1. IndexGrid mechanism

• Inputs:
• i,j,ki, j, ki,j,k are the signed coordinates representing a position within a grid or node.
• mOffsetmOffsetmOffset is the starting point in an external array where values are stored.
• m∈{0,511}m \in \{0, 511\}m∈{0,511} represents the linear index inside a 512-element array (since 512=8×8×8).
512=8×8×8512 = 8 \times 8 \times 8
• n∈{0,7}n \in \{0, 7\}n∈{0,7} is the offset into the 64-bit array mBitMask, which stores the active status of the 512 values.
mBitMaskmBitMask
• mBitMaskmBitMaskmBitMask is an array where each bit represents whether a corresponding value in the dense 512-element array is "on" (active) or "off" (inactive).

• Active Value Mapping:
• The 512 values are stored in a compressed format, where only the active values (those "on" in mBitMask) are stored in the external array.
mBitMaskmBitMask
• www is a specific 64-bit word from mBitMask, which corresponds to the block containing the value at position (i,j,k).
mBitMaskmBitMask
(i,j,k)(i, j, k)
• The value of w determines which positions are active (i.e., bits set to 1) in the 64-bit block.
ww

• Masking Process:
• The variable mask is used to filter out higher-order bits in w, ensuring that only the bits corresponding to positions before (i,j,k) are considered. This is important to determine the number of "on" values before the current position.
ww
(i,j,k)(i, j, k)

• Checking for Activity (Line 6):
• If (i,j,k) corresponds to an inactive value (i.e., the bit for (i,j,k) in w is 0), the function returns a zero offset. This likely maps to a "background index" or a default value, indicating no data at that position.
(i,j,k)(i, j, k)
(i,j,k)(i, j, k)
ww

• Preceding Active Values (Line 7):
• If w is not the first word in mBitMask, the code uses mPrefixSum to extract the count of active values from previous 64-bit words in mBitMask. mPrefixSum encodes the prefix sums, providing a cumulative count of the active values in the preceding words.
ww
mBitMaskmBitMask
mPrefixSummPrefixSum
mBitMaskmBitMask
mPrefixSummPrefixSum
• Each prefix sum covers a block of 64 values, so the code extracts the count of "on" values before the current word, speeding up the calculation of the linear index.

• Counting Active Values (Line 8):
• The code counts how many bits in w (representing the current word) are "on" before the bit corresponding to the position (i,j,k). This gives the count of active values in the current word up to (i,j,k).
ww
(i,j,k)(i, j, k)
(i,j,k)(i, j, k)
• The sum of the active values from the preceding words (from mPrefixSum) and the current word gives the final linear index of the position (i,j,k).
mPrefixSummPrefixSum
(i,j,k)(i, j, k)

2. GridBatch & JaggedTensor

1. Concatenating along dimension 0 is a concatenation along the batch dimension.
If J1 has 10 member grids in the batch and J2 has 20 members in the batch, then VDBTensor.cat([J1,J2], dim=0) will have 30 members.
All input VDBTensors must have the same number of features.

2. Concatenating along dimension 1 concatenates the features of the VDBTensors together.
If J1 has 3 features and J2 has 4 features, then VDBTensor.cat([J1,J2], dim=1) will have 7 features.
All input VDBTensors must have the same number of grids in the batch and number of voxels in each grid.

3. Actually training

• data.grid.grid_to_world(): This function likely transforms the grid coordinates from the grid space to world space (real-world coordinates). It maps (i,j,k)(i, j, k)(i,j,k) grid positions to actual world coordinates using some transformation (e.g., scaling, rotation).

4. Perform Conv