ReverseBiorthogonalSplineWavelet
✖
ReverseBiorthogonalSplineWavelet
represents a reverse biorthogonal spline wavelet of order 4 and dual order 2.
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 allBasic Examples (6)Summary of the most common use cases

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


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


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


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


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


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

Scope (17)Survey of the scope of standard use cases
Basic Uses (10)
Compute primal lowpass filter coefficients:

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

Dual lowpass filter coefficients:

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

Primal highpass filter coefficients:

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

Dual highpass filter coefficients:

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


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


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

Generate a function to compute lifting wavelet transform:

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


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


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


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


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

Plot scaling function using different levels of recursion:

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


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


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

Plot wavelet function at different refinement scales:

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

Wavelet Transforms (5)
Compute a DiscreteWaveletTransform:

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

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


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

View the tree of wavelet coefficients:

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

Get the dimensions of wavelet coefficients:

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

Plot the wavelet coefficients:

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

Compute a DiscreteWaveletPacketTransform:

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

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

View the tree of wavelet coefficients:

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

Get the dimensions of wavelet coefficients:

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

Plot the wavelet coefficients:

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

Compute a StationaryWaveletTransform:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-lgbd1m
View the tree of wavelet coefficients:

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

Get the dimensions of wavelet coefficients:

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

Plot the wavelet coefficients:

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

Compute a StationaryWaveletPacketTransform:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-4fxpz9
View the tree of wavelet coefficients:

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

Get the dimensions of wavelet coefficients:

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

Plot the wavelet coefficients:

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

Compute a LiftingWaveletTransform:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-6vbut6
View the tree of wavelet coefficients:

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

Get the dimensions of wavelet coefficients:

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

Plot the wavelet coefficients:

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

Higher Dimensions (2)
Multivariate scaling and wavelet functions are products of univariate ones:

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

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


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


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


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

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

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

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


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


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


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

Properties & Relations (19)Properties of the function, and connections to other functions
ReverseBiorthogonalSplineWavelet[1,1] is equivalent to HaarWavelet:

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


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

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet
:

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

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet
:

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

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet
:

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

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet
:

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

Lowpass filter coefficients sum to unity; :

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

Highpass filter coefficients sum to zero; :

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

Dual filter coefficients sum to unity; :

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

Dual highpass filter coefficients sum to zero; :

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

Scaling function integrates to unity; :

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

Dual scaling function integrates to unity; :

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

Wavelet function integrates to zero; :

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

Dual wavelet function integrates to zero; :

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

Scaling function has compact support {n1,n2}:

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


https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-k737yp
Dual scaling function has compact support {nd1,nd2}:

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


https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-wot97e
Corresponding wavelet function has support {(n1– nd2+1)/2, (n2– nd1+1)/2}:

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


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

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

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


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

satisfies the recursion equation
:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-xgoie0
Plot the components and the sum of the recursion:

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

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

satisfies the recursion equation
:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-lfizfd
Plot the components and the sum of the recursion:

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

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

satisfies the recursion equation
:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-dp3ed3
Plot the components and the sum of the recursion:

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

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

satisfies the recursion equation
:

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

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-seqsk7
Plot the components and the sum of the recursion:

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

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

Frequency response for is given by
:

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-y5x0mm
The filter is a lowpass filter:

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

Fourier transform of is given by
:

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

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

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

Frequency response for is given by
:

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-n4u9y2
The filter is a dual lowpass filter:

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

Fourier transform of is given by
:

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

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

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

Frequency response for is given by
:

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-6u603k
The filter is a lowpass filter:

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

Fourier transform of is given by
:

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

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

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

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

Frequency response for is given by
:

https://wolfram.com/xid/07xrch6bm83hz2nmzfa6f1wqrqu-sjcfi0
The filter is a lowpass filter:

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

Fourier transform of is given by
:

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

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

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

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

Neat Examples (2)Surprising or curious use cases
Plot translates and dilations of scaling function:

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

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


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

Plot translates and dilations of wavelet function:

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

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


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

Wolfram Research (2010), ReverseBiorthogonalSplineWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html.
Text
Wolfram Research (2010), ReverseBiorthogonalSplineWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html.
Wolfram Research (2010), ReverseBiorthogonalSplineWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html.
CMS
Wolfram Language. 2010. "ReverseBiorthogonalSplineWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html.
Wolfram Language. 2010. "ReverseBiorthogonalSplineWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html.
APA
Wolfram Language. (2010). ReverseBiorthogonalSplineWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html
Wolfram Language. (2010). ReverseBiorthogonalSplineWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html
BibTeX
@misc{reference.wolfram_2025_reversebiorthogonalsplinewavelet, author="Wolfram Research", title="{ReverseBiorthogonalSplineWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html}", note=[Accessed: 07-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_reversebiorthogonalsplinewavelet, organization={Wolfram Research}, title={ReverseBiorthogonalSplineWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/ReverseBiorthogonalSplineWavelet.html}, note=[Accessed: 07-June-2025
]}