On characterizations of dual frames and approximation of the canonical dual via dual frames on every iteration
This webpage provides the source files for the figures in the paper:
On compactly supported dual frames of Gabor frames
Diana T. Stoeva
Abstract
The main purpose of the paper is to give a characterization of all compactly supported dual windows of a Gabor frame. As an application, we consider an iterative procedure for approximation of the canonical dual window via compactly supported dual windows on every step. In particular, the procedure allows to have approximation of the canonical dual window via dual windows from certain modulation spaces or from the Schwartz space.
I. Source files for Figure 1 in the paper
The main result of the paper (Theorem 1.1) gives a characterization of all compactly suppored dual windows of a Gabor frame based on a given compactly supported dual window (when such one exists). As an illustration, using a known compactly supported dual window of a Gabor frame with a B-spline window, new compactly supported dual windows are determined based on Theorem 1.1 and they are visualized in Figure 1.
To produce Fig.1(a)(b), run the script dualwindowFig1.m (it involves the functions B2.m and h2.m in the input). The output figure-files named fig1_lambda1over10_real_compact_dual_window_7.png and fig1_lambda1over10_real_compact_dual_window_8.png are the graphics visualized in Fig.1(a)(b).
To produce Fig.1(c)(d), modify the script dualwindowFig1.m as follows: replace the line "lambda=1/10;" from the script with "lambda=1/2;". Then run the modified script (it involves the functions B2.m and h2.m). The output figure-files named fig1_lambda1over2_real_compact_dual_window_7.png and fig1_lambda1over2_real_compact_dual_window_8.png are the graphics visualized in Fig.1(c)(d).
The zip-file containing the above mentioned scripts producing Fig. 1 can be downloaded from here.
II. Source files for Figure 2 in the paper
As an application of the above mentioned Theorem 1.1, one obtains an algorithm for approximation of the canonical dual window of a Gabor frame via compactly suppored dual windows on every iteration (Proposition 1.2 in the paper). A procedure applying to general frames, for approximation of the canonical dual via dual frames on every iteration, is also considered (Prop. 3.1 in the paper). A simple illustration is given in Example 3.2 in the paper, including first evaluation of efficieny varying the initial dual frame and a parameter lambda. Fig. 2 vizualizes part of the cases considered in Ex. 3.2 - for certain values of lambda, it shows the given frame G, the initial dual frame G^d, the new dual frame F^1 from the first iteration step of the algorithm, and the approximation F^p of the canonical dual up to 4 digits after the decimal point, where p represents the number of the needed iterations for the desired precision.
The script which was used to produce Fig. 2 is dualFiniteDimCaseGeneralFr_Canonical_Run.m. It involves the script dualFiniteDimCaseGeneralFr.m which is an implementation of Prop. 2.1 in the paper, as well as the initialization-files dualFiniteDimCaseGeneralFr_Input1.m, dualFiniteDimCaseGeneralFr_Input1_2.m, dualFiniteDimCaseGeneralFr_Input1_3.m, dualFiniteDimCaseGeneralFr_Input2.
To produce Fig.2(a), modify the script dualFiniteDimCaseGeneralFr_Canonical_Run.m following the instructions in the input-paragraph concerning RUN1.
To produce Fig.2(b), modify the script dualFiniteDimCaseGeneralFr_Canonical_Run.m following the instructions in the input-paragraph concerning RUN2.
To produce Fig.2(c), modify the script dualFiniteDimCaseGeneralFr_Canonical_Run.m following the instructions in the input-paragraph concerning RUN3.
To produce Fig.2(d), modify the script dualFiniteDimCaseGeneralFr_Canonical_Run.m following the instructions in the input-paragraph concerning RUN4.
To produce Fig.2(e), modify the script dualFiniteDimCaseGeneralFr_Canonical_Run.m following the instructions in the input-paragraph concerning RUN8.
To produce Fig.2(f), modify the script dualFiniteDimCaseGeneralFr_Canonical_Run.m following the instructions in the input-paragraph concerning RUN7.
The zip-file containing the above mentioned scripts producing Fig. 2 can be downloaded from here.