Resize#

Overview#

The resize function in the splineops library enables the resizing (scaling) of N-dimensional data arrays through advanced spline-based methods [1], [2], [3].

Three resizing methods are available:

  • Standard Interpolation: Smooth, continuous interpolation.

  • Least-Squares Projection: Optimal resizing with minimal approximation error. Slower than standard interpolation but with better interpolation quality. It requires 64-bit float precision.

  • Oblique Projection: Similar to least-squares projection, with tunable speed at the expense of quality. It works well with with 32-bit float precision.

It is recommended to use standard interpolation for most cases; in case higher accuracy is required and float64 precision is used, least-squares projection is recommended. Oblique projection provides a finer balance of performance, speed, and accuracy.

Standard Interpolation#

B-spline interpolation reconstructs a smooth function from samples using a B-spline as basis function. The interpolation function is defined as

\[s(x) = \sum_k c_k \beta^{n}(x - k),\]

where:

  • the B-spline of degree \(n\) is \(\beta^{n}(x)\);

  • the interpolation coefficients \(c_k\) are obtained by the application of a prefilter to the samples.

The key property of B-splines is their compact support, which ensures efficient computation while maintaining high smoothness. Given a discrete sequence \(\{f_k\}\) of samples, the interpolation requirement

\[s(k) = f_k\]

is satisfied by a proper choice of the coefficients \(c_k\). We establish them through a digital prefiltering step that involves the application of a recursive IIR filter to the sequence \(\{f_k\}\).

Least-Squares Projection#

Least-squares projection provides an optimal approximation of a function. Instead of direct interpolation, the least-squares approach seeks to find the function \(s(x)\) in a spline space \(V_n\) that best approximates a given function \(f(x)\) in the sense of

\[\min_{s \in V_n} \int |f(x) - s(x)|^2 \mathrm{d}x.\]

The least-squares approximation is obtained by projecting \(f(x)\) onto the space spanned by the basis functions. This projection is given by

\[s(x) = \sum_k \langle f, \varphi_k \rangle \tilde{\varphi}_k(x),\]

where

  • the basis functions are \(\varphi_k(x)\) (typically, B-splines);

  • the duals of the basis functions are \(\tilde{\varphi}_k(x)\), which ensures biorthonormality.

This method effectively reduces aliasing and blocking artifacts. It improves image quality, especially for downsampling.

Oblique Projection#

Oblique projection is a generalization of least-squares projection where the approximation space and the analysis space are allowed to differ. Instead of computing an orthonormal projection, we use an auxiliary analysis function \(\psi(x)\), which leads to the approximation

\[s(x) = \sum_k \langle f, \psi_k \rangle \tilde{\varphi}_k(x).\]

If \(\psi_k = \tilde{\varphi}_k\), we obtain the orthonormal projection (least-squares solution). Otherwise, when \(\psi_k\) differs from \(\tilde{\varphi}_k\), the projection is oblique.

The computational complexity of an oblique projection is lower than that of a least-squares projection, as the former avoids the explicit computation of the optimal prefilter. However, some approximation error arises, depending on the angle between the analysis and synthesis spaces.

Resize Example#

References#