Implementation of algorithms for signal recovery from frame coefficients with erasures

This webpage provides the resource files for implementation of the algorithms and the tests for their efficiency for the paper:

"Dual frames compensating for erasures - non canonical case"

Ljiljana Arambašić and Diana T. Stoeva

Abstract

In this paper we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set E. Starting from a frame (x_n)_{n=1}^\infty and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of (x_n)_{n\in E^c} so that the perfect reconstruction can be obtained from preserved frame coefficients. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated - in the canonical case some of the required assumptions are fulfilled automatically. We also give several ways of computing a dual of the reduced frame. For the methods based on iterative procedures, we implemented the algorithms and tested them for computational efficiency.

I. Implementation of the algorithm of Proposition 2.2 in the paper and tests for its efficiency (in Section 2 in the paper)

Table 1: Tests for the algorithm based on Proposition 2.2, measuring the time for single runs of the algorithm.

The script which was used to perform the tests reflected in Table 1 is TestTable1.m and it involves the script Prop22Efirst.m.

Table 2: Tests for the algorithm of Proposition 2.2, for each test - running the algorithm num-times varying randomly the frame and the number of
elements in the set E.

The script which was used to perform the tests reflected in Table 2 is TestTable2.m and it involves the script Prop22Efirst.m.

Table 3: Test for the algorithm of Proposition 2.2, running the algorithm 9 times using a same fixed frame and varying the set E.

The script which was used to perform the tests reflected in Table 3 is TestTable3.m and it involves the script Prop22Efirst.m.

Table 4: Tests for the algorithm of Proposition 2.2 using multiple runs of the algorithm with a same frame for all the num-runs and including the time
for the initial computation of the canonical dual of this frame.

The script which was used to perform the tests reflected in Table 4 is TestTable4.m and it involves the script Prop22Efirst.m.

Table 5: Tests for the algorithm of Proposition 2.2 when the elements of the set E are not necessarily the first k integer numbers.

The script which was used to perform the tests reflected in Table 5 is TestTable5.m and it involves the script Prop22Efirst.m.

II. Implementation of the algorithm of Proposition 3.6 in the paper and tests for its efficiency (in Section 3 in the paper)

Table 6: Tests for the algorithm based on Proposition 3.6

The scripts which were used to perform the tests reflected in Table 6 are Table6Test1i2.m (for Tests 1 and 2),  Table6Test3.m (for Test 3), Table6Test4.m (for Test 4), Table6Test567.m (for Tests 5, 6, and 7), Table6Test8.m (for Test 8), and Table6Test9.m (for Test 9), and each of these scripts involves the use of Prop36Efirst.m and Prop22Efirst.m.

The zip-file containing all the scripts mentioned above can be downloaded from here. The scripts are done under the Matlab environment, using also functions from the Large Time-Frequency Analysis Toolbox (LTFAT, freely available at https://ltfat.github.io/).