format code

Author: yuehhua

src/Transform/polynomials.jl

@@ -16,47 +16,47 @@ function legendre_ϕ_ψ(k)
     p = Polynomial([-1, 2])  # 2x-1
     p2 = Polynomial([-1, 4])  # 4x-1
 
-    for ki in 0:(k-1)
+    for ki in 0:(k - 1)
         l = convert(Polynomial, gen_poly(Legendre, ki))  # Legendre of n=ki
-        ϕ_coefs[ki+1, 1:(ki+1)] .= sqrt(2*ki+1) .* coeffs(l(p))
-        ϕ_2x_coefs[ki+1, 1:(ki+1)] .= sqrt(2*(2*ki+1)) .* coeffs(l(p2))
+        ϕ_coefs[ki + 1, 1:(ki + 1)] .= sqrt(2 * ki + 1) .* coeffs(l(p))
+        ϕ_2x_coefs[ki + 1, 1:(ki + 1)] .= sqrt(2 * (2 * ki + 1)) .* coeffs(l(p2))
     end
-    
+
     ψ1_coefs = zeros(k, k)
     ψ2_coefs = zeros(k, k)
-    for ki in 0:(k-1)
-        ψ1_coefs[ki+1, :] .= ϕ_2x_coefs[ki+1, :]
-        for i in 0:(k-1)
-            a = ϕ_2x_coefs[ki+1, 1:(ki+1)]
-            b = ϕ_coefs[i+1, 1:(i+1)]
+    for ki in 0:(k - 1)
+        ψ1_coefs[ki + 1, :] .= ϕ_2x_coefs[ki + 1, :]
+        for i in 0:(k - 1)
+            a = ϕ_2x_coefs[ki + 1, 1:(ki + 1)]
+            b = ϕ_coefs[i + 1, 1:(i + 1)]
             proj_ = proj_factor(a, b)
-            ψ1_coefs[ki+1, :] .-= proj_ .* view(ϕ_coefs, i+1, :)
-            ψ2_coefs[ki+1, :] .-= proj_ .* view(ϕ_coefs, i+1, :)
+            ψ1_coefs[ki + 1, :] .-= proj_ .* view(ϕ_coefs, i + 1, :)
+            ψ2_coefs[ki + 1, :] .-= proj_ .* view(ϕ_coefs, i + 1, :)
         end
 
-        for j in 0:(k-1)
-            a = ϕ_2x_coefs[ki+1, 1:(ki+1)]
-            b = ψ1_coefs[j+1, :]
+        for j in 0:(k - 1)
+            a = ϕ_2x_coefs[ki + 1, 1:(ki + 1)]
+            b = ψ1_coefs[j + 1, :]
             proj_ = proj_factor(a, b)
-            ψ1_coefs[ki+1, :] .-= proj_ .* view(ψ1_coefs, j+1, :)
-            ψ2_coefs[ki+1, :] .-= proj_ .* view(ψ2_coefs, j+1, :)
+            ψ1_coefs[ki + 1, :] .-= proj_ .* view(ψ1_coefs, j + 1, :)
+            ψ2_coefs[ki + 1, :] .-= proj_ .* view(ψ2_coefs, j + 1, :)
         end
 
-        a = ψ1_coefs[ki+1, :]
+        a = ψ1_coefs[ki + 1, :]
         norm1 = proj_factor(a, a)
 
-        a = ψ2_coefs[ki+1, :]
-        norm2 = proj_factor(a, a, complement=true)
+        a = ψ2_coefs[ki + 1, :]
+        norm2 = proj_factor(a, a, complement = true)
         norm_ = sqrt(norm1 + norm2)
-        ψ1_coefs[ki+1, :] ./= norm_
-        ψ2_coefs[ki+1, :] ./= norm_
+        ψ1_coefs[ki + 1, :] ./= norm_
+        ψ2_coefs[ki + 1, :] ./= norm_
         zero_out!(ψ1_coefs)
         zero_out!(ψ2_coefs)
     end
 
-    ϕ = [Polynomial(ϕ_coefs[i,:]) for i in 1:k]
-    ψ1 = [Polynomial(ψ1_coefs[i,:]) for i in 1:k]
-    ψ2 = [Polynomial(ψ2_coefs[i,:]) for i in 1:k]
+    ϕ = [Polynomial(ϕ_coefs[i, :]) for i in 1:k]
+    ψ1 = [Polynomial(ψ1_coefs[i, :]) for i in 1:k]
+    ψ2 = [Polynomial(ψ2_coefs[i, :]) for i in 1:k]
 
     return ϕ, ψ1, ψ2
 end
@@ -68,14 +68,14 @@ function chebyshev_ϕ_ψ(k)
     p = Polynomial([-1, 2])  # 2x-1
     p2 = Polynomial([-1, 4])  # 4x-1
 
-    for ki in 0:(k-1)
+    for ki in 0:(k - 1)
         if ki == 0
-            ϕ_coefs[ki+1, 1:(ki+1)] .= sqrt(2/π)
-            ϕ_2x_coefs[ki+1, 1:(ki+1)] .= sqrt(4/π)
+            ϕ_coefs[ki + 1, 1:(ki + 1)] .= sqrt(2 / π)
+            ϕ_2x_coefs[ki + 1, 1:(ki + 1)] .= sqrt(4 / π)
         else
             c = convert(Polynomial, gen_poly(Chebyshev, ki))  # Chebyshev of n=ki
-            ϕ_coefs[ki+1, 1:(ki+1)] .= 2/sqrt(π) .* coeffs(c(p))
-            ϕ_2x_coefs[ki+1, 1:(ki+1)] .= sqrt(2) * 2/sqrt(π) .* coeffs(c(p2))
+            ϕ_coefs[ki + 1, 1:(ki + 1)] .= 2 / sqrt(π) .* coeffs(c(p))
+            ϕ_2x_coefs[ki + 1, 1:(ki + 1)] .= sqrt(2) * 2 / sqrt(π) .* coeffs(c(p2))
         end
     end
 
@@ -87,43 +87,43 @@ function chebyshev_ϕ_ψ(k)
     # x_m[x_m==0.5] = 0.5 + 1e-8 # add small noise to avoid the case of 0.5 belonging to both phi(2x) and phi(2x-1)
     # not needed for our purpose here, we use even k always to avoid
     wm = π / k_use / 2
-    
+
     ψ1_coefs = zeros(k, k)
     ψ2_coefs = zeros(k, k)
 
-    ψ1 = Array{Any,1}(undef, k)
-    ψ2 = Array{Any,1}(undef, k)
+    ψ1 = Array{Any, 1}(undef, k)
+    ψ2 = Array{Any, 1}(undef, k)
 
-    for ki in 0:(k-1)
-        ψ1_coefs[ki+1, :] .= ϕ_2x_coefs[ki+1, :]
-        for i in 0:(k-1)
-            proj_ = sum(wm .* ϕ[i+1].(x_m) .* sqrt(2) .* ϕ[ki+1].(2*x_m))
-            ψ1_coefs[ki+1, :] .-= proj_ .* view(ϕ_coefs, i+1, :)
-            ψ2_coefs[ki+1, :] .-= proj_ .* view(ϕ_coefs, i+1, :)
+    for ki in 0:(k - 1)
+        ψ1_coefs[ki + 1, :] .= ϕ_2x_coefs[ki + 1, :]
+        for i in 0:(k - 1)
+            proj_ = sum(wm .* ϕ[i + 1].(x_m) .* sqrt(2) .* ϕ[ki + 1].(2 * x_m))
+            ψ1_coefs[ki + 1, :] .-= proj_ .* view(ϕ_coefs, i + 1, :)
+            ψ2_coefs[ki + 1, :] .-= proj_ .* view(ϕ_coefs, i + 1, :)
         end
 
-        for j in 0:(ki-1)
-            proj_ = sum(wm .* ψ1[j+1].(x_m) .* sqrt(2) .* ϕ[ki+1].(2*x_m))
-            ψ1_coefs[ki+1, :] .-= proj_ .* view(ψ1_coefs, j+1, :)
-            ψ2_coefs[ki+1, :] .-= proj_ .* view(ψ2_coefs, j+1, :)
+        for j in 0:(ki - 1)
+            proj_ = sum(wm .* ψ1[j + 1].(x_m) .* sqrt(2) .* ϕ[ki + 1].(2 * x_m))
+            ψ1_coefs[ki + 1, :] .-= proj_ .* view(ψ1_coefs, j + 1, :)
+            ψ2_coefs[ki + 1, :] .-= proj_ .* view(ψ2_coefs, j + 1, :)
         end
 
-        ψ1[ki+1] = ϕ_(ψ1_coefs[ki+1,:]; lb=0., ub=0.5)
-        ψ2[ki+1] = ϕ_(ψ2_coefs[ki+1,:]; lb=0.5, ub=1.)
-    
-        norm1 = sum(wm .* ψ1[ki+1].(x_m) .* ψ1[ki+1].(x_m))
-        norm2 = sum(wm .* ψ2[ki+1].(x_m) .* ψ2[ki+1].(x_m))
-    
+        ψ1[ki + 1] = ϕ_(ψ1_coefs[ki + 1, :]; lb = 0.0, ub = 0.5)
+        ψ2[ki + 1] = ϕ_(ψ2_coefs[ki + 1, :]; lb = 0.5, ub = 1.0)
+
+        norm1 = sum(wm .* ψ1[ki + 1].(x_m) .* ψ1[ki + 1].(x_m))
+        norm2 = sum(wm .* ψ2[ki + 1].(x_m) .* ψ2[ki + 1].(x_m))
+
         norm_ = sqrt(norm1 + norm2)
-        ψ1_coefs[ki+1, :] ./= norm_
-        ψ2_coefs[ki+1, :] ./= norm_
+        ψ1_coefs[ki + 1, :] ./= norm_
+        ψ2_coefs[ki + 1, :] ./= norm_
         zero_out!(ψ1_coefs)
         zero_out!(ψ2_coefs)
-    
-        ψ1[ki+1] = ϕ_(ψ1_coefs[ki+1,:]; lb=0., ub=0.5+1e-16)
-        ψ2[ki+1] = ϕ_(ψ2_coefs[ki+1,:]; lb=0.5+1e-16, ub=1.)
+
+        ψ1[ki + 1] = ϕ_(ψ1_coefs[ki + 1, :]; lb = 0.0, ub = 0.5 + 1e-16)
+        ψ2[ki + 1] = ϕ_(ψ2_coefs[ki + 1, :]; lb = 0.5 + 1e-16, ub = 1.0)
     end
-    
+
     return ϕ, ψ1, ψ2
 end
 
@@ -137,22 +137,26 @@ function legendre_filter(k)
     l = convert(Polynomial, gen_poly(Legendre, k))
     x_m = roots(l(Polynomial([-1, 2])))  # 2x-1
     m = 2 .* x_m .- 1
-    wm = 1 ./ k ./ legendre_der.(k, m) ./ gen_poly(Legendre, k-1).(m)
-    
-    for ki in 0:(k-1)
-        for kpi in 0:(k-1)
-            H0[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ϕ[ki+1].(x_m/2) .* ϕ[kpi+1].(x_m))
-            G0[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ψ(ψ1, ψ2, ki, x_m/2) .* ϕ[kpi+1].(x_m))
-            H1[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ϕ[ki+1].((x_m.+1)/2) .* ϕ[kpi+1].(x_m))
-            G1[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ψ(ψ1, ψ2, ki, (x_m.+1)/2) .* ϕ[kpi+1].(x_m))
+    wm = 1 ./ k ./ legendre_der.(k, m) ./ gen_poly(Legendre, k - 1).(m)
+
+    for ki in 0:(k - 1)
+        for kpi in 0:(k - 1)
+            H0[ki + 1, kpi + 1] = 1 / sqrt(2) *
+                                  sum(wm .* ϕ[ki + 1].(x_m / 2) .* ϕ[kpi + 1].(x_m))
+            G0[ki + 1, kpi + 1] = 1 / sqrt(2) *
+                                  sum(wm .* ψ(ψ1, ψ2, ki, x_m / 2) .* ϕ[kpi + 1].(x_m))
+            H1[ki + 1, kpi + 1] = 1 / sqrt(2) *
+                                  sum(wm .* ϕ[ki + 1].((x_m .+ 1) / 2) .* ϕ[kpi + 1].(x_m))
+            G1[ki + 1, kpi + 1] = 1 / sqrt(2) * sum(wm .* ψ(ψ1, ψ2, ki, (x_m .+ 1) / 2) .*
+                                      ϕ[kpi + 1].(x_m))
         end
     end
 
     zero_out!(H0)
     zero_out!(H1)
     zero_out!(G0)
     zero_out!(G1)
-        
+
     return H0, H1, G0, G1, I(k), I(k)
 end
 
@@ -172,14 +176,19 @@ function chebyshev_filter(k)
     # not needed for our purpose here, we use even k always to avoid
     wm = π / k_use / 2
 
-    for ki in 0:(k-1)
-        for kpi in 0:(k-1)
-            H0[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ϕ[ki+1].(x_m/2) .* ϕ[kpi+1].(x_m))
-            H1[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ϕ[ki+1].((x_m.+1)/2) .* ϕ[kpi+1].(x_m))
-            G0[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ψ(ψ1, ψ2, ki, x_m/2) .* ϕ[kpi+1].(x_m))
-            G1[ki+1, kpi+1] = 1/sqrt(2) * sum(wm .* ψ(ψ1, ψ2, ki, (x_m.+1)/2) .* ϕ[kpi+1].(x_m))
-            Φ0[ki+1, kpi+1] = 2*sum(wm .* ϕ[ki+1].(2x_m) .* ϕ[kpi+1].(2x_m))
-            Φ1[ki+1, kpi+1] = 2*sum(wm .* ϕ[ki+1].(2 .* x_m .- 1) .* ϕ[kpi+1].(2 .* x_m .- 1))
+    for ki in 0:(k - 1)
+        for kpi in 0:(k - 1)
+            H0[ki + 1, kpi + 1] = 1 / sqrt(2) *
+                                  sum(wm .* ϕ[ki + 1].(x_m / 2) .* ϕ[kpi + 1].(x_m))
+            H1[ki + 1, kpi + 1] = 1 / sqrt(2) *
+                                  sum(wm .* ϕ[ki + 1].((x_m .+ 1) / 2) .* ϕ[kpi + 1].(x_m))
+            G0[ki + 1, kpi + 1] = 1 / sqrt(2) *
+                                  sum(wm .* ψ(ψ1, ψ2, ki, x_m / 2) .* ϕ[kpi + 1].(x_m))
+            G1[ki + 1, kpi + 1] = 1 / sqrt(2) * sum(wm .* ψ(ψ1, ψ2, ki, (x_m .+ 1) / 2) .*
+                                      ϕ[kpi + 1].(x_m))
+            Φ0[ki + 1, kpi + 1] = 2 * sum(wm .* ϕ[ki + 1].(2x_m) .* ϕ[kpi + 1].(2x_m))
+            Φ1[ki + 1, kpi + 1] = 2 * sum(wm .* ϕ[ki + 1].(2 .* x_m .- 1) .*
+                                      ϕ[kpi + 1].(2 .* x_m .- 1))
         end
     end
 
@@ -189,6 +198,6 @@ function chebyshev_filter(k)
     zero_out!(G1)
     zero_out!(Φ0)
     zero_out!(Φ1)
-        
+
     return H0, H1, G0, G1, Φ0, Φ1
 end

src/Transform/utils.jl

@@ -1,46 +1,46 @@
-function ϕ_(ϕ_coefs; lb::Real=0., ub::Real=1.)
+function ϕ_(ϕ_coefs; lb::Real = 0.0, ub::Real = 1.0)
     function partial(x)
-        mask = (lb ≤ x ≤ ub) * 1.
+        mask = (lb ≤ x ≤ ub) * 1.0
         return Polynomial(ϕ_coefs)(x) * mask
     end
     return partial
 end
 
 function ψ(ψ1, ψ2, i, inp)
     mask = (inp .> 0.5) .* 1.0
-    return ψ1[i+1].(inp) .* mask .+ ψ2[i+1].(inp) .* mask
+    return ψ1[i + 1].(inp) .* mask .+ ψ2[i + 1].(inp) .* mask
 end
 
-zero_out!(x; tol=1e-8) = (x[abs.(x) .< tol] .= 0)
+zero_out!(x; tol = 1e-8) = (x[abs.(x) .< tol] .= 0)
 
 function gen_poly(poly, n)
-    x = zeros(n+1)
+    x = zeros(n + 1)
     x[end] = 1
     return poly(x)
 end
 
 function convolve(a, b)
     n = length(b)
-    y = similar(a, length(a)+n-1)
+    y = similar(a, length(a) + n - 1)
     for i in 1:length(a)
-        y[i:(i+n-1)] .+= a[i] .* b
+        y[i:(i + n - 1)] .+= a[i] .* b
     end
     return y
 end
 
-function proj_factor(a, b; complement::Bool=false)
+function proj_factor(a, b; complement::Bool = false)
     prod_ = convolve(a, b)
     r = collect(1:length(prod_))
     s = complement ? (1 .- 0.5 .^ r) : (0.5 .^ r)
     proj_ = sum(prod_ ./ r .* s)
     return proj_
 end
 
-_legendre(k, x) = (2k+1) * gen_poly(Legendre, k)(x)
+_legendre(k, x) = (2k + 1) * gen_poly(Legendre, k)(x)
 
 function legendre_der(k, x)
     out = 0
-    for i in k-1:-2:-1
+    for i in (k - 1):-2:-1
         out += _legendre(i, x)
     end
     return out

src/Transform/wavelet_transform.jl

@@ -1,15 +1,15 @@
 export WaveletTransform
 
-struct WaveletTransform{N, S}<:AbstractTransform
-    ec_d
-    ec_s
+struct WaveletTransform{N, S} <: AbstractTransform
+    ec_d::Any
+    ec_s::Any
     modes::NTuple{N, S} # N == ndims(x)
 end
 
 Base.ndims(::WaveletTransform{N}) where {N} = N
 
 function transform(wt::WaveletTransform, 𝐱::AbstractArray)
-    N = size(X, ndims(wt)-1)
+    N = size(X, ndims(wt) - 1)
     # 1d
     Xa = vcat(view(𝐱, :, :, 1:2:N, :), view(𝐱, :, :, 2:2:N, :))
     # 2d
@@ -24,9 +24,7 @@ function transform(wt::WaveletTransform, 𝐱::AbstractArray)
     return d, s
 end
 
-function inverse(wt::WaveletTransform, 𝐱_fwt::AbstractArray)
-    
-end
+function inverse(wt::WaveletTransform, 𝐱_fwt::AbstractArray) end
 
 # function truncate_modes(wt::WaveletTransform, 𝐱_fft::AbstractArray)
 #     return view(𝐱_fft, map(d->1:d, wt.modes)..., :, :) # [ft.modes..., in_chs, batch]