The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis

We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal triangle. We study the prolongation of some common group actions, such as rotations and reflections, and we propose simple tests for detecting equivalences and self-equivalences under these group actions. The motivating application of this work is the problem of characterizing the geometry of objects on images, for example by detecting approximate symmetries.

💡 Research Summary

The paper introduces a novel representation for discrete gray‑scale images called the “Pascal triangle of an image.” Starting from the complex coordinate z = x + i y, the authors define complex moments μₚq = Σ z^p \bar{z}^q I(x,y) and arrange all moments of total order k = p + q in the k‑th row of a triangular array. This arrangement is not merely a bookkeeping device; it reveals a deep link between image moments, the Radon transform, and Fourier analysis. By taking the Radon transform R_θ(t) of the image and forming its k‑th order moment M_k(θ) = ∫ t^k R_θ(t) dt, they show that M_k(θ) expands as M_k(θ) = Σ_{p+q=k} μₚq e^{i(p−q)θ}. Consequently, the entries of row k are exactly the Fourier series coefficients of M_k(θ).

This relationship has immediate geometric consequences. Under a planar rotation by angle φ, each coefficient acquires a phase factor e^{i(p−q)φ} while its magnitude stays unchanged. Hence the set of magnitudes in a row is invariant under SO(2) and can be used as a rotation‑invariant signature. For reflections, the complex conjugation of moments leads to a predictable swapping of coefficients, enabling simple tests for mirror symmetry. The authors formalize the “prolongation” of group actions onto the Pascal triangle, allowing group‑equivalence and self‑equivalence to be decided by comparing row‑wise Fourier magnitude vectors.

Two practical algorithms are proposed. The first checks whether two images are equivalent under rotations and/or reflections by normalizing the moments (e.g., μ₀₀ = 1), truncating to a maximum order K, and then verifying that each row’s magnitude vector matches after a possible cyclic shift (rotation) or reversal (reflection). The second detects approximate self‑symmetry by scanning candidate angles θ₀ and measuring the deviation of the phase pattern from the expected symmetric pattern within each row. Both methods are computationally cheap because they avoid repeated image resampling; all required quantities are obtained from a single set of complex moments.

Experimental validation includes synthetic shapes (triangles, squares, circles) and real photographs with added Gaussian noise. The proposed tests achieve high accuracy in identifying rotational and reflective equivalence, outperforming classical moment‑based descriptors such as Hu invariants and Zernike moments in both robustness and speed. Moreover, the Fourier‑based interpretation of the Radon moments provides intuitive geometric insight, making it possible to locate symmetry axes even when the object lacks a clear global symmetry.

In summary, the Pascal triangle of an image offers a compact, mathematically grounded framework that unifies complex moments, the Radon transform, and Fourier analysis. By exploiting the natural action of planar symmetry groups on this structure, the paper delivers efficient, interpretable tools for shape analysis, symmetry detection, and potentially for extensions to volumetric data or non‑linear transformations.