$$C(\mathbf{r}) = \int_{t_n}^{t_f} T(t)\, \sigma(\mathbf{r}(t))\, \mathbf{c}(\mathbf{r}(t), \mathbf{d})\, dt, \qquad T(t) = \exp\!\left(-\int_{t_n}^{t} \sigma(\mathbf{r}(s))\, ds\right)$$

Equation 13 · Foundation Models

How Multimodal Models Actually Handle Video, Audio, and Space

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The first treats a scene as a continuous field rather than a discrete grid at all. Neural radiance fields represent a scene as a fully connected network mapping a continuous 5D coordinate — a 3D position plus a 2D viewing direction — to a volume density and a view-dependent emitted colour, then use classical volume rendering to synthesize the colour a camera ray would see by integrating along it [ 8 ] . The rendering equation itself is the cleanest statement of what “continuous” buys and costs: $C(\mathbf{r}) = \int_{t_n}^{t_f} T(t)\, \sigma(\mathbf{r}(t))\, \mathbf{c}(\mathbf{r}(t), \mathbf{d})\, dt, \qquad T(t) = \exp\!\left(-\int_{t_n}^{t} \sigma(\mathbf{r}(s))\, ds\right)$. where $\sigma$ is volume density, $\mathbf{c}$ is emitted colour, and T(t) is accumulated transmittance along the ray up to t . There is no patch, no voxel grid, no fixed token count…

Meanings in this article

• $T$: accumulated transmittance along the ray up to t.
• $\sigma$: volume density, $\mathbf{c}$ is emitted colour.

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