ReverseBiorthogonalSplineWavelet

✖
`ReverseBiorthogonalSplineWavelet`

✖

ReverseBiorthogonalSplineWavelet[]

represents a reverse biorthogonal spline wavelet of order 4 and dual order 2.

✖

ReverseBiorthogonalSplineWavelet[n,m]

represents a reverse biorthogonal spline wavelet of order n and dual order m.

Details

ReverseBiorthogonalSplineWavelet defines a family of biorthogonal wavelets.
ReverseBiorthogonalSplineWavelet[n,m] is defined for positive integers m and n where m+n is even.
The scaling function () and wavelet function () have compact support. The functions are symmetric.
ReverseBiorthogonalSplineWavelet can be used with such functions as DiscreteWaveletTransform and WaveletPhi, etc.

Examples

open allclose all

Basic Examples (6)Summary of the most common use cases

Primal scaling function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-bag7qy

Out[1]=1

Primal wavelet function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-dy6au5

Out[1]=1

Dual scaling function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4uuqre

Out[1]=1

Dual wavelet function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-n0f2g4

Out[1]=1

Primal filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-d2rdl1

Out[1]=1

Dual filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-uiuc73

Out[1]=1

Scope (17)Survey of the scope of standard use cases

Basic Uses (10)

Compute primal lowpass filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-x5t4az

Out[1]=1

Dual lowpass filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-8wgk6z

Out[1]=1

Primal highpass filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-tovf4h

Out[1]=1

Dual highpass filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-5utoqa

Out[1]=1

Lifting filter coefficients:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-u5x0a9

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-1x7frz

Out[2]=2

Generate a function to compute lifting wavelet transform:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-nfjqzn

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-ol7ejo

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-7bq5gu

Out[3]=3

Primal scaling function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vl9zbf

Out[1]=1

Dual scaling function:

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4ybkxk

Out[2]=2

Plot scaling function using different levels of recursion:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vovrxw

Out[1]=1

Primal wavelet function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-zncr2c

Out[1]=1

Dual wavelet function:

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-duatsf

Out[2]=2

Plot wavelet function at different refinement scales:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-hjxsba

Out[1]=1

Wavelet Transforms (5)

Compute a DiscreteWaveletTransform:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-t8skl0

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-2kxsqf

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-xhbmxi

Out[3]=3

View the tree of wavelet coefficients:

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-gtqy6c

Out[4]=4

Get the dimensions of wavelet coefficients:

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-mrjnjd

Out[5]=5

Plot the wavelet coefficients:

In[6]:=6

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vkcd9f

Out[6]=6

Compute a DiscreteWaveletPacketTransform:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-o34j9k

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vtrxi2

Out[2]=2

View the tree of wavelet coefficients:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-j3n0f1

Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-wuwmcl

Out[4]=4

Plot the wavelet coefficients:

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-15wf00

Out[5]=5

Compute a StationaryWaveletTransform:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-h4s71h

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-lgbd1m

View the tree of wavelet coefficients:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-jbb9sz

Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-msff83

Out[4]=4

Plot the wavelet coefficients:

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-yakm1p

Out[5]=5

Compute a StationaryWaveletPacketTransform:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-gbl37f

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4fxpz9

View the tree of wavelet coefficients:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-9rgpx

Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-ngfrzh

Out[4]=4

Plot the wavelet coefficients:

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-q5dqbp

Out[5]=5

Compute a LiftingWaveletTransform:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4t2boe

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-6vbut6

View the tree of wavelet coefficients:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-xwj8z0

Out[3]=3

Get the dimensions of wavelet coefficients:

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-56ffkm

Out[4]=4

Plot the wavelet coefficients:

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vyacdo

Out[5]=5

Higher Dimensions (2)

Multivariate scaling and wavelet functions are products of univariate ones:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-rk8e1w

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-tvf11

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-yf2o9

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-s16yjj

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-gmiius

Out[5]=5

Multivariate dual scaling and wavelet functions are products of univariate ones:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4ea1y

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-d90lhk

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-u7njzj

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-j0zc13

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-bqg2ki

Out[5]=5

Properties & Relations (19)Properties of the function, and connections to other functions

ReverseBiorthogonalSplineWavelet[1,1] is equivalent to HaarWavelet:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-rkckkj

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-hodnve

Out[2]=2

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-6qvgrk

Out[1]=1

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-i76i8a

Out[2]=2

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-0a9npb

Out[3]=3

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4bos9p

Out[4]=4

Lowpass filter coefficients sum to unity; :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-v1t5fi

Out[1]=1

Highpass filter coefficients sum to zero; :

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-jrlqqz

Out[2]=2

Dual filter coefficients sum to unity; :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-80xeur

Out[1]=1

Dual highpass filter coefficients sum to zero; :

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vggc8o

Out[2]=2

Scaling function integrates to unity; :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-3k44bm

Out[1]=1

Dual scaling function integrates to unity; :

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-50secc

Out[2]=2

Wavelet function integrates to zero; :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-7te20t

Out[1]=1

Dual wavelet function integrates to zero; :

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-cqbh6v

Out[2]=2

Scaling function has compact support {n₁,n₂}:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-xgo8uj

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-k737yp

Dual scaling function has compact support {nd₁,nd₂}:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-wj0r6e

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-wot97e

Corresponding wavelet function has support {(n₁– nd₂+1)/2, (n₂– nd₁+1)/2}:

In[5]:=5

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-t31uwp

Out[5]=5

In[6]:=6

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-rwecki

Out[6]=6

Dual wavelet function has support {(nd₁– n₂+1)/2, (nd₂– n₁+1)/2}:

In[7]:=7

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-2xdvyc

Out[7]=7

In[8]:=8

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-yadoas

Out[8]=8

satisfies the recursion equation :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-yjbxzh

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-xgoie0

Plot the components and the sum of the recursion:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-ghxj25

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-566qyu

Out[4]=4

satisfies the recursion equation :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-32ph7g

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-lfizfd

Plot the components and the sum of the recursion:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-pv65od

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-siy44p

Out[4]=4

satisfies the recursion equation :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4tgwmy

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-dp3ed3

Plot the components and the sum of the recursion:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-byu78x

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-m8329z

Out[4]=4

satisfies the recursion equation :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4xlw98

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-seqsk7

Plot the components and the sum of the recursion:

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-or3keu

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-n5sx2h

Out[4]=4

Frequency response for is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-y5x0mm

The filter is a lowpass filter:

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-za6vn3

Out[2]=2

Fourier transform of is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-idptzz

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4mxsvj

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-i4smhk

Out[3]=3

Frequency response for is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-n4u9y2

The filter is a dual lowpass filter:

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-jod7ab

Out[2]=2

Fourier transform of is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-le5e5v

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-qp24bt

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-ofeok2

Out[3]=3

Frequency response for is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-6u603k

The filter is a lowpass filter:

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-3n6ywg

Out[2]=2

Fourier transform of is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-b8inb0

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-vsncry

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-h8wifj

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-rzs5ol

Out[4]=4

Frequency response for is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-sjcfi0

The filter is a lowpass filter:

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-my0oxb

Out[2]=2

Fourier transform of is given by :

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-69ezed

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-go2zbw

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-53vj8h

In[4]:=4

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-eytnvi

Out[4]=4

Neat Examples (2)Surprising or curious use cases

Plot translates and dilations of scaling function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-yz9dxl

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-evsjio

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-p5xgws

Out[3]=3

Plot translates and dilations of wavelet function:

In[1]:=1

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-iu0uje

In[2]:=2

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-b9ooxs

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-hts69i

Out[3]=3

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