*5.1. Robustness Analysis*

In this subsection, we illustrate the robustness of DRTPF by training DRTPF using the image 'Barbara' and testing DRTPF for denoising the same image with Gaussian white noise added. The noise level (standard deviation) *δ* ranged from 20 to 60 with a step size of 2. In the experiment, the frames **Φ** and **Ψ** of size 100 × 200 were initialized as 1D overcomplete DCT (ODCT) and 10 × 10 overlapping mean-subtracted patches were used. The patch size was set as 8 × 8 with stripe 1. We set the parameters *η*<sup>1</sup> = 1.1 and *η*<sup>3</sup> = 1*e* + 7, and *η*<sup>2</sup> was replaced by the -<sup>0</sup> thresholding 0.6*σ* (i.e., **Y**<sup>0</sup> ≤ 0.6*σ*). For comparison, our proposed algorithm was compared with K-SVD [14]. The size of dictionary learnt from K-SVD is 8 × 256 at its optimal state, according to the previous work.

We show the denoising result in Figure 1, from which it is apparent that with higher noise, our DRTPF method outperformed K-SVD more and more. In other words, our proposed model has good robustness. In fact, in our model, the sparse coefficients are calculated accurately by the inner product of the signals and the frame **Ψ**, and are limited to a certain range. Theoretically, it should be more robust. The learnt transforms **Φ** and **Ψ** are illustrated in Figure 2. These figures show that our frame learning method can capture the features in both analysis and synthesis ways. Figure 3 shows two exemplified visual results on the images 'Babara' at noise level *σ* = 30 and *σ* = 50. From Figure 3 we know that our proposed DRTPF can obtain more clearer features than K-SVD [14].

**Figure 1.** Robustness Analysis. DRTPF is trained and tested using the image 'Barbara'. The *X*-label is the noise level *δ* and the *Y*-label is the PSNR. It can be seen that DRTPF performs more robustly than K-SVD.

**Figure 2.** The learnt operators **Φ** (**left**) and **Ψ** (**right**) for *barbara*.

**Figure 3.** Reconstruction of *barbara* using DRTPF (**left**) and K-SVD [14] (**right**). Top: *σ* = 30; bottom: *σ* = 50.
