Refine rendering equation

diffdrr/renderers.py

@@ -127,10 +127,10 @@ def __init__(
         self.mode = mode
 
     def dims(self, volume):
-        return torch.tensor(volume.shape).to(volume) + 1
+        return torch.tensor(volume.shape).to(volume) - 1
 
     def forward(
-        self, volume, spacing, source, target, n_points=100, align_corners=True
+        self, volume, spacing, source, target, n_points=250, align_corners=True
     ):
         # Get the raylength and reshape sources
         raylength = (source - target + self.eps).norm(dim=-1)
@@ -151,5 +151,5 @@ def forward(
         vol = volume[None, None, :, :, :].expand(batch_size, -1, -1, -1, -1)
         drr = grid_sample(vol, rays, mode=self.mode, align_corners=align_corners)
         drr = drr[:, 0, 0].sum(dim=-1)
-        drr *= raylength
+        drr *= raylength / n_points
         return drr

notebooks/api/01_renderers.ipynb

@@ -226,7 +226,13 @@
    "source": [
     "## Trilinear interpolation\n",
     "\n",
-    "Instead of computing the exact line integral over the voxel grid (i.e., Siddon's method), we can sample colors at points along the each ray using trilinear interpolation."
+    "Instead of computing the exact line integral over the voxel grid (i.e., Siddon's method), we can sample colors at points along the each ray using trilinear interpolation.\n",
+    "\n",
+    "Now, the rendering equation is\n",
+    "\\begin{equation}\n",
+    "    E(R) = \\frac{\\|\\mathbf p - \\mathbf s\\|_2}{M} \\sum_{m=1}^{M} \\mathbf V \\left[ \\mathbf s + \\alpha_m (\\mathbf p - \\mathbf s) \\right] \\,,\n",
+    "\\end{equation}\n",
+    "where $\\mathbf V[\\cdot]$ is the trilinear interpolation function and $M$ is the number of points sampled per ray."
    ]
   },
   {
@@ -254,9 +260,9 @@
     "        self.mode = mode\n",
     "\n",
     "    def dims(self, volume):\n",
-    "        return torch.tensor(volume.shape).to(volume) + 1\n",
+    "        return torch.tensor(volume.shape).to(volume) - 1\n",
     "\n",
-    "    def forward(self, volume, spacing, source, target, n_points=100, align_corners=True):        \n",
+    "    def forward(self, volume, spacing, source, target, n_points=250, align_corners=True):        \n",
     "        # Get the raylength and reshape sources\n",
     "        raylength = (source - target + self.eps).norm(dim=-1)\n",
     "        source = source[:, None, :, None, :]\n",
@@ -276,7 +282,7 @@
     "        vol = volume[None, None, :, :, :].expand(batch_size, -1, -1, -1, -1)\n",
     "        drr = grid_sample(vol, rays, mode=self.mode, align_corners=align_corners)\n",
     "        drr = drr[:, 0, 0].sum(dim=-1)\n",
-    "        drr *= raylength\n",
+    "        drr *= raylength / n_points\n",
     "        return drr"
    ]
   },