Multivariate Fake Nodes Approach

Davide Poggiali (FAR Networks srl)

On behalf of prof. Stefano De Marchi (DM@UniPD)

FAATNA20>22 - July 5th, 2022

Summary:

Motivations
Fake Nodes Approach (FNA)
Application: Image resampling
Error bounds and Gibbs Effect
Experiments
Conclusions

1. Motivations

Let's start with some vastly-known example.

Motivation #1. The Runge effect is taught in every Num. Calculus course. Basically, we avoid it by resampling to Chebyshev or Chebyshev-Lobatto nodes...

what is the take-home message?

Motivation #2. The Gibbs effect.

Gibbs effect (in a nutshell):

Appears at discontinuities as oscillatory;
Does not vanish as $N\to\infty$;
As $N\to\infty$ their frequency grows, the amplitude reaches a plateau.

3D T1-weighted MRI:

usually ~1 mm $^3$ voxel

Segmentation/parcellation at full resolution

PET image:

usually ~8mm $^3$ voxel

How to apply segmentation?

1. Undersampling T1 segmentation

2. Upsampling PET image

2. Fake Nodes Approach (FNA)

Interpolation on Fake Nodes, or interpolation via Mapped Bases is a technique that applies to any interpolation method in any dimension. It has shown in many cases the ability of obtaining a better interpolation without getting new samples.

This means that, provided that the original nodes are a sufficient sample of the given signal, it can be possible in principle to provide an interpolation of smaller error.

Given a function

f: \Omega \subseteq \mathbb{R}^d \longrightarrow \mathbb{R}\;\;\;\;\; {d \in \mathbb{N}},

a set of nodes

X = \{ x_0, \dots, x_N \in \Omega \},

and a set of basis functions

\mathcal{B} = \{ b_i: \Omega \longrightarrow \mathbb{R} \}_{i = 0, \ldots, N}

The interpolant is given by a function

\displaystyle \mathcal{P}(\bm{x}) = \sum_{i=0}^{N} c_i b_i(\bm{x})

s.t.

\mathcal{P}(\bm{x}_i)=y_i:=f(\bm{x}_i) \;\;\forall i

\mathcal{B} = \{ b_i \circ S: \Omega \longrightarrow \mathbb{R}^d \}_{i = 0, \ldots, N}

\displaystyle \mathcal{R}^s(\bm{x}) = \sum_{i=0}^{N} c_i b_i(S(\bm{x}))

\mathcal{R}^s(\bm{x}_i)=y_i:=f(\bm{x}_i) \;\;\forall i

FNA is an interpolation over the original basis $\mathcal{B}$ of a function $g: S(\Omega) \longrightarrow\mathbb{R}$

on the fake nodes

\[S(X) = \{S(x_i) \}_{i=0, \dots, N}.\]

$g\in C^s$ must satisfy $g(S(x_i)) = f(x_i)\;\;\forall i$.

If $\mathcal{P} $ is an interpolant on $g$ on the fake nodes, then $\mathcal{P}(S(x)) = g(S(x)) $ for all $x\in X$,

then $\mathcal{R}^s := \mathcal{P} \circ S$ is an interpolant on $f$. In fact $\mathcal{R}^s (x)= \mathcal{P} (S(x)) = f(x) $ for all $x\in X$.

Common hypothesis from now on:

(1.) The mapping $S$ is injective $\longrightarrow$ Fake Nodes do not overlap

(2.) We interpolate a function $g$ s.t. $ g\circ S =f$. Such function exists as (1.) holds

Proposition 2.1 [Equivalence of the Lebesgue constant]

The Lebesgue constant associated to the mapped nodes $\Lambda^s(\Omega)$ is so that

\[\Lambda^s(\Omega) = \Lambda(S(\Omega)).\]

This means that the interpolation via mapped basis inherits the Lebesgue constant of the original basis in the Fake Nodes, and (by consequence) the stability.

Lemma 2.1 [Equivalence of a cardinal basis]

If $\{u_i\}_{i=0,\dots,N}$ is a cardinal basis on the Fake Nodes $S(X)$ then

$ \{u_i\circ S\}_{i=0,\dots,N}$ is cardinal too on $X$

Proposition 2.2 [Stability]
Let $\mathcal{R}^s_{\tilde{f}}$ be the interpolant of the function values

$\tilde{f}(x_i)$, $i=1,\ldots,n$. Then,
\[ \lvert\lvert \mathcal{R}_f^s - \mathcal{R}_{\tilde f}^s \rvert\rvert_{\infty,\Omega} \leq \Lambda^s(\Omega) \; \| f-{\tilde f} \|_{\infty, X_n}.\]

Summing up: suppose that if we know that a basis is more stable over a different distribution of interpolation nodes $X$, but all we have on our dataset is the samples on some other nodes $Y$. All we have to do is finding a mapping $S$ such that $S(Y) =X$, as regular as possible and apply the FNA with such mapping.

3. Application: Image resampling

A three-dimesional MRI can be ~ $10^8$ voxels (256x256x256 takes ~800Mb in double precision, uncompressed) and counting. Hence a fast, lightweight interpolation method is needed.

Let us take a $10^8$ voxels image. The interpolation matrix would be $10^8 \times 10^8$ .... ~80Pb in double precision!

What is the fastest interpolation method?

Choose a basis;
Compute Interpolation/Evaluation matrices;
Solve a linear system;
Multiply Eval. matrix by coefficients.

So we need to choose a basis:

Separable → the interpolation is computed on each axis sequentially;
Cardinal → no need to compute or invert the Interpolation matrix;
Compact supported → Evaluation matrix is sparse.

To achieve that we have to suppose that the interpolation nodes are on a equispaced grid.

We can choose

\[\mathcal{P}_f (\bm{x}) = \sum_{ijk} f(\bm{x}_{ijk})W_{ijk}(\bm{x})\]

with

\[W_{ijk}(x, y, z) = w(x-x_i)\; w(y-y_i) \;w(z-z_i)\]

the separable basis with $w(t-t_i)$ a cardinal and compact-supported basis.

This is called interpolation by convolution.

4. Error bounds and Gibbs Effect

Definition 4.1 Let $f: \Omega \subseteq \mathbb{R}^d \longrightarrow \mathbb{R}$ be a piecewise continuous function. The function $f$ admits a (local) modulus of continuity

$$\omega_{\bm x_0}(\delta): \mathbb{R}^{+} \longrightarrow \mathbb{R}^{+}$$
in $\bm x_0\in\Omega$ if
\[ |f(\bm x) - f(\bm x_0) | \leq \omega_{\bm x_0}(\delta) \;\;\forall \bm x \in \Omega \text{ s.t. } \|\bm x- \bm x_0\|\leq \delta, \]
where $\|\cdot\|$ is an arbitrary norm.

In order to provide an estimate of the interpolation error for the image interpolation by convolution, we need to provide some background theory.

Theorem 4.1 Let $f$ be a d-variate and bounded function admitting a local modulus of continuity $\omega_{\bm x} (\delta)$. Let $\mathcal{P}f$ be its interpolant by convolution. Then, there exits some $\delta^* > 0$ such that
\[ |\mathcal{P}f (\bm x) - f(\bm x)| \leq \omega_{\bm x} (\delta^*)\,. \]

Corollary 4.1 Let $f$ be a d-variate and bounded function admitting a global modulus of continuity $\omega_{\bm x} (\delta)$. Let $\mathcal{P}f$ be its interpolant by convolution. Then, there exitsdome $\delta^* > 0$ such that
\[ |\mathcal{P}f (\bm x) - f(\bm x)| \leq \omega (\delta^*)\,. \]

Lemma 4.1 A d-variate, Lipschitz-continuous function $f$ admits a modulus of continuity of the form
\[\omega(\delta) = K\delta \]
being $K$ the Lipschitz constant.

Lemma 4.2 A piecewise d-variate, Lipschitz-continuous function $f$ admits a modulus of continuity of the form
\[\omega(\delta) = K\delta + D \]

5. Experiments

In silico validation: the Shepp-Logan 3D phantom

Using a phantom the gold standard is known and we can analyse the errors of approximation.

High resolution "morphological" image

Segmentation mask

Low resolution "functional" image

1mm^3

8mm^3

1mm^3

$(2mm)^3$

$1mm^3$

Fake image

In vitro validation:

The RIDER PET/CT phantom images is a dataset of repeated scans of a dummy (20 scans of the same phantom).

Axial

Sagittal

Axial

CT segm.

PET

6. Conclusions

We have defined the FNA as a powerful instrument to cheat on the nodes;
applications in imaging have been studied;
we have proven the presence of Gibbs effect in image oversampling;
we have proposed a Fake-Nodes based workaround in the context of Multimodal medical imaging;
we have evaluated such method both in silico and in vitro.

[1] Burger, W.; Burge, M.J., Principles of digital image processing: core algorithms; Springer, 2009.

[2] Dumitrescu.; Boiangiu. A Study of Image Upsampling and Downsampling Filters. Computers doi:10.3390/computers8020030.4139

[3] S. De Marchi, F. Marchetti, E. Perracchione, Jumping with Variably Scaled Discontinuous Kernels (VSDKs), BIT Numerical Mathematics 2019.

[4] S. De Marchi, F. Marchetti, E. Perracchione, D. Poggiali, Polynomial interpolation via mapped bases without resampling, JCAM 2020.

[5] S. De Marchi, F. Marchetti, E. Perracchione, D. Poggiali, Multivariate approximation at Fake Nodes, AMC 2021.

[5] S. De Marchi, G. Elefante, E. Perracchione, D. Poggiali, Quadrature at Fake Nodes, DRNA 2021.
[6] D. Poggiali, D. Cecchin, C. Campi, S. De Marchi, Oversampling errors in multimodal medical imaging are due to the Gibbs effect, Mathematics 2021.

[7] D. Poggiali, D. Cecchin, S. De Marchi, Reducing the Gibbs effect in multimodal medical imaging by the Fake Nodes Approach, JCMDS 2022.

[8] C. Runge, Uber empirische Funktionen und die Interpolation zwischen aquidistanten Ordinaten, Zeit. Math. Phys, 1901.

Thanks for your attention!