Transient Detection and Classification

For a sonar transient occurring in a noisy environment, the transient is localized by amplitude and level by the TI best basis. That is, the noise occurs at the higher levels which are more frequency specific. The transient occurs at lower levels, reflecting the better time localization. Selecting coefficients with absolute value (energy) greater than a tolerance and restricted to levels less than a reference level in the best basis we can reconstruct the transient.

We can minimize the entropy simultaneously over the basis and the translates of the signal. However, this raises the computational complexity by large factor that depends on the length of the signal. In the worst (generic) case it is equivalent to applying the redundant (noncritical, continuous) wavelet transform. We use the algorithm that preserves the critical sampling and is translationally invariant as described previously.

The TI algorithm is not equivalent to an algorithm based on minimizing an entropy over all circulant shifts. We find a better result for a lower computational cost. This has an obvious application to transient classification.

Again, the best basis expansion puts energy of the transient into the lower levels (rows) associated with greater time localization (narrower support). The energy terms at the levels (rows) associated with broader support and greater frequency localization are dominated by the noise. The TI best basis has a lower entropy and better time-scale localization.

We also consider the effect of an additional symmetry on the entropy and best basis construction. We also consider the representation of the same transient in a noiser environment. The additional symmetry, beside lowering the entropy, provides for a better localization of the transient for this lower signal to noise ratio.

To illustrate these comments we present the representative results of specific calculations.

Figure 1 shows a transient sonar signal in a moderate noise background. This is termed signal A1. Figure 2 shows the same transient in a high noise environment. This is termed signal A5.

Figure 3 shows the energy of the coefficients in a standard best basis for signal A1. Figure 4 shows the time-scale distribution of these coefficients.

Figure 5 shows the energy of the coefficients in a standard best basis for signal A1 shifted by 477. Figure 4 shows the time-scale distribution of these coefficients. The energy and distribution is clearly different from the unshifted signal.

We examined the best bases for every shift of the signal. The basis with the minimal entropy for a shift of the signal occurs at a shift of 476. Figure 7 shows the energy of the coefficients in a standard best basis for signal A1 shifted by 476. Figure 8 shows the time-scale distribution of these coefficients. The entropy is lower and the energy and distribution of the coefficients has a better localization. This procedure is also, by definition, translation invariant. Even using the redundancy of certain coefficients under shifts the cost of this transformation is a factor ofmore expensive than the standard best basis procedure.

We compare this result to an application of our (TI) translation invariant algorithm, which is factor of 2 more expensive than the standard best basis procedure. Figure 9 shows the energy of the coefficients in a TI best basis for signal A1. Figure 10 shows the time-scale distribution of these coefficients. The entropy for the TI transform is lower than the result for minimizing entropy over shifts and the cost is much less. The energy and distribution of the coefficients also have a better localization. This procedure is translation invariant.

We consider the effect of an additional symmetry. The transform invariant under shifts and time reversals is termed the TRI transformation. This is a factor of 4 more expensive than the standard wavelet transformation. Figure 11 shows the energy of the coefficients in a TRI best basis for signal A1. Figure 12 shows the time-scale distribution of these coefficients. The entropy for the TRI transform is lower than the result for the TI transformatiion. The energy and distribution of the coefficients also have a better localization. This procedure is translation and time reversal invariant.

Signal A5 has a high noise environment. The standard best basis does not find a good localization of the transient. Figure 13 shows the energy of the coefficients in a standard best basis for signal A5. Figure 14 shows the time-scale distribution of these coefficients.

Figure 15 shows the energy of the coefficients in a TI best basis for signal A5. Figure 16 shows the time-scale distribution of these coefficients. The entropy is lower and localization in energy is improved. The localization in time-scale is still poor.

Figure 17 shows the energy of the coefficients in a TRI best basis for signal A5. Figure 18 shows the time-scale distribution of these coefficients.The entropy is lower and localization in energy is improved. The localization in time-scale is also improved.

To calculate the best bases we first calculate a uniform tree of expansion coefficients. To compare standard, TI and TRI expansions we find the entrophy in each row of the uniform tree for each expansion ad plot the results. Figures 19 and 20 show these results for signals A1 and A5, resp. At higher rows, the entrophy for the TI case in each row is significantly lower than the standard case. At higher rows, the entrophy for the TRI case in each row is significantly lower than the TI case.