otnodes
Order terminal nodes of binary wavelet packet tree
Syntax
[Tn_Pal,Tn_Seq]
= otnodes(WPT)
[Tn_Pal,Tn_Seq,I,J]
= otnodes(WPT)
[DP_Pal,DP_Seq]
= otnodes(WPT,'dp')
Description
[Tn_Pal,Tn_Seq]
= otnodes( returns the terminal
nodes of the binary wavelet packet tree, WPT)WPT,
in Paley (natural) ordering, Tn_Pal, and sequency
(frequency) ordering, Tn_Seq. Tn_Pal and Tn_Seq are N-by-1
column vectors where N is the number of terminal
nodes.
[Tn_Pal,Tn_Seq,I,J]
= otnodes( returns the permutations
of the terminal node indices such that WPT)Tn_Seq = Tn_Pal(I) and Tn_Pal
= Tn_Seq(J).
[DP_Pal,DP_Seq]
= otnodes( returns
the Paley and frequency-ordered terminal nodes in node depth-position
format. WPT,'dp')DP_Pal and DP_Seq are N-by-2
matrices. The first column contains the depth index, and the second
column contains the position index.
Input Arguments
|
Binary wavelet packet tree. You can use |
|
Character vector indicating that the Paley-ordered or sequency-ordered nodes are returned in depth-position format. |
Output Arguments
|
Terminal nodes in Paley (natural) ordering |
|
Terminal nodes in sequency ordering |
|
Paley-ordered terminal nodes in depth-position format. This
output argument only applies when you use the |
|
Sequency-ordered terminal nodes in depth-position format. This
output argument only applies when you use the |
Examples
Order Terminal Nodes
Order terminal nodes with Paley and frequency ordering.
x = randn(8,1);
wpt = wpdec(x,2,'haar');
[Tn_Pal,Tn_Seq] = otnodes(wpt)Tn_Pal = 4×1
3
4
5
6
Tn_Seq = 4×1
3
4
6
5
Return Permutations for Ordering
Return permutations for Paley and frequency ordering.
load noisdopp; wpt = wpdec(noisdopp,6,'sym4'); [Tn_Pal,Tn_Seq,I,J] = otnodes(wpt); isequal(Tn_Seq(J),Tn_Pal)
ans = logical
1
isequal(Tn_Seq,Tn_Pal(I))
ans = logical
1
Order Terminal Nodes by Depth and Position
Order terminal nodes by depth and position.
x = randn(8,1); wpt = wpdec(x,2,'haar'); [DP_Pal,DP_Seq] = otnodes(wpt,'dp')
DP_Pal = 4×2
2 0
2 1
2 2
2 3
DP_Seq = 4×2
2 0
2 1
2 3
2 2
Order Terminal Nodes from Wavelet Packet Tree
Order terminal nodes from a modified wavelet packet tree.
t = wptree(2,2,rand(1,512),'haar');
t = wpsplt(t,4);
t = wpsplt(t,5);
t = wpsplt(t,10);
plot(t);
[tn_Pal,tn_Seq,I,J] = otnodes(t)
tn_Pal = 7×1
3
9
21
22
11
12
6
tn_Seq = 7×1
3
21
22
9
6
12
11
I = 7×1
1
3
4
2
7
6
5
J = 7×1
1
4
2
3
7
6
5
More About
Paley (Natural) and Sequency (Frequency) Ordering
The discrete wavelet packet transform iterates on both approximation and detail coefficients at each level. In this transform, A denotes the lowpass (approximation) filter followed by downsampling. D denotes the highpass (detail) filter followed by downsampling. The following figure represents a wavelet packet transform in Paley ordering acting on a time series of length 8. The transform has a depth of two.
Because of aliasing introduced by downsampling, the frequency content extracted by the operator AD is higher than the frequency content extracted by the DD operator. Therefore, the terminal nodes in frequency (sequency) order are: AA,DA,DD,AD. The terminal nodes in Paley order have the following indices: 3,4,5,6. The frequency order has the indices: 3,4,6,5.
References
Wickerhauser, M.V. Lectures on Wavelet Packet Algorithms, Technical Report, Washington University, Department of Mathematics, 1992.