**5. Quantitative Test Measures**

In the following, several performance measures are introduced to test the quality of the *PHMS*. The quality measure is the Monte Carlo estimate for the different operator norm by generating a sequence of five random images *ui*, *i* = 1, ..., 5 on *CPG* for *R* = 256, - = 8 with standard normally distributed entries.

1. *Isometry of CFT*:

$$\text{f(a)} \qquad \text{Closness to tight: } M\_{clo} = \max\_{i=1,\dots,5} \frac{||\mathcal{F}\_p^\* \mathcal{F}\_p u\_i - u\_i||\_2}{||u\_i||\_2}.$$

(b) Quality of preconditioning. *Mqua* = *<sup>λ</sup>*max(F∗*p* <sup>F</sup>*p*) *<sup>λ</sup>*min(F∗*p <sup>w</sup>*F*p*).

	- (a) Thresholding: Let *u* be the regular sampling of a Gaussian function with mean 0 and variance 512 on [257]<sup>2</sup> generating an 512 × 512 image. Two types of robustness are considered, for *k* = 1, 2, and *Mpk* = P∗*Tpk*<sup>P</sup>*<sup>u</sup>*−*<sup>u</sup>* 2 *u* 2.

*Tp*1 **:** *Tp*1 discards 100(1 − 2−*p*<sup>1</sup> ) percent of coefficient, with *p*1 = [2 : 2 : <sup>20</sup>].

*Tp*2 **:** *Tp*2 keeps the absolute value of coefficients bigger than *m*/2*<sup>p</sup>*<sup>2</sup> with *m* is the maximal absolute value of all coefficients, where *p*2 = [0.5 : 0.5 : 5].

(b) Quantization: The quality measure is given as *Mp* = P∗*Qq*P*<sup>u</sup>*−*<sup>u</sup>* 2 *u* 2 , where *Qq*(*c*) = *round*(*c*/(*m*/2*<sup>q</sup>*)) · (*m*/2*<sup>q</sup>*), and *q* ∈ [5 : −0.5 : 0.5].
