# 4.1 Perfect reconstruction (pr)

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This module discusses perfect reconstruction (PR).

We are now able to generalize our analysis for arbitrary filters ${H}_{0}$ , ${H}_{1}$ , ${G}_{0}$ and ${G}_{1}$ . Substituting this equation and this equation in our discussion of 2-band filter bank into this earlier equation and then using this equation and this equation from the same discussion, we get:

$\stackrel{^}{X}(z)=\frac{1}{2}{G}_{0}(z)({Y}_{0}(z)+{Y}_{0}(-z))+\frac{1}{2}{G}_{1}(z)({Y}_{1}(z)+{Y}_{1}(-z))=\frac{1}{2}{G}_{0}(z){H}_{0}(z)X(z)+\frac{1}{2}{G}_{0}(z){H}_{0}(-z)X(-z)+\frac{1}{2}{G}_{1}(z){H}_{1}(z)X(z)+\frac{1}{2}{G}_{1}(z){H}_{1}(-z)X(-z)=\frac{1}{2}X(z)({G}_{0}(z){H}_{0}(z)+{G}_{1}(z){H}_{1}(z))+\frac{1}{2}X(-z)({G}_{0}(z){H}_{0}(-z)+{G}_{1}(z){H}_{1}(-z))$
If we require $\stackrel{^}{X}(z)\equiv X(z)$ - the Perfect Reconstruction (PR) condition - then:
${G}_{0}(z){H}_{0}(z)+{G}_{1}(z){H}_{1}(z)\equiv 2$
and
${G}_{0}(z){H}_{0}(-z)+{G}_{1}(z){H}_{1}(-z)\equiv 0$
Identity is known as the anti-aliasing condition because the term in $X(-z)$ in is the unwanted aliasing term caused by down-sampling ${y}_{0}$ and ${y}_{1}$ by 2.

It is straightforward to show that the expression for ${H}_{0}$ , ${H}_{1}$ , ${G}_{0}$ and ${G}_{1}$ , given in this equation , this equation , this equation and this equation for the filters based on the Haar transform, satisfy and . They are the simplest set of filters which do.

Before we look at more complicated PR filters, we examine how the filter structures of this figure may be extended to form a binary filter tree (and the discrete wavelet transform).

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Source:  OpenStax, Image coding. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10206/1.3
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