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| {{otheruses4|the transfer matrix in wavelet theory|the transfer matrix method in statistical physics|Transfer-matrix method|the transfer matrix method in optics|Transfer-matrix method (optics)}}
| | Jayson Berryhill is how I'm called and my wife doesn't like it at all. I've always cherished living in Mississippi. The preferred hobby for him and his children is style and he'll be beginning something else along with it. He is an order clerk and it's something he truly enjoy.<br><br>My weblog ... real psychic ([http://bigpolis.com/blogs/post/6503 resources]) |
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| In [[applied mathematics]], the '''transfer matrix''' is a formulation in terms of a [[block-Toeplitz matrix]] of the two-scale equation, which characterizes [[refinable function]]s. Refinable functions play an important role in [[wavelet]] theory and [[finite element]] theory.
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| For the mask <math>h</math>, which is a vector with component indexes from <math>a</math> to <math>b</math>,
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| the transfer matrix of <math>h</math>, we call it <math>T_h</math> here, is defined as
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| :<math>
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| (T_h)_{j,k} = h_{2\cdot j-k}.
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| </math>
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| More verbosely
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| :<math>
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| T_h =
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| \begin{pmatrix}
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| h_{a } & & & & & \\
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| h_{a+2} & h_{a+1} & h_{a } & & & \\
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| h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & \\
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| \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\
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| & h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\
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| & & & h_{b } & h_{b-1} & h_{b-2} \\
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| & & & & & h_{b }
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| \end{pmatrix}.
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| </math>
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| The effect of <math>T_h</math> can be expressed in terms of the [[downsampling]] operator "<math>\downarrow</math>": | |
| :<math>T_h\cdot x = (h*x)\downarrow 2.</math>
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| ==Properties==
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| * <math>T_h\cdot x = T_x\cdot h</math>.
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| * If you drop the first and the last column and move the odd-indexed columns to the left and the even-indexed columns to the right, then you obtain a transposed [[Sylvester matrix]].
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| * The determinant of a transfer matrix is essentially a resultant.
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| :More precisely:
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| :Let <math>h_{\mathrm{e}}</math> be the even-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{e}})_k = h_{2k}</math>) and let <math>h_{\mathrm{o}}</math> be the odd-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{o}})_k = h_{2k+1}</math>).
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| :Then <math>\det T_h = (-1)^{\lfloor\frac{b-a+1}{4}\rfloor}\cdot h_a\cdot h_b\cdot\mathrm{res}(h_{\mathrm{e}},h_{\mathrm{o}})</math>, where <math>\mathrm{res}</math> is the [[resultant]].
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| :This connection allows for fast computation using the [[Euclidean algorithm]].
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| * For the [[Trace (linear algebra)|trace]] of the transfer matrix of [[convolution|convolved]] masks holds
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| :<math>\mathrm{tr}~T_{g*h} = \mathrm{tr}~T_g \cdot \mathrm{tr}~T_h</math>
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| * For the [[determinant]] of the transfer matrix of convolved mask holds
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| :<math>\det T_{g*h} = \det T_g \cdot \det T_h \cdot \mathrm{res}(g_-,h)</math>
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| :where <math>g_-</math> denotes the mask with alternating signs, i.e. <math>(g_-)_k = (-1)^k \cdot g_k</math>.
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| * If <math>T_{h}\cdot x = 0</math>, then <math>T_{g*h}\cdot (g_-*x) = 0</math>.
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| : This is a concretion of the determinant property above. From the determinant property one knows that <math>T_{g*h}</math> is [[Singular matrix|singular]] whenever <math>T_{h}</math> is singular. This property also tells, how vectors from the [[null space]] of <math>T_{h}</math> can be converted to null space vectors of <math>T_{g*h}</math>.
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| * If <math>x</math> is an eigenvector of <math>T_{h}</math> with respect to the eigenvalue <math>\lambda</math>, i.e.
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| : <math>T_{h}\cdot x = \lambda\cdot x</math>,
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| :then <math>x*(1,-1)</math> is an eigenvector of <math>T_{h*(1,1)}</math> with respect to the same eigenvalue, i.e.
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| : <math>T_{h*(1,1)}\cdot (x*(1,-1)) = \lambda\cdot (x*(1,-1))</math>.
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| * Let <math>\lambda_a,\dots,\lambda_b</math> be the eigenvalues of <math>T_h</math>, which implies <math>\lambda_a+\dots+\lambda_b = \mathrm{tr}~T_h</math> and more generally <math>\lambda_a^n+\dots+\lambda_b^n = \mathrm{tr}(T_h^n)</math>. This sum is useful for estimating the [[spectral radius]] of <math>T_h</math>. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small <math>n</math>.
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| :Let <math>C_k h</math> be the periodization of <math>h</math> with respect to period <math>2^k-1</math>. That is <math>C_k h</math> is a circular filter, which means that the component indexes are [[Modular arithmetic#Ring of congruence classes|residue class]]es with respect to the modulus <math>2^k-1</math>. Then with the [[upsampling]] operator <math>\uparrow</math> it holds
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| :<math>\mathrm{tr}(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^{n-1})\right)_{[0]_{2^n-1}}</math>
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| :Actually not <math>n-2</math> convolutions are necessary, but only <math>2\cdot \log_2 n</math> ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the [[Fast Fourier transform]]. | |
| * From the previous statement we can derive an estimate of the [[spectral radius]] of <math>\varrho(T_h)</math>. It holds
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| :<math>\varrho(T_h) \ge \frac{a}{\sqrt{\# h}} \ge \frac{1}{\sqrt{3\cdot \# h}}</math>
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| :where <math>\# h</math> is the size of the filter and if all eigenvalues are real, it is also true that
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| :<math>\varrho(T_h) \le a</math>,
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| :where <math>a = \Vert C_2 h \Vert_2</math>.
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| ==See also==
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| * [[Transfer matrix method]]
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| * [[Hurwitz determinant]]
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| ==References==
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| * {{cite article
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| |first=Gilbert|last=Strang
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| |author-link=Gilbert Strang
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| |title=Eigenvalues of <math>(\downarrow 2){H}</math> and convergence of the cascade algorithm
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| |journal=IEEE Transactions on Signal Processing
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| |volume=44
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| |pages=233–238
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| |year=1996
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| }}
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| * {{cite thesis
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| |first=Henning
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| |last=Thielemann
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| |url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131
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| |title=Optimally matched wavelets
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| |type=PhD thesis
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| |year=2006
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| }} (contains proofs of the above properties)
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| [[Category:Wavelets]]
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| [[Category:Numerical analysis]]
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Jayson Berryhill is how I'm called and my wife doesn't like it at all. I've always cherished living in Mississippi. The preferred hobby for him and his children is style and he'll be beginning something else along with it. He is an order clerk and it's something he truly enjoy.
My weblog ... real psychic (resources)