Use much more accurate filtsum estimation polynomials

cuburn/code/filtering.py

@@ -124,6 +124,9 @@ void logscale(float4 *pixbuf, float4 *outbuf, float k1, float k2) {
     outbuf[i] = pix;
 }
 
+#define MIN_SD 0.23299530
+#define MAX_SD 4.33333333
+
 __global__
 void density_est(float4 *pixbuf, float4 *outbuf, float *denbuf,
                  float est_sd, float neg_est_curve, float est_min,
@@ -147,37 +150,60 @@ void density_est(float4 *pixbuf, float4 *outbuf, float *denbuf,
             in.z *= ls;
             in.w *= ls;
 
+            // Base index of destination for writes
+            int si = (threadIdx.y + W2) * FW + threadIdx.x + W2;
+
             // Calculate standard deviation of Gaussian kernel. The base SD is
             // then scaled in inverse proportion to the density of the point
             // being scaled.
             float sd = est_sd * powf(den+1.0f, neg_est_curve);
             // Clamp the final standard deviation. Things will go badly if the
             // minimum is undershot.
-            sd = fmaxf(sd, est_min);
+            sd = fminf(MAX_SD, fmaxf(sd, est_min));
 
-            // This five-term polynomial approximates the sum of the filters
+            // Below a certain threshold, only one coeffecient would be
+            // retained anyway; we hop right to it.
+            if (sd <= MIN_SD) {
+                de_add(si, 0,  0, in);
+            } else {
+                // These polynomials approximates the sum of the filters
                 // with the clamping logic used here. See helpers/filt_err.py.
                 float filtsum;
-            filtsum = -0.20885075f  * sd +  0.90557721f;
+                if (sd < 0.75) {
-            filtsum = filtsum       * sd +  5.28363054f;
+                    filtsum = -352.25061035f;
-            filtsum = filtsum       * sd + -0.11733533f;
+                    filtsum = filtsum * sd +    1117.09680176f;
-            filtsum = filtsum       * sd +  0.35670333f;
+                    filtsum = filtsum * sd +   -1372.48864746f;
-            float filtscale = 1 / filtsum;
+                    filtsum = filtsum * sd +     779.15478516f;
+                    filtsum = filtsum * sd +    -164.04229736f;
+                    filtsum = filtsum * sd +     -12.04892635f;
+                    filtsum = filtsum * sd +       9.04126644f;
+                    filtsum = filtsum * sd +       0.10304667f;
+                } else {
+                    filtsum = -0.00403376f;
+                    filtsum = filtsum * sd +       0.06608720f;
+                    filtsum = filtsum * sd +      -0.38924992f;
+                    filtsum = filtsum * sd +       0.84797901f;
+                    filtsum = filtsum * sd +       0.34173131f;
+                    filtsum = filtsum * sd +      -4.67077589f;
+                    filtsum = filtsum * sd +      14.34595776f;
+                    filtsum = filtsum * sd +      -5.80082798f;
+                    filtsum = filtsum * sd +       1.54098487f;
+                }
+                float filtscale = 1.0f / filtsum;
 
-            // The reciprocal SD scaling coeffecient in the Gaussian exponent.
+                // The reciprocal SD scaling coeffecient in the Gaussian
-            // exp(-x^2/(2*sd^2)) = exp2f(x^2*rsd)
+                // exponent: exp(-x^2/(2*sd^2)) = exp2f(x^2*rsd)
                 float rsd = -0.5f * CUDART_L2E_F / (sd * sd);
 
-            int si = (threadIdx.y + W2) * FW + threadIdx.x + W2;
                 for (int jj = 0; jj <= W2; jj++) {
                     float jj2f = jj;
                     jj2f *= jj2f;
 
                     float iif = 0;
                     for (int ii = 0; ii <= jj; ii++) {
-                    iif += 1;
 
-                    float coeff = exp2f((jj2f + iif * iif) * rsd) * filtscale;
+                        float coeff = exp2f((jj2f + iif * iif) * rsd)
+                                    * filtscale;
                         if (coeff < 0.0001f) break;
 
                         float4 scaled;
@@ -201,10 +227,12 @@ void density_est(float4 *pixbuf, float4 *outbuf, float *denbuf,
                         de_add(si, -jj, -ii, scaled);
                         de_add(si,  jj, -ii, scaled);
 
+                        iif += 1;
                         // TODO: validate that the above avoids bank conflicts
                     }
                 }
             }
+        }
 
         __syncthreads();
         // TODO: could coalesce this, but what a pain
@@ -262,7 +290,7 @@ void density_est(float4 *pixbuf, float4 *outbuf, float *denbuf,
             # (0.5/1.5)=1/3.
             est_sd = np.float32(cp.estimator / 3.)
             neg_est_curve = np.float32(-cp.estimator_curve)
-            est_min = np.float32(max(cp.estimator_minimum / 3., 0.4))
+            est_min = np.float32(cp.estimator_minimum / 3.)
             fun = mod.get_function("density_est")
             fun(abufd, obufd, dbufd, est_sd, neg_est_curve, est_min, k1, k2,
                 block=(32, 32, 1), grid=(self.features.acc_width/32, 1),

helpers/filt_err.py

@@ -9,38 +9,96 @@ F2 = int(FWIDTH/2)
 # The maximum size of any one coeffecient to be retained
 COEFF_EPS = 0.0001
 
-dists = np.fromfunction(lambda i, j: np.hypot(i-F2, j-F2), (FWIDTH, FWIDTH))
+dists2d = np.fromfunction(lambda i, j: np.hypot(i-F2, j-F2), (FWIDTH, FWIDTH))
-dists = dists.flatten()
+dists = dists2d.flatten()
 
 # A flam3 estimator radius corresponds to a Gaussian filter with a standard
 # deviation of 1/3 the radius. We choose 13 as an arbitrary upper bound for the
-# max filter radius. Larger radii will work without
+# max filter radius. The filter should reject larger radii.
 MAX_SD = 13 / 3.
 
-# The minimum estimator radius is 1. In flam3, this is effectively no
+# The minimum estimator radius can be set as low as 0, but below a certain
-# filtering, but since the cutoff structure is defined by COEFF_EPS in cuburn,
+# radius only one coeffecient is retained. Since things get unstable near 0,
-# we undershoot it a bit to make the polyfit behave better at high densities.
+# we explicitly set a minimum threshold below which no coeffecients are
-MIN_SD = 0.3
+# retained.
+MIN_SD = np.sqrt(-1 / (2 * np.log(COEFF_EPS)))
 
-sds = np.logspace(np.log10(MIN_SD), np.log10(MAX_SD), num=100)
+# Using two predicated three-term approximations is much more accurate than
+# using a very large number of terms, due to nonlinear behavior at low SD.
+# Everything above this SD uses one approximation; below, another.
+SPLIT_SD = 0.75
 
-# Calculate the filter sums at each coordinate
+# The lower endpoints are undershot by this proportion to reduce error
-sums = []
+UNDERSHOOT = 0.98
-for sd in sds:
+
+sds_hi = np.linspace(SPLIT_SD * UNDERSHOOT, MAX_SD, num=1000)
+sds_lo = np.linspace(MIN_SD * UNDERSHOOT, SPLIT_SD, num=1000)
+
+print 'At MIN_SD = %g, these are the coeffs:' % MIN_SD
+print np.exp(dists2d**2 / (-2 * MIN_SD ** 2))
+
+def eval_sds(sds, name, nterms):
+    # Calculate the filter sums at each coordinate
+    sums = []
+    for sd in sds:
         coeffs = np.exp(dists**2 / (-2 * sd ** 2))
-    sums.append(np.sum(filter(lambda v: v / np.sum(coeffs) > COEFF_EPS, coeffs)))
+        # Note that this sum is the sum of all coordinates, though it should
-print sums
+        # actually be the result of the polynomial approximation. We could do
+        # a feedback loop to improve accuracy, but I don't think the difference
+        # is worth worrying about.
+        sum = np.sum(coeffs)
+        sums.append(np.sum(filter(lambda v: v / sum > COEFF_EPS, coeffs)))
+    print 'Evaluating %s:' % name
+    poly, resid, rank, sing, rcond = np.polyfit(sds, sums, nterms, full=True)
+    print 'Fit for %s:' % name, poly, resid, rank, sing, rcond
+    return sums, poly
 
 import matplotlib.pyplot as plt
 
-poly, resid, rank, sing, rcond = np.polyfit(sds, sums, 4, full=True)
+sums_hi, poly_hi = eval_sds(sds_hi, 'hi', 8)
-print poly, resid, rank, sing, rcond
+sums_lo, poly_lo = eval_sds(sds_lo, 'lo', 7)
+
+num_undershoots = len(filter(lambda v: v < SPLIT_SD, sds_hi))
+sds_hi = sds_hi[num_undershoots:]
+sums_hi = sums_hi[num_undershoots:]
+
+num_undershoots = len(filter(lambda v: v < MIN_SD, sds_lo))
+sds_lo = sds_lo[num_undershoots:]
+sums_lo = sums_lo[num_undershoots:]
+
+polyf_hi = np.float32(poly_hi)
+vals_hi = np.polyval(polyf_hi, sds_hi)
+polyf_lo = np.float32(poly_lo)
+vals_lo = np.polyval(polyf_lo, sds_lo)
+
+def print_filt(filts):
+    print '    filtsum = %4.8ff;' % filts[0]
+    for f in filts[1:]:
+        print '    filtsum = filtsum * sd + % 16.8ff;' % f
+
+print '\n\nFor your convenience:'
+print '#define MIN_SD %.8f' % MIN_SD
+print '#define MAX_SD %.8f' % MAX_SD
+print 'if (sd < %g) {' % SPLIT_SD
+print_filt(polyf_lo)
+print '} else {'
+print_filt(polyf_hi)
+print '}'
+
+sds = np.concatenate([sds_lo, sds_hi])
+sums = np.concatenate([sums_lo, sums_hi])
+vals = np.concatenate([vals_lo, vals_hi])
+
+fig = plt.figure()
+ax = fig.add_subplot(1,1,1)
+ax.plot(sds, sums)
+ax.plot(sds, vals)
+ax.set_xlabel('stdev')
+ax.set_ylabel('filter sum')
+
+ax = ax.twinx()
+ax.plot(sds, [abs((s-v)/v) for s, v in zip(sums, vals)])
+ax.set_ylabel('rel err')
 
-polyf = np.float32(poly)
-plt.plot(sds, sums)
-plt.plot(sds, np.polyval(polyf, sds))
 plt.show()
 
-print np.polyval(poly, 1.1)
-
-# TODO: calculate error more fully, verify all this logic