Moments
Moments in image processing are numerical descriptors used to characterize the shape characteristics of 2D image functions.
Mathematical Definition
The moment of a function f(x,y) is defined as the weighted average of the function, where the weights are determined by the position (x,y) in the function domain. The general formula for the (p,q)th moment of a continuous function f(x,y) is given by:
m_pq = ∫∫ x^p * y^q * f(x,y) dx dy
where 'p' and 'q' are the non-negative integers that represent the order of the moment.
Types of Moments
Raw Moments
Raw moments are the moments calculated using the original pixel coordinates, without any shifting or normalization. The (p,q)th raw moment of a digital image f(x,y) is defined as:
m_pq = Σ Σ x^p * y^q * f(x,y)where summations are taken over the entire image domain.
Central Moments
Central moments are a variation of raw moments, where the moment calculation is done with respect to the image centroid. The (p,q)th central moment of a digital image f(x,y) is defined as:
μ_pq = Σ Σ (x - x_bar)^p * (y - y_bar)^q * f(x,y)where
(x_bar, y_bar)is the image centroid, calculated as:x_bar = m_10 / m_00 y_bar = m_01 / m_00Normalized Moments
Normalized moments are central moments that are further divided by appropriate powers of the zeroth-order central moment (m_00) to make them scale-invariant. The (p,q)th normalized moment of a digital image f(x,y) is defined as:
η_pq = μ_pq / (m_00)^((p+q)/2 + 1)
Moment Invariants
Moment invariants are a class of image moments that are invariant to certain geometric transformations, such as translation, rotation, and scaling. These invariant properties make moment invariants particularly useful in applications like object recognition, image registration, and shape analysis, where the goal is to identify and compare objects or shapes regardless of their position, orientation, or size in the image.
The key concept behind moment invariants is to construct moment-based features that remain unchanged under specific geometric transformations. This is achieved by deriving mathematical expressions for the moments that exhibit the desired invariance properties.
Some commonly used moment invariants are:
Hu Moment (seven rotation, translation, and scale-invariant moments)
Zernike Moments (orthogonal moments that exhibit rotation and scale invariance)
Legendre Moments (orthogonal moments that can be made invariant to translation, rotation, and scaling)
Key Aspects of Moment-based Image Analysis
Feature Extraction
- Image moments can be used to extract features from images that are robust to geometric transformations, such as translation, rotation, and scaling
Object Recognition and Classification
- The invariance properties of moments can allow for accurate identification and categorization of objects, even in the presence of geometric transformations
Image Registration and Alignment
Moment-based techniques can be employed for image registration, where the goal is to align two or more images of the same scene or object
Moment invariants can be used to find corresponding features between images, enabling the estimation of the geometric transformation required to align them
Shape Analysis and Description
- Moment invariants provide a compact and informative representation of the object's shape, which can be used for tasks like shape matching, shape classification, and shape-based retrieval
Image Reconstruction and Interpolation
- Moment-based techniques can be applied to image reconstruction and interpolation problems, where the goal is to reconstruct or estimate the original image from a limited set of data or a lower-resolution version
Moment Computation Algorithms
Direct Moment Computation
The most straightforward approach to compute the (p,q)th moment of a digital image f(x,y) is the direct summation method, which is defined as:
m_pq = Σ Σ x^p * y^q * f(x,y)where the summations are taken over the entire image domain. This approach is simple to implement but can be computationally expensive, especially for higher-order moments and large image sizes.
Recursive Moment Computation
To improve the computational efficiency of moment calculation, recursive algorithms have been developed. These algorithms take advantage of the recurrence relations between moments of different orders, allowing for the incremental computation of moments.
One of the most well-known recursive algorithms is the Hu-Xiang algorithm, which computes the raw moments of an image using the following recurrence relations:
m_p,q = Σ x^p * Σ y^q * f(x,y) = Σ x^p * (m_p-1,q + x * m_p,q-1)This recursive formulation allows for the efficient computation of raw moments, with a computational complexity of O(N^2), where N is the image size.
Fast Moment Computation
To further improve the computational efficiency of moment calculation, researchers have proposed various fast moment computation algorithms. These techniques leverage advanced mathematical and computational methods to achieve even faster moment computation.
One example is the use of the Discrete Tchebichef Transform (DTT) for moment computation. The DTT-based algorithm exploits the orthogonal properties of Tchebichef polynomials to compute moments in a more efficient manner, with a computational complexity of O(N log N).
Another fast moment computation technique is based on the use of the Discrete Fourier Transform (DFT). By leveraging the properties of the DFT, it is possible to compute moments in the frequency domain, reducing the computational complexity to O(N log N).
Parallel and Hardware-Accelerated Moment Computation
To handle the computational demands of moment calculation, especially for large images or real-time applications, researchers have explored parallel and hardware-accelerated approaches.
Parallel algorithms for moment computation have been developed to take advantage of multi-core processors, GPU acceleration, and distributed computing architectures. These parallel implementations can significantly reduce the computation time by distributing the workload across multiple processing units.
Furthermore, hardware-accelerated approaches, such as the use of dedicated moment computation hardware units or FPGA-based implementations, have been proposed to achieve even higher computational speeds and enable real-time moment-based image processing applications.