Rename the Rosenblatt method; add the double CDF method. Add a simple example in notebook.

EtaInterpolationFunctions.py

@@ -5,7 +5,7 @@ Functions for eta interpolation
     2. bilinear_xy_lookup(etaX, etaY, X_tab, Y_tab, x_centers, y_centers)
     3. map_xy_no_interp(etaX, etaY, X_tab, Y_tab, x_edges, y_edges)
 '''
-def build_xy_lut_from_hist(H):
+def build_xy_lut_Rosenblatt(H):
     """
     Generate two 2D tables X_tab, Y_tab from a 2D histogram H
     such that (X_tab[i,j], Y_tab[i,j]) is the (x,y) value at the center of bin (i,j).
@@ -44,6 +44,44 @@ def build_xy_lut_from_hist(H):
 
     return (X_tab, Y_tab)
 
+def build_xy_lut_DoubleCDF(H):
+    """
+    Generate two 2D tables X_tab, Y_tab from a 2D histogram H
+    such that (X_tab[i,j], Y_tab[i,j]) is the (x,y) value at the center of bin (i,j).
+    Input:
+        H: 2D histogram of charge centroid (eta_x, eta_y) distribution, shape (nBinX, nBinY), ranging from 0 to 1.
+    Output:
+        X_tab, Y_tab: two 2D tables of interpolated (x,y) values in position space, shape (nBinX, nBinY), ranging from 0 to 1.
+    """
+    H = np.asarray(H, dtype=float)
+    nBinX, nBinY = H.shape
+    etaX_edges = np.linspace(0, 1, nBinX+1)
+    etaY_edges = np.linspace(0, 1, nBinY+1)
+
+    # slight smoothing to avoid zero-count rows/columns causing flat CDF
+    eps=1e-9
+    alpha = max(eps, 1e-12 * max(1.0, H.sum()))
+    Hs = H + alpha
+
+    # marginal 1D CDF F_EtaX
+    pEtaX = Hs.sum(axis=1)                                  # Marginal PDF (nBinX,)
+    F_EtaX_edges = np.concatenate([[0.0], np.cumsum(pEtaX)])# CDF at bin edges (nBinX+1,)
+    F_EtaX_edges = F_EtaX_edges / F_EtaX_edges[-1]                   
+
+    # marginal 1D CDF F_EtaY
+    pEtaY = Hs.sum(axis=0)                                  # Marginal PDF (nBinY,)
+    F_EtaY_edges = np.concatenate([[0.0], np.cumsum(pEtaY)])# CDF at bin edges (nBinY+1,)
+    F_EtaY_edges = F_EtaY_edges / F_EtaY_edges[-1]                   
+
+    # calculate (u,v) at the center of each original bin
+    Xc_1d = 0.5 * (F_EtaX_edges[:-1] + F_EtaX_edges[1:])
+    Yc_1d = 0.5 * (F_EtaY_edges[:-1] + F_EtaY_edges[1:])
+
+    X_tab = np.repeat(Xc_1d[:, None], nBinY, axis=1)        # (nBinX, nBinY)
+    Y_tab = np.repeat(Yc_1d[None, :], nBinX, axis=0)        # (nBinX, nBinY)
+
+    return (X_tab, Y_tab)
+
 def bilinear_xy_lookup(etaX, etaY, X_tab, Y_tab):
     """
     bilinear interpolation, from (eta_x, eta_y) to (u,v):