Multiresolution analysis

<h2 id="definition">Definition</h2>
A multiresolution analysis of the <a href="/facts/Lp_space/gbad0Rv1">Lebesgue space</a> 
 
 
 
 
 L
 
 2
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle L^{2}(\mathbb {R} )}
 
 consists of a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of nested <a href="/facts/Linear_subspace/2wtKTgHL">subspaces</a>

{
        0
        }
        ⊂
        ⋯
        ⊂
        
          V
          
            1
          
        
        ⊂
        
          V
          
            0
          
        
        ⊂
        
          V
          
            −
            1
          
        
        ⊂
        ⋯
        ⊂
        
          V
          
            −
            n
          
        
        ⊂
        
          V
          
            −
            (
            n
            +
            1
            )
          
        
        ⊂
        ⋯
        ⊂
        
          L
          
            2
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle \{0\}\subset \dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset \dots \subset V_{-n}\subset V_{-(n+1)}\subset \dots \subset L^{2}(\mathbb {R} )}

that satisfies certain <a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a> relations in time-space and scale-frequency, as well as <a href="/facts/Complete_metric_space/lsckmU8G">completeness</a> and regularity relations.

<ul><li>Self-similarity in time demands that each subspace Vk is invariant under shifts by <a href="/facts/Integer/PUwwrawx">integer</a> <a href="/facts/Multiple_(mathematics)/tkte6NXa">multiples</a> of 2k. That is, for each 
 
 
 
 f
 ∈
 
 V
 
 k
 
 
 ,
 
 m
 ∈
 
 Z
 
 
 
 {\displaystyle f\in V_{k},\;m\in \mathbb {Z} }
 
 the function g defined as 
 
 
 
 g
 (
 x
 )
 =
 f
 (
 x
 −
 m
 
 2
 
 k
 
 
 )
 
 
 {\displaystyle g(x)=f(x-m2^{k})}
 
 also contained in 
 
 
 
 
 V
 
 k
 
 
 
 
 {\displaystyle V_{k}}
 
.</li>
<li>Self-similarity in scale demands that all subspaces 
 
 
 
 
 V
 
 k
 
 
 ⊂
 
 V
 
 l
 
 
 ,
 
 k
 >
 l
 ,
 
 
 {\displaystyle V_{k}\subset V_{l},\;k>l,}
 
 are time-scaled versions of each other, with <a href="/facts/Scaling_(geometry)/pFm6fvpu">scaling</a> respectively <a href="/facts/Dilation_(metric_space)/ruV5IB7I">dilation</a> factor 2k-l. I.e., for each 
 
 
 
 f
 ∈
 
 V
 
 k
 
 
 
 
 {\displaystyle f\in V_{k}}
 
 there is a 
 
 
 
 g
 ∈
 
 V
 
 l
 
 
 
 
 {\displaystyle g\in V_{l}}
 
 with 
 
 
 
 ∀
 x
 ∈
 
 R
 
 :
 
 g
 (
 x
 )
 =
 f
 (
 
 2
 
 k
 −
 l
 
 
 x
 )
 
 
 {\displaystyle \forall x\in \mathbb {R} :\;g(x)=f(2^{k-l}x)}
 
.</li>
<li>In the sequence of subspaces, for k>l the space resolution 2l of the l-th subspace is higher than the resolution 2k of the k-th subspace.</li>
<li>Regularity demands that the model <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> V0 be generated as the <a href="/facts/Linear_hull/vPjclEtb">linear hull</a> (<a href="/facts/Algebraic_closure/yjewG1mM">algebraically</a> or even <a href="/facts/Topologically_closed/9x2wlgUW">topologically closed</a>) of the integer shifts of one or a finite number of generating functions 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 or 
 
 
 
 
 ϕ
 
 1
 
 
 ,
 …
 ,
 
 ϕ
 
 r
 
 
 
 
 {\displaystyle \phi _{1},\dots ,\phi _{r}}
 
. Those integer shifts should at least form a frame for the subspace 
 
 
 
 
 V
 
 0
 
 
 ⊂
 
 L
 
 2
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle V_{0}\subset L^{2}(\mathbb {R} )}
 
, which imposes certain conditions on the decay at <a href="/facts/Infinity/cF506uD7">infinity</a>. The generating functions are also known as <a href="/facts/Wavelet/3XcX4uIP">scaling functions</a> or <a href="/facts/Father_wavelets/3XcX4uIP">father wavelets</a>. In most cases one demands of those functions to be <a href="/facts/Piecewise_continuous/nlQwptpe">piecewise continuous</a> with <a href="/facts/Compact_support/BTuMGbsb">compact support</a>.</li>
<li>Completeness demands that those nested subspaces fill the whole space, i.e., their union should be <a href="/facts/Dense_set/nPT9Yxao">dense</a> in 
 
 
 
 
 L
 
 2
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle L^{2}(\mathbb {R} )}
 
, and that they are not too redundant, i.e., their <a href="/facts/Intersection/hVW6QFWD">intersection</a> should only contain the <a href="/facts/Zero_element/PvH2qhxh">zero element</a>.</li></ul>
<h2 id="important-conclusions">Important conclusions</h2>
In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to <a href="/facts/Ingrid_Daubechies/vjRQuWwo">Ingrid Daubechies</a>.
Assuming the scaling function has compact support, then 
 
 
 
 
 V
 
 0
 
 
 ⊂
 
 V
 
 −
 1
 
 
 
 
 {\displaystyle V_{0}\subset V_{-1}}
 
 implies that there is a finite sequence of coefficients 
 
 
 
 
 a
 
 k
 
 
 =
 2
 ⟨
 ϕ
 (
 x
 )
 ,
 ϕ
 (
 2
 x
 −
 k
 )
 ⟩
 
 
 {\displaystyle a_{k}=2\langle \phi (x),\phi (2x-k)\rangle }
 
 for 
 
 
 
 
 |
 
 k
 
 |
 
 ≤
 N
 
 
 {\displaystyle |k|\leq N}
 
, and 
 
 
 
 
 a
 
 k
 
 
 =
 0
 
 
 {\displaystyle a_{k}=0}
 
 for 
 
 
 
 
 |
 
 k
 
 |
 
 >
 N
 
 
 {\displaystyle |k|>N}
 
, such that

ϕ
        (
        x
        )
        =
        
          ∑
          
            k
            =
            −
            N
          
          
            N
          
        
        
          a
          
            k
          
        
        ϕ
        (
        2
        x
        −
        k
        )
        .
      
    
    {\displaystyle \phi (x)=\sum _{k=-N}^{N}a_{k}\phi (2x-k).}

Defining another function, known as mother wavelet or just the wavelet

ψ
        (
        x
        )
        :=
        
          ∑
          
            k
            =
            −
            N
          
          
            N
          
        
        (
        −
        1
        
          )
          
            k
          
        
        
          a
          
            1
            −
            k
          
        
        ϕ
        (
        2
        x
        −
        k
        )
        ,
      
    
    {\displaystyle \psi (x):=\sum _{k=-N}^{N}(-1)^{k}a_{1-k}\phi (2x-k),}

one can show that the space 
 
 
 
 
 W
 
 0
 
 
 ⊂
 
 V
 
 −
 1
 
 
 
 
 {\displaystyle W_{0}\subset V_{-1}}
 
, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to 
 
 
 
 
 V
 
 0
 
 
 
 
 {\displaystyle V_{0}}
 
 inside 
 
 
 
 
 V
 
 −
 1
 
 
 
 
 {\displaystyle V_{-1}}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> Or put differently, 
 
 
 
 
 V
 
 −
 1
 
 
 
 
 {\displaystyle V_{-1}}
 
 is the <a href="/facts/Orthogonal_direct_sum/Y6PBJnjg">orthogonal sum</a> (denoted by 
 
 
 
 ⊕
 
 
 {\displaystyle \oplus }
 
) of 
 
 
 
 
 W
 
 0
 
 
 
 
 {\displaystyle W_{0}}
 
 and 
 
 
 
 
 V
 
 0
 
 
 
 
 {\displaystyle V_{0}}
 
. By self-similarity, there are scaled versions 
 
 
 
 
 W
 
 k
 
 
 
 
 {\displaystyle W_{k}}
 
 of 
 
 
 
 
 W
 
 0
 
 
 
 
 {\displaystyle W_{0}}
 
 and by completeness one has

L
          
            2
          
        
        (
        
          R
        
        )
        =
        
          
            closure of 
          
        
        
          ⨁
          
            k
            ∈
            
              Z
            
          
        
        
          W
          
            k
          
        
        ,
      
    
    {\displaystyle L^{2}(\mathbb {R} )={\mbox{closure of }}\bigoplus _{k\in \mathbb {Z} }W_{k},}

thus the set

{
        
          ψ
          
            k
            ,
            n
          
        
        (
        x
        )
        =
        
          
            
              2
            
          
          
            −
            k
          
        
        ψ
        (
        
          2
          
            −
            k
          
        
        x
        −
        n
        )
        :
        
        k
        ,
        n
        ∈
        
          Z
        
        }
      
    
    {\displaystyle \{\psi _{k,n}(x)={\sqrt {2}}^{-k}\psi (2^{-k}x-n):\;k,n\in \mathbb {Z} \}}

is a countable complete <a href="/facts/Orthonormal_wavelet/4vk3W5be">orthonormal wavelet</a> basis in 
 
 
 
 
 L
 
 2
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle L^{2}(\mathbb {R} )}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Multigrid_method/6FTEwkja">Multigrid method</a></li>
<li><a href="/facts/Multiscale_modeling/PkxUJ6FN">Multiscale modeling</a></li>
<li><a href="/facts/Scale_space/XJNvTH1N">Scale space</a></li>
<li><a href="/facts/Time%E2%80%93frequency_analysis/oeysrIKN">Time–frequency analysis</a></li>
<li><a href="/facts/Wavelet/3XcX4uIP">Wavelet</a></li></ul>

<ul><li>Chui, Charles K. (1992). An Introduction to Wavelets. San Diego: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-585-47090-1.</li>
<li><a href="/facts/Ali_Akansu/KIN9yxlm">Akansu, A.N.</a>; Haddad, R.A. (1992). Multiresolution signal decomposition: transforms, subbands, and wavelets. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-047141-6.</li>
<li>Crowley, J. L., (1982). <a href="http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Thesis81.pdf">A Representations for Visual Information</a>, Doctoral Thesis, Carnegie-Mellon University, 1982.</li>
<li><a href="/facts/C._Sidney_Burrus/E4tsUg4S">Burrus, C.S.</a>; Gopinath, R.A.; Guo, H. (1997). Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice-Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-489600-9.</li>
<li>Mallat, S.G. (1999). <a href="http://www.cmap.polytechnique.fr/~mallat/book.html">A Wavelet Tour of Signal Processing</a>. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-466606-X.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Mallat, S.G. "A Wavelet Tour of Signal Processing". www.di.ens.fr. Retrieved 2019-12-30. <a href="http://www.cmap.polytechnique.fr/~mallat/book.html" target="_blank">http://www.cmap.polytechnique.fr/~mallat/book.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Multiresolution analysis open-in-new

Multiresolution analysis