Fig. 5

Illustration of the various algorithms necessary to create the Haar breakpoint array in-place. The top figure represents the transformation of an input array \(\mathbf{y }\) at level 0 into various other forms. The terms \(c_{j,k}\) and \(w_{j,k}\) represent values associated with the scale and detail coefficients of the wavelet transform, respectively. The wavelet tree (bold lines) represents the nested nature of the support intervals: the horizontal position of \(\psi _{j,k}\) represents the position t of central discontinuity \(\mathbf{b }_{j,k}^\pm\) of \({\varvec{\psi }} _{j,k}\), and its vertical position represents the resolution level i. The support interval for each wavelet corresponds to all descendants at level 0. The tree nodes contain the output arrays of the various transforms. Dotted lines indicate the recursive relations in the lifting scheme, as used by the in-place Haar wavelet transform and the maxlet transform. The solid lines (including tree edges) indicate the dependencies in the Haar boundary transform. In the bottom figure, white bullets represent maxlet coefficients, black bullets represent their changed values after the Haar boundary transform, and lines indicate breakpoint array pointers