Add UMAs for numerical approximation of table-based transfer functions.

ui/gfx/skia_color_space_util.cc

Index: ui/gfx/skia_color_space_util.cc
diff --git a/ui/gfx/skia_color_space_util.cc b/ui/gfx/skia_color_space_util.cc
index 2dda075fe65c1b129c6488b6019aea778c29132d..a4cfa3b1a5f2c80e46ea204b90edd680623ef4a8 100644
--- a/ui/gfx/skia_color_space_util.cc
+++ b/ui/gfx/skia_color_space_util.cc
@@ -7,11 +7,341 @@
 #include <algorithm>
 #include <cmath>
 
+#include "base/logging.h"
+
 namespace gfx {
 
 namespace {
+
+// Evaluate the gradient of the linear component of fn
+void SkTransferFnEvalGradientLinear(const SkColorSpaceTransferFn& fn,
+                                    float x,
+                                    float* d_fn_d_fC_at_x,
+                                    float* d_fn_d_fF_at_x) {
+  *d_fn_d_fC_at_x = x;
+  *d_fn_d_fF_at_x = 1.f;
+}
+
+// Solve for the parameters fC and fF, given the parameter fD.
+void SkTransferFnSolveLinear(SkColorSpaceTransferFn* fn,
+                             const float* x,
+                             const float* t,
+                             size_t n) {
+  // If this has no linear segment, don't try to solve for one.
+  if (fn->fD <= 0) {
+    fn->fC = 1;
+    fn->fF = 0;
+    return;
+  }
+
+  // Let ne_lhs be the left hand side of the normal equations, and let ne_rhs
+  // be the right hand side. This is a 4x4 matrix, but we will only update the
+  // upper-left 2x2 sub-matrix.
+  SkMatrix44 ne_lhs;
+  SkVector4 ne_rhs;
+  for (int row = 0; row < 2; ++row) {
+    for (int col = 0; col < 2; ++col) {
+      ne_lhs.set(row, col, 0);
+    }
+  }
+  for (int row = 0; row < 4; ++row)
+    ne_rhs.fData[row] = 0;
+
+  // Add the contributions from each sample to the normal equations.
+  float x_linear_max = 0;
+  for (size_t i = 0; i < n; ++i) {
+    // Ignore points in the nonlinear segment.
+    if (x[i] >= fn->fD)
+      continue;
+    x_linear_max = std::max(x_linear_max, x[i]);
+
+    // Let J be the gradient of fn with respect to parameters C and F, evaluated
+    // at this point.
+    float J[2];
+    SkTransferFnEvalGradientLinear(*fn, x[i], &J[0], &J[1]);
+
+    // Let r be the residual at this point.
+    float r = t[i];
+
+    // Update the normal equations left hand side with the outer product of J
+    // with itself.
+    for (int row = 0; row < 2; ++row) {
+      for (int col = 0; col < 2; ++col) {
+        ne_lhs.set(row, col, ne_lhs.get(row, col) + J[row] * J[col]);
+      }
+      // Update the normal equations right hand side the product of J with the
+      // residual
+      ne_rhs.fData[row] += J[row] * r;
+    }
+  }
+
+  // If we only have a single x point at 0, that isn't enough to construct a
+  // linear segment, so add an additional point connecting to the nonlinear
+  // segment.
+  if (x_linear_max == 0) {
+    float J[2];
+    SkTransferFnEvalGradientLinear(*fn, fn->fD, &J[0], &J[1]);
+    float r = SkTransferFnEval(*fn, fn->fD);
+    for (int row = 0; row < 2; ++row) {
+      for (int col = 0; col < 2; ++col) {
+        ne_lhs.set(row, col, ne_lhs.get(row, col) + J[row] * J[col]);
+      }
+      ne_rhs.fData[row] += J[row] * r;
+    }
+  }
+
+  // Solve the normal equations.
+  SkMatrix44 ne_lhs_inv;
+  bool invert_result = ne_lhs.invert(&ne_lhs_inv);
+  DCHECK(invert_result);
+  SkVector4 solution = ne_lhs_inv * ne_rhs;
+
+  // Update the transfer function.
+  fn->fC = solution.fData[0];
+  fn->fF = solution.fData[1];
+}
+
+// Evaluate the gradient of the nonlinear component of fn
+void SkTransferFnEvalGradientNonlinear(const SkColorSpaceTransferFn& fn,
+                                       float x,
+                                       float* d_fn_d_fA_at_x,
+                                       float* d_fn_d_fB_at_x,
+                                       float* d_fn_d_fE_at_x,
+                                       float* d_fn_d_fG_at_x) {
+  float base = fn.fA * x + fn.fB;
+  if (base > 0.f) {
+    *d_fn_d_fA_at_x = fn.fG * x * std::pow(base, fn.fG - 1.f);
+    *d_fn_d_fB_at_x = fn.fG * std::pow(base, fn.fG - 1.f);
+    *d_fn_d_fE_at_x = 1.f;
+    *d_fn_d_fG_at_x = std::pow(base, fn.fG) * std::log(base);
+  } else {
+    *d_fn_d_fA_at_x = 0.f;
+    *d_fn_d_fB_at_x = 0.f;
+    *d_fn_d_fE_at_x = 0.f;
+    *d_fn_d_fG_at_x = 0.f;
+  }
 }
 
+// Take one Gauss-Newton step updating fA, fB, fE, and fG, given fD.
+bool SkTransferFnGaussNewtonStepNonlinear(SkColorSpaceTransferFn* fn,
+                                          float* error_Linfty_before,
+                                          const float* x,
+                                          const float* t,
+                                          size_t n) {
+  float kEpsilon = 1.f / 1024.f;
+  // Let ne_lhs be the left hand side of the normal equations, and let ne_rhs
+  // be the right hand side.
+  SkMatrix44 ne_lhs;
+  SkVector4 ne_rhs;
+  for (int row = 0; row < 4; ++row) {
+    for (int col = 0; col < 4; ++col) {
+      ne_lhs.set(row, col, 0);
+    }
+    ne_rhs.fData[row] = 0;
+  }
+
+  // Add the contributions from each sample to the normal equations.
+  *error_Linfty_before = 0;
+  for (size_t i = 0; i < n; ++i) {
+    // Ignore points in the linear segment.
+    if (x[i] < fn->fD)
+      continue;
+    // Let J be the gradient of fn with respect to parameters A, B, E, and G,
+    // evaulated at this point.
+    float J[4];
+    SkTransferFnEvalGradientNonlinear(*fn, x[i], &J[0], &J[1], &J[2], &J[3]);
+    // Let r be the residual at this point;
+    float r = t[i] - SkTransferFnEval(*fn, x[i]);
+    *error_Linfty_before += std::abs(r);
+    // Update the normal equations left hand side with the outer product of J
+    // with itself.
+    for (int row = 0; row < 4; ++row) {
+      for (int col = 0; col < 4; ++col) {
+        ne_lhs.set(row, col, ne_lhs.get(row, col) + J[row] * J[col]);
+      }
+      // Update the normal equations right hand side the product of J with the
+      // residual
+      ne_rhs.fData[row] += J[row] * r;
+    }
+  }
+
+  // Note that if fG = 1, then the normal equations will be singular (because,
+  // when fG = 1, because fA and fE are equivalent parameters).  To avoid
+  // problems, fix fE (row/column 3) in these circumstances.
+  if (std::abs(fn->fG - 1) < kEpsilon) {
+    for (int row = 0; row < 4; ++row) {
+      float value = (row == 2) ? 1.f : 0.f;
+      ne_lhs.set(row, 2, value);
+      ne_lhs.set(2, row, value);
+    }
+    ne_rhs.fData[2] = 0.f;
+  }
+
+  // Solve the normal equations.
+  SkMatrix44 ne_lhs_inv;
+  if (!ne_lhs.invert(&ne_lhs_inv))
+    return false;
+  SkVector4 step = ne_lhs_inv * ne_rhs;
+
+  // Update the transfer function.
+  fn->fA += step.fData[0];
+  fn->fB += step.fData[1];
+  fn->fE += step.fData[2];
+  fn->fG += step.fData[3];
+  return true;
+}
+
+// Solve for fA, fB, fE, and fG, given fD. The initial value of |fn| is the
+// point from which iteration starts.
+bool SkTransferFnSolveNonlinear(SkColorSpaceTransferFn* fn,
+                                const float* x,
+                                const float* t,
+                                size_t n) {
+  // Take a maximum of 16 Gauss-Newton steps.
+  const size_t kNumSteps = 16;
+
+  // The L-infinity error before each step.
+  float step_error[kNumSteps] = {0};
+
+  for (size_t step = 0; step < kNumSteps; ++step) {
+    // If the normal equations are singular, we can't continue.
+    if (!SkTransferFnGaussNewtonStepNonlinear(fn, &step_error[step], x, t, n))
+      return false;
+
+    // If the error is inf or nan, we are clearly not converging.
+    if (std::isnan(step_error[step]) || std::isinf(step_error[step]))
+      return false;
+
+    // If our error is non-negligable and increasing then we are not in the
+    // region of convergence.
+    const float kNonNegligbleErrorEpsilon = 1.f / 256.f;
+    const float kGrowthFactor = 1.25f;
+    if (step > 2 && step_error[step] > kNonNegligbleErrorEpsilon) {
+      if (step_error[step - 1] * kGrowthFactor < step_error[step] &&
+          step_error[step - 2] * kGrowthFactor < step_error[step - 1]) {
+        return false;
+      }
+    }
+  }
+
+  // We've converged to a reasonable solution. If some of the parameters are
+  // extremely close to 0 or 1, set them to 0 or 1.
+  const float kRoundEpsilon = 1.f / 1024.f;
+  if (std::abs(fn->fA - 1.f) < kRoundEpsilon)
+    fn->fA = 1.f;
+  if (std::abs(fn->fB) < kRoundEpsilon)
+    fn->fB = 0;
+  if (std::abs(fn->fE) < kRoundEpsilon)
+    fn->fE = 0;
+  if (std::abs(fn->fG - 1.f) < kRoundEpsilon)
+    fn->fG = 1.f;
+  return true;
+}
+
+void SkApproximateTransferFnInternal(const float* x,
+                                     const float* t,
+                                     size_t n,
+                                     SkColorSpaceTransferFn* fn,
+                                     float* error,
+                                     bool* nonlinear_fit_converged) {
+  // First, guess at a value of fD. Assume that the nonlinear segment applies
+  // to all x >= 0.1. This is generally a safe assumption (fD is usually less
+  // than 0.1).
+  fn->fD = 0.1f;
+
+  // Do a nonlinear regression on the nonlinear segment. Use a number of guesses
+  // for the initial value of fG, because not all values will converge.
+  *nonlinear_fit_converged = false;
+  {
+    const size_t kNumInitialGammas = 4;
+    float initial_gammas[kNumInitialGammas] = {1.f, 2.f, 3.f, 0.5f};
+    for (size_t i = 0; i < kNumInitialGammas; ++i) {
+      fn->fG = initial_gammas[i];
+      fn->fA = 1;
+      fn->fB = 0;
+      fn->fC = 1;
+      fn->fE = 0;
+      fn->fF = 0;
+      if (SkTransferFnSolveNonlinear(fn, x, t, n)) {
+        *nonlinear_fit_converged = true;
+        break;
+      }
+    }
+  }
+
+  // Now walk back fD from our initial guess to the point where our nonlinear
+  // fit no longer fits (or all the way to 0 if it fits).
+  if (*nonlinear_fit_converged) {
+    // Find the L-infinity error of this nonlinear fit (using our old fD value).
+    float max_error_in_nonlinear_fit = 0;
+    for (size_t i = 0; i < n; ++i) {
+      if (x[i] < fn->fD)
+        continue;
+      float error_at_xi = std::abs(t[i] - SkTransferFnEval(*fn, x[i]));
+      max_error_in_nonlinear_fit =
+          std::max(max_error_in_nonlinear_fit, error_at_xi);
+    }
+
+    // Now find the maximum x value where this nonlinear fit is no longer
+    // accurate or no longer defined.
+    fn->fD = 0.f;
+    float max_x_where_nonlinear_does_not_fit = -1.f;
+    for (size_t i = 0; i < n; ++i) {
+      bool nonlinear_fits_xi = true;
+      if (fn->fA * x[i] + fn->fB < 0) {
+        // The nonlinear segment is undefined when fA * x + fB is less than 0.
+        nonlinear_fits_xi = false;
+      } else {
+        // Define "no longer accurate" as "has more than 10% more error than
+        // the maximum error in the fit segment".
+        float error_at_xi = std::abs(t[i] - SkTransferFnEval(*fn, x[i]));
+        if (error_at_xi > 1.1f * max_error_in_nonlinear_fit)
+          nonlinear_fits_xi = false;
+      }
+      if (!nonlinear_fits_xi) {
+        max_x_where_nonlinear_does_not_fit =
+            std::max(max_x_where_nonlinear_does_not_fit, x[i]);
+      }
+    }
+
+    // Now let fD be the highest sample of x that is above the threshold where
+    // the nonlinear segment does not fit.
+    fn->fD = 1.f;
+    for (size_t i = 0; i < n; ++i) {
+      if (x[i] > max_x_where_nonlinear_does_not_fit)
+        fn->fD = std::min(fn->fD, x[i]);
+    }
+  } else {
+    // If the nonlinear fit didn't converge, then try a linear fit on the full
+    // range. Note that fD must be strictly greater than 1 because the nonlinear
+    // segment starts at fD.
+    const float kEpsilon = 1.f / 256.f;
+    fn->fD = 1.f + kEpsilon;
+  }
+
+  // Compute the linear segment, now that we have our definitive fD.
+  SkTransferFnSolveLinear(fn, x, t, n);
+
+  // If the nonlinear fit didn't converge, re-express the linear fit in
+  // canonical form as a nonlinear fit with fG as 1.
+  if (!*nonlinear_fit_converged) {
+    fn->fA = fn->fC;
+    fn->fB = fn->fF;
+    fn->fC = 1;
+    fn->fD = 0;
+    fn->fE = 0;
+    fn->fF = 0;
+    fn->fG = 1;
+  }
+
+  // Return the L-infinity error of the total approximation.
+  *error = 0.f;
+  for (size_t i = 0; i < n; ++i)
+    *error = std::max(*error, std::abs(t[i] - SkTransferFnEval(*fn, x[i])));
+}
+
+}  // namespace
+
 float SkTransferFnEval(const SkColorSpaceTransferFn& fn, float x) {
   if (x < 0.f)
     return 0.f;
@@ -61,6 +391,15 @@ bool SkTransferFnIsApproximatelyIdentity(const SkColorSpaceTransferFn& a) {
   return true;
 }
 
+void SkApproximateTransferFn(const float* x,
+                             const float* t,
+                             size_t n,
+                             SkColorSpaceTransferFn* fn,
+                             float* error,
+                             bool* nonlinear_fit_converged) {
+  SkApproximateTransferFnInternal(x, t, n, fn, error, nonlinear_fit_converged);
+}
+
 bool SkMatrixIsApproximatelyIdentity(const SkMatrix44& m) {
   const float kEpsilon = 1.f / 256.f;
   for (int i = 0; i < 4; ++i) {