Representing Noisy Image Without Denoising

TL;DR

This work introduces Fractional-order Moments in Radon space (FMR), a learning-free representation for robust recognition in noisy images. By extending moments to fractional order and operating in the Radon domain, FMR achieves rotation invariance, noise robustness, and time-frequency discriminability through a flexible parameter $\alpha$, with both implicit and explicit formulations. The authors provide Fourier-based and recursive implementations, prove key invariance and robustness properties, and demonstrate superior performance across histogram analysis, image reconstruction, pattern recognition, template matching, and zero-watermarking compared to classical moments and many learning-based approaches. The approach offers a principled, efficient alternative to data augmentation and denoising, with strong potential for small-scale robust vision and security applications.

Abstract

A long-standing topic in artificial intelligence is the effective recognition of patterns from noisy images. In this regard, the recent data-driven paradigm considers 1) improving the representation robustness by adding noisy samples in training phase (i.e., data augmentation) or 2) pre-processing the noisy image by learning to solve the inverse problem (i.e., image denoising). However, such methods generally exhibit inefficient process and unstable result, limiting their practical applications. In this paper, we explore a non-learning paradigm that aims to derive robust representation directly from noisy images, without the denoising as pre-processing. Here, the noise-robust representation is designed as Fractional-order Moments in Radon space (FMR), with also beneficial properties of orthogonality and rotation invariance. Unlike earlier integer-order methods, our work is a more generic design taking such classical methods as special cases, and the introduced fractional-order parameter offers time-frequency analysis capability that is not available in classical methods. Formally, both implicit and explicit paths for constructing the FMR are discussed in detail. Extensive simulation experiments and an image security application are provided to demonstrate the uniqueness and usefulness of our FMR, especially for noise robustness, rotation invariance, and time-frequency discriminability.

Figures (13)

• Figure 1: Illustration for the Radon transform. (a) The image function $f(x,y)$ and a straight line $L(r,\theta )$ with distance $r$ to the origin and angle $\theta$ to the $y$-axis. (b) The line integral along such straight line, $\int_{L(r,\theta )} {f(x,y)dxdy}$, as the corresponding Radon transform ${{\cal R}_f}(r,\theta )$.
• Figure 2: Illustration for the effect of fractional parameter $\alpha \in {\mathbb{R}^ + }$ on a pair of radial independent variables: ${r_{{\rm{new}}}} \in [0,1]$ and ${r_{{\rm{old}}}} = {r_{{\rm{new}}}}^\alpha \in [0,1]$.
• Figure 3: Illustration for the time-frequency discriminability. Under different parameter settings, the phases for the angular basis function ${A_m}(\theta )$, the harmonic radial basis function $R_n^\alpha (r)$ in (7), and the 2D basis function $V_{nm}^\alpha (r,\theta )$ (as the product of the two) are plotted with red, blue, and black borders, respectively. Note that $n$ and $m$ encode the radial and angular frequency properties, respectively, while the newly introduced $\alpha$ encodes the radial spatial properties which is not available in the previous integer-order approach (i.e., with $\alpha = 1$).
• Figure 4: Illustration for the implementation of FMR. The original image $f$ is first projected into the Radon space as ${{\cal R}_f}$. Then, the inner product of ${{\cal R}_f}$ and the basis function $V_{nm}^\alpha$ with different parameters $(\alpha ,n,m)$ is computed as FMR $M_{nm}^\alpha = \left< {{\cal R}_f},V_{nm}^\alpha \right>$. Here, the computation of the inner product and the estimation of the basis function can be efficiently executed by the Fourier and recursive based strategies, respectively. Note that the above transformation is invertible.
• Figure 5: Illustration for the relationship between the implicit and explicit paths for defining FMR. The explicit path enables a direct transformation from the original image $f$ to the FMR $M_{nm}^\alpha$ without the Radon transform ${{\cal R}_f}$ as an intermediary.

Theorems & Definitions (12)

• Definition 1
• Proposition 1
• Proposition 2
• Example 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Lemma 1
• Theorem 2