Incorporate alphaminimax in trilinear integral (#327)

* Incorporate alphaminimax in trilinear integral

* Rerun trilinear tutorial

Author: eigenvivek

diffdrr/renderers.py

@@ -217,7 +217,8 @@ def forward(
                 .scatter_add_(1, channels.transpose(-1, -2), img.transpose(-1, -2))
             )
 
-        # Multiply by raylength and return the drr
+        # Compute the line integral with the rectangular rule and return the DRR
         raylength = (target - source + self.eps).norm(dim=-1).unsqueeze(1)
-        img *= raylength / n_points
+        step_size = (alphamax - alphamin) / (n_points - 1)
+        img *= raylength * step_size
         return img

notebooks/api/01_renderers.ipynb

@@ -278,7 +278,7 @@
     "\n",
     "Now, the rendering equation is\n",
     "\\begin{equation}\n",
-    "    E(R) = \\|\\mathbf p - \\mathbf s\\|_2\\frac{1}{M} \\sum_{m=1}^{M} \\mathbf V \\left[ \\mathbf s + \\alpha_m (\\mathbf p - \\mathbf s) \\right] \\,,\n",
+    "    E(R) = \\|\\mathbf p - \\mathbf s\\|_2\\frac{\\alpha_{\\max} - \\alpha_{\\min}}{M-1} \\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."
    ]
@@ -348,9 +348,10 @@
     "                .scatter_add_(1, channels.transpose(-1, -2), img.transpose(-1, -2))\n",
     "            )\n",
     "\n",
-    "        # Multiply by raylength and return the drr\n",
+    "        # Compute the line integral with the rectangular rule and return the DRR\n",
     "        raylength = (target - source + self.eps).norm(dim=-1).unsqueeze(1)\n",
-    "        img *= raylength / n_points\n",
+    "        step_size = (alphamax - alphamin) / (n_points - 1)\n",
+    "        img *= raylength * step_size\n",
     "        return img"
    ]
   },