Integer-Order Discrete Hankel Transform (DHT)

The Integer-Order Discrete Hankel Transform (DHT) is an integral transformation method widely used in polar or cylindrical coordinate systems, particularly suited for solving problems with axial symmetry. It has significant applications in physics, engineering, and mathematics, such as optical diffraction analysis, acoustic wave propagation simulation, and solving radial problems in quantum mechanics.

Unlike the continuous Hankel transform, the discrete Hankel transform achieves efficient computation through discretized kernel functions and sampling points. Its core concept involves projecting functions onto an orthogonal basis of Bessel functions, typically sampling based on the zeros of Bessel functions of specific orders (e.g., integer order n). This transformation preserves the duality of the continuous version, meaning the forward and inverse transforms exhibit symmetric mathematical forms.

For practical computation, the Fast Hankel Transform (FHT) significantly reduces computational complexity through optimized algorithms (such as leveraging mapping relationships with Fast Fourier Transforms or recurrence formulas), improving traditional O(N²) operations to nearly O(N log N). This characteristic enables efficient processing of large-scale data, for example, rapid frequency-domain conversion of radial data in medical imaging or seismic signal analysis. Code implementations often utilize precomputed Bessel function zeros and recurrence relations to optimize transformation matrices.

It is important to note that implementing the integer-order discrete Hankel transform requires careful handling of sampling intervals and truncation errors, especially as the oscillatory nature of high-order Bessel functions may cause numerical instability. Modern algorithms commonly incorporate normalization factors or weight functions to enhance accuracy, ensuring the closure property of the transforms (i.e., inverse transformation can restore the original signal). Implementation typically involves validating transform pairs with known analytical solutions to verify numerical precision.