A Study of the
Discretization Errors
in Volume Rendering



Tiago Etiene | University of Utah
Mike Kirby | University of Utah
Cláudio Silva | NYU-Poly

What?

Physical Phenomena
Mathematical Model
Simulation
Visualization



What?

Visualization
Mathematical Model
Code
Image Generation



\( I(x,y)=\int_0^D C(\lambda)\tau(\lambda) \times \exp\left(-\int_0^\lambda\tau(\lambda') \mathrm{d}\lambda'\right)\mathrm{d}\lambda\)

What?


Given a mathematical model and its discretization, we are interested in the following error:

Example: \( I(x,y)=\int_0^D C(\lambda)\tau(\lambda) \times \exp\left(-\int_0^\lambda\tau(\lambda') \mathrm{d}\lambda'\right)\mathrm{d}\lambda \)
\( \qquad \qquad \quad~ = \sum_i^n C_i \tau_i d \prod_j^{i-1} (1 - \tau_i d) + \) \( O(d) \)

Generally: \( I(x,y)= \tilde{I}(x,y) + \) \( O(d^k) \)

Why?

How?






How the order of accuracy will change when different approximation schemes are used?






We will not talk about how the discretization errors affect the quality of the rendered image (the evaluation question)

Discretization

Assumptions


Ground truth: \( C, \tau, s, \) and the analytical solution \( I \) of the VRI are known

Approximation: given \( C, \tau, \) and \( s \), \( \tilde{I} \) is numerically computed

Approximation errors are computed as: \( |\tilde{I}-I| \)

Example:
Standard discretization
a.k.a., "The Volume Rendering algorithm"

The standard discretization of the VRI uses Riemann summation for both inner and outer integrals, and second order error for the exponential

\( I = \sum_i^n C_i \tau_i d \prod_j^{i-1} (1 - \tau_i d) + O(d) \)

Example:
Controlled Precision Volume Integration

\( I(x,y) = \int_0^D \alpha(s) \exp \left( -\int_0^s \epsilon(s') \mathrm{d}s' \right) \mathrm{d}s \)

Assuming that \( \alpha(s) \) and \( \epsilon(s) \) are approximated by polynomials:

\( I(x,y) = \int_0^D p(s) \exp \left( q(s) \right) \mathrm{d}s \)

The order of accuracy will depend on \( p, q, \) and the approximation of the outer integral: Trapezoid, Simpson, or Power Series.

Example:
Pre-Integration

Pre-compute \( C_i \) and \( \alpha_i \) for pairs \( (s_f, s_b) \)

\( \alpha_i = 1-\exp{\left( -\int_{0}^{1} \tau( (1-w)s_f + ws_b )d \text{d}w \right)} \)
\( C_i =\int_{0}^{1}C((1-w)s_f + ws_b)\tau((1-w)s_f + ws_b) \\ \qquad\qquad \times \exp\left( -\int_{0}^{w}\tau((1-w')s_f + w's_b)d\text{d}w' \right) d \text{d}w \)

Example:
Other combinations

Now, assume that we can choose any method for inner, outer, and exponential terms. What are the expected convergence rates?

Results

Convergence rate is given by \( \min(p, q, r-1) \), where:

\( f_{\text{inner}} = \tilde{f}_{\text{inner}} + O(d^p) \)

\( f_{\text{outer}} = \tilde{f}_{\text{outer}} + O(d^q) \)

\( e^{-f_{inner}} = \tilde{e}^{-f_{inner}} + O(d^r) \)

Results:
Exact Computation of \( e^{-x} \)



Results:
\( e^{-x} = 1 - x + O(x^2)\)



Next steps

Thanks!
Questions?

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