_{0}$ be a generalized

filter. Then there exist a pseudo-scaling function and an MRA PFMW
$\Psi=\\{\psi_{1},\psi_{2},...,\psi_{p-1}\\}$ such that they satisfy (9).

###### Proof.

Suppose that $M\in\tilde{F}_{0}$ a generalized

filter. For $m_{0}\in M$, define the signum function $\mu$ such that

$\mu(\omega)=\left\\{\begin{matrix}\frac{m_{0}(\omega)}{|m_{0}(\omega)|}&m_{0}(\omega)\neq
0;\\\ 1&m_{0}(\omega)=0.\end{matrix}\right.$

Clearly, $\mu$ is a measurable unimodular function and we see the following
equation holds for a.e. $\omega$

$m_{0}(\omega)=\mu(\omega)|m_{0}(\omega)|.$

By Lemma (2.10), there exists a unimodular measurable function $v$ such that

$\mu(\omega)=v(B\omega)\overline{v(\omega)},\quad\text{ a.e. }\omega.$

By (7), we have the function $\hat{\phi}_{|m_{0}|}(\omega)$ from $|m_{0}|$.
Let

$\hat{\phi}(\omega)=v(\omega)\hat{\phi_{|m_{0}|}}(\omega),$ (11)

then, $\phi\in L^{2}(G)$. Using (7),(10), (11) and the definition of the
signum function $\mu$, we have

$\displaystyle\hat{\phi}(B\omega)$
$\displaystyle=v(B\omega)\hat{\phi_{|m_{0}|}}(\omega)$
$\displaystyle=v(B\omega)|m_{0}(\omega)||\hat{\phi_{|m_{0}|}}(\omega)$
$\displaystyle==v(B\omega)|m_{0}(\omega)|\overline{v(\omega)}\hat{\phi}(\omega)$
$\displaystyle=\mu(\omega)|m_{0}(\omega)|\hat{\phi}(\omega)$
$\displaystyle=m_{0}(\omega)\hat{\phi}(\omega).$

Thus, $\phi$ is a pseudo-scaling function. For
$\hat{\Psi}=\\{\psi_{1},\psi_{2},...,\psi_{p-1}\\}$, let
$\hat{\psi}_{i}=W_{\alpha}(\omega)s_{i}(B\omega)\overline{M(\omega)}\hat{\phi}(\omega)$,
$1\leq i\leq p-1$, a.e. $\omega$, where $s_{i}(\omega)\in L^{2}(G),m_{i}\in
M$. By Theorem 2.3, we see that $\Psi$ is an MRA PFMW. Note that $m_{0}$ is a
generalized low pass

filter, that implies $lim_{j\rightarrow\infty}|\hat{\phi}(B^{-j}\omega)|=1$,
a.e. $\omega$ holds. By using Lemma (2.7), we get

$\displaystyle\sum_{i=1}^{p-1}\sum_{j\in\mathbb{Z}}|\hat{\psi}_{i}(B^{j}\omega)|^{2}$
$\displaystyle=\sum_{i=1}^{p-1}\sum_{j\in\mathbb{Z}}|m_{i}(B^{j-1}\omega)|^{2}|\hat{\phi}_{i}(B^{j-1}\omega)|^{2}$
$\displaystyle=lim_{n\rightarrow\infty}\sum_{j=-n}^{n}(\sum_{i=1}^{p-1}|m_{i}(B^{j-1}\omega)|^{2})|\hat{\phi}(B^{j-1}\omega)|^{2}$
$\displaystyle=lim_{n\rightarrow\infty}\sum_{j=-n}^{n}(1-|m_{0}(B^{j-1}\omega)|^{2})|\hat{\phi}(B^{j-1}\omega)|^{2}$
$\displaystyle=lim_{n\rightarrow\infty}\\{|\hat{\phi}(B^{-n-1}\omega)|^{2}-|\hat{\phi}(B^{n}\omega)|^{2}\\}.$

Using $\phi\in L^{2}(G)$, Lemma $\eqref{21}$ implies
$lim_{n\rightarrow\infty}|\hat{\phi}(B^{n}\omega)|^{2}=0$ for a.e. $\omega$.
Then for a.e $\omega$,
$lim_{n\rightarrow\infty}|\hat{\phi}(B^{-n-1}\omega)|^{2}=1$ holds. Now to
show that $\Psi$ given above satisfies the second condition of Theorem 2.3.
Fix

the $\omega$ and $q=Bh\oplus\gamma,\text{ where }h\in H,\gamma\in
H^{\perp}/BH^{\perp}$ and split the equation as
$\sum_{i=1}^{p-1}\sum_{j=0}^{\infty}\hat{\psi}_{i}(B^{j}\omega)\overline{\hat{\psi}_{i}(B^{j}(\omega+q))}$

$=\sum_{i=1}^{p-1}\hat{\psi}_{i}(\omega)\overline{\hat{\psi}_{i}(\omega+q)}+\sum_{i=1}^{p-1}\sum_{j=1}^{\infty}\hat{\psi}_{i}(B^{j}\omega)\overline{\hat{\psi}_{i}(B^{j}(\omega+q)}$
(12)

Using (5), (6) and Lemma (2.7), the

first term on the right-hand side of (12) have the following equation.

$\displaystyle\sum_{i=1}^{p-1}\hat{\psi}_{i}(\omega)\overline{\hat{\psi}_{i}(\omega+q)}$
$\displaystyle=\sum_{i=1}^{p-1}W_{\alpha}(B^{-1}\omega)s_{i}(\omega)m_{i}(B^{-1}\omega)\hat{\phi}(B^{-1}\omega)$
$\displaystyle*\overline{W_{\alpha}(B^{-1}(\omega+q))}\overline{s_{i}(\omega+q)m_{i}(B^{-1}\omega+q)\hat{\phi}(B^{-1}\omega+q)}$
$\displaystyle=\overline{W_{\alpha}(B^{-1}q)}(\sum_{i=1}^{p-1}|s_{i}(\omega)|^{2}m_{i}(B^{-1}\omega)\overline{m_{i}(B^{-1}(\omega+q))})*\hat{\phi}(B^{-1}\omega)\overline{\hat{\phi}(B^{-1}(\omega+q))}$
$\displaystyle=\overline{W_{\alpha}(B^{-1}q)}\overline{m_{0}(B^{-1}(\omega+B^{-1}q)})*\hat{\phi}(B^{-1}\omega)\overline{\hat{\phi}(B^{-1}(\omega+q))}$
$\displaystyle=-\hat{\phi}(\omega)\overline{\hat{\phi}(\omega+q)}.$

By (4), (6), (9) and Lemma (2.7), for the second term on the right-hand side
of (12), we have

$\displaystyle\sum_{i=1}^{p-1}\sum_{j=1}^{\infty}\hat{\psi}_{i}(B^{j}\omega)\overline{\hat{\psi}_{i}(B^{j}(\omega+q))}$
$\displaystyle=\sum_{i=1}^{p-1}\sum_{j=1}^{\infty}W_{\alpha}(B^{j-1}\omega)s_{i}(B^{j}\omega)m_{i}(B^{j-1}\omega)\hat{\phi}(B^{j-1}\omega)$
$\displaystyle*\overline{W_{\alpha}(B^{j-1}(\omega+q))}\overline{s_{i}(B^{j}(\omega+q))m_{i}(B^{j-1}(\omega+q))\hat{\phi}(B^{j-1}(\omega+q))}$
$\displaystyle=\sum_{j=1}^{\infty}\overline{W_{\alpha}(B^{j-1}q)}(\sum_{i=1}^{p-1}|s_{i}(B^{j}\omega)|^{2}m_{i}(B^{j-1}\omega)\overline{m_{i}(B^{j-1}(\omega+q))})$
$\displaystyle*\hat{\phi}(B^{j-1}\omega)\overline{\hat{\phi}(B^{j-1}(\omega+q))}$
$\displaystyle=\sum_{j=1}^{\infty}(1-|m_{0}(B^{j-1}(\omega))|^{2})\hat{\phi}(B^{j-1}\omega)\overline{\hat{\phi}(B^{j-1}(\omega+q))}$
$\displaystyle=\sum_{j=1}^{\infty}\\{\hat{\phi}(B^{j-1}\omega)\overline{\hat{\phi}(B^{j-1}(\omega+q))}-\hat{\phi}(B^{j}\omega)\overline{\hat{\phi}(B^{j}(\omega+q))}\\}$
$\displaystyle=\hat{\phi}(\omega)\overline{\hat{\phi}(\omega+q)}-lim_{p\rightarrow\infty}\hat{\phi}(B^{p}\omega)\overline{\hat{\phi}(B^{p}(\omega+q))}$
$\displaystyle=\hat{\phi}(\omega)\overline{\hat{\phi}(\omega+q)}.$

Adding both the result that we obtained above, right hand side of (12) is
equal to 0. Hence, $\Psi$ is a PFMW by theorem 2.3. ∎

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