**6. Test Results**

In this section, the test results of *PHMS* on *CPG* for quantitative measure in (1)–(3) are shown. First the performance with respect to quantitative measures in (1), (2) are presented in Table 1.

**Table 1.** Results of Labels (1)–(2).


From Table 1, the error of tightness *Mtig* for the *PHMS* transform is about 0.1, which confirms that the multilevel system is indeed not a tight frame. The main reason is the redundancy of the multilevel structure due to the radius in (36). The quantity *Mclo* ≈ 0.0095 and *Mqua* ≈ 1.934 suggests that the circular-polar Fourier transform provides good properties in terms of isometry, which allows us to employ the conjugate gradient method to compute the inverse of <sup>F</sup>*p*. The weight *w* should also be chosen carefully.

Second, the robustness measurements in (3) displayed in Table 2:


**Table 2.** Results of Label (3).

Table 2 shows the robustness of *PHMS*. Even discarding 100(1 − <sup>2</sup>−<sup>10</sup>) ≈ 99.9% of the coefficients, the image still can be recovered with error *MP*1 = 0.9 × 10−2. The second row suggests that just the coefficient greater than thresholding value *m*(<sup>1</sup> − 1/20.001) ≈ 0.1% can give a good reconstruction with *Mp*2 = 0.009. In the third row, the quantization of robustness *Mp* is displayed.
