Isotropic Kernels for Two-Dimensional Image Interpolation

Abstract

The paper proposes a technique of the composition of two-dimensional interpolation kernels possessing the approximately isotropic spectral characteristics. The application of these kernels makes it possible to weaken many artifacts that appear at interpolation procedures implemented by the traditional techniques. The paper presents the results of the mathematical simulation that confirm the advantages of the proposed technique.

Appendix: Proof of the Theorems

Theorem 1

The finite isotropic approximation kernel conforming to the conditions (4) does not exist.

Proof

For the arbitrary real function f(t) let us designate as z(f,a,b) the number of function f zeroes at the (a,b) interval. Let us introduce a zeroes’ density at the (a,b) interval as follows:

$$\rho( f,a,b ) = \frac{z( f,a,b )}{| a - b |}.$$

Let us consider the isotropic function χ(r), compactly supported at [−r _max,r _max]. The spectrum $\hat{\chi} ( \Omega)$ can be expressed as a superposition of two finite 1D Fourier integrals:

(12)

There is known [16] the following estimation of the zeroes’ density for the finite Fourier integrals like (12). In the above-introduced terms it can be written as follows:

$$\forall a > 0\qquad\rho( \hat{\chi} ,a, + \infty) \le\frac{a}{\pi} . $$

(13)

But, according to the Strang-Fix conditions, the spectrum $\hat{\chi} (\Omega)$ must have zeroes at all $\Omega = 2\pi\sqrt{m^{2} + n^{2}}$, for any m,n∈N. This condition means that the following obvious estimation is true:

$$\forall a > 1,b > a + 1\qquad\rho\bigl( \hat{\chi} ( \Omega / 2\pi),a,b \bigr) \ge 2a+ 1. $$

(14)

But conditions (13) and (14) are incompatible, since the latter estimation grows infinitely. Theorem 1 is proved. □

Corollary

It obviously follows from this theorem, that isotropic strict interpolators in terms of conditions (5) are also impossible.

Theorem 2

Given a function $h( \vec{r} )$ and some integer value N≥1. If the function $h( \vec{r} )$ possesses the following properties:

• (1) $h( \vec{r} ) = 0$, if |x|>r _max or |y|>r _max;

• (2) $\hat{h}( 0,0 ) = 1$;

• (3) ∀x,y∈R∑ _m,n∈Z h(x+2mr _max/N,y+2nr _max/N)≠0

then on the uniform grid with the step δx=2r _max/N the function

$$\tilde{h}( \vec{r} ) = \frac{h( \vec{r} )}{\sum_{m,n \in \mathbf{Z}} h(x + m\delta x,y + n\delta y )} , $$

(15)

will possess the approximation order L, which is not less than 1 in terms of the condition (4).

Proof

Let us designate the denominator of (15) as follows:

$$g( \vec{r} ) = \frac{1}{\sum_{m,n \in \mathbf{Z}} h( x +m\delta x,y + n\delta y )}$$

The spectrum of this function possesses the following property:

$$\hat{g}( \vec{\Omega} )*\sum_{m,n \in \mathbf{Z}} \delta\biggl(\Omega_{x} - \frac{2\pi m}{\delta x},\Omega_{y} -\frac{2\pi n}{\delta x} \biggr)\hat{h}( \vec{\Omega} ) = \delta( \vec{\Omega} ). $$

(16)

Let us continue the functions $h( \vec{r} )$ and $\tilde{h}( \vec{r} )$ periodically onto the whole plane R ² as follows:

(17)

The both functions h and $\tilde{h}$ can be definitely reconstructed by u and $\tilde{u}$. The spectrum of $\hat{u}$ can be expressed through the spectrum of $\hat{h}$ as follows:

$$\hat{u}( \vec{\Omega} ) = \sum_{m,n \in \mathbf{Z}} \delta\biggl(\Omega_{x} - \frac{\pi m}{r_{\max}} ,\Omega_{y} -\frac{\pi n}{r_{\max}} \biggr)\hat{h}( \Omega). $$

(18)

The spectrum of $\hat{\tilde{u}}$ is expressed through the spectrum of $\hat{\tilde{h}}$ by the similar way. From the property (16) it follows, that the following expression is valid:

$$\hat{\tilde{u}}( 2\pi m / \delta x,2\pi n / \delta x ) = \{ \hat{g}*\hat{u} \}(2\pi m / \delta x,2\pi n / \delta x ) = \delta_{mn} $$

(19)

at the introduced by us condition r _max=Nδx/2. Therefore, the statement that $\hat{\tilde{h}}( 2\pi m / \delta x,2\pi n / \delta x) = \delta_{mn}$ is also true. The function $\hat{\tilde{h}}$ is continuous at all the values of the arguments as it is a Fourier transform of the compact function $\tilde{h}$. This means the implementation of the condition (4) at L≥1. So, the theorem is proved. □

Remark 1

Since the proof is performed only for L=1, it must be not ruled out, that greater L values can be possible, providing additional conditions. Some specific cases were investigated numerically, but the theoretical proof of this fact now is unavailable.

Remark 2

This theorem allows constructing the approximation kernel using as h any function, which would satisfy the conditions (1)–(3). This function can be applied not only to the quasi-spline kernels but also to the functions of the form (7), multiplied by any finite window.