Rician Noise Removal via a Learned Dictionary

This paper proposes a new effective model for denoising images with Rician noise. The sparse representations of images have been shown to be efficient approaches for image processing. Inspired by this, we learn a dictionary from the noisy image and then combine the MAP model with it for Rician noise removal. For solving the proposed model, the primal-dual algorithm is applied and its convergence is studied. The computational results show that the proposed method is promising in restoring images with Rician noise

On small bases for which 1 has countably many expansions

Let $q\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $(\delta_i)_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ satisfying $$x=\sum_{i=1}^\infty\frac{\delta_i}{q^i}.$$ Let $\mathcal{B}_{\aleph_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let $\mathcal{B}_{1, \aleph_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \cite{Sidorov6} it was shown that $\min\mathcal{B}_{\aleph_0}=\min\mathcal{B}_{1,\aleph_0}=\frac{1+\sqrt{5}}{2},$ and in \cite{Baker} it was shown that $\mathcal{B}_{\aleph_0}\cap(\frac{1+\sqrt{5}}{2}, q_1]=\{ q_1\}$, where $q_1(\approx1.64541)$ is the positive root of $x^6-x^4-x^3-2x^2-x-1=0$. In this paper we show that the second smallest point of $\mathcal{B}_{1,\aleph_0}$ is $q_3(\approx1.68042)$, the positive root of $x^5-x^4-x^3-x+1=0$. Enroute to proving this result we show that $\mathcal{B}_{\aleph_0}\cap(q_1, q_3]=\{ q_2, q_3\}$, where $q_2(\approx1.65462)$ is the positive root of $x^6-2x^4-x^3-1=0$.Comment: 14 pages, 2 figure

Metric results for numbers with multiple q-expansions

Let M be a positive integer and q ∈ (1,M +1]. A q-expansion of a real number x is a sequence (ci) = c1c2 · · · with ci ∈ {0, 1, . . . ,M} such that x = [equation here]. In this paper we study the set Ujq consisting of those real numbers having exactly j q-expansions. Our main result is that for Lebesgue almost every q ∈ (qKL,M + 1), we have  dimH Ujq ≤ max{0, 2 dimH Uq − 1} for all j ∈ {2, 3, . . .}. Here qKL is the Komornik-Loreti constant. As a corollary of this result, we show that for any j ∈ {2, 3, . . .}, the function mapping q to dimH Ujq is not continuous.</p

A Robust Fractal Color Image Watermarking Algorithm

One of the main objectives of watermarking is to achieve a better tradeoff between robustness and high visual quality of a host image. In recent years, there has been a significant development in gray-level image watermarking using fractal-based method. This paper presents a human visual system (HVS) based fractal watermarking method for color images. In the proposed method, a color pixel is considered as a 3-D vector in RGB space. And a general form of 3 × 3 matrix is utilized as the scaling operator. Meanwhile, the luminance offset vector is substituted by the range block mean vector. Then an orthogonalization fractal color coding method is achieved to obtain very high image quality. We also show that the orthogonalization fractal color decoding is a mean vector-invariant iteration. So, the range block mean vector is a good place for hiding watermark. Furthermore, for consistency with the characteristics of the HVS, we carry out the embedding process in the CIE La*b* space and incorporate a just noticeable difference (JND) profile to ensure the watermark invisibility. Experimental results show that the proposed method has good robustness against various typical attacks, at the same time, with an imperceptible change in image quality

Rician Noise Removal via a Learned Dictionary

This paper proposes a new effective model for denoising images with Rician noise. The sparse representations of images have been shown to be efficient approaches for image processing. Inspired by this, we learn a dictionary from the noisy image and then combine the MAP model with it for Rician noise removal. For solving the proposed model, the primal-dual algorithm is applied and its convergence is studied. The computational results show that the proposed method is promising in restoring images with Rician noise