Lensing a flat-sky map

using CMBLensing, PyPlot

First we load a simulated unlensed field, $f$, and lensing potential, $\phi$,

@unpack f,ϕ = load_sim(
    θpix  = 2,       # size of the pixels in arcmin
    Nside = 256,     # number of pixels per side in the map
    T     = Float32, # Float32 or Float64 (former is ~twice as fast)
    pol   = :I       # :I for Intensity, :P for polarization, or :IP for both
);

We can lense the map with LenseFlow,

f̃ = LenseFlow(ϕ) * f;

And flip between lensed and unlensed maps,

animate([f,f̃], fps=1)

The difference between lensed and unlensed,

plot(f-f̃);

png

Loading your own data

CMBLensing flat-sky Field objects like f or ϕ are just thin wrappers around arrays. You can get the underlying data arrays for $I(\mathbf{x})$, $Q(\mathbf{x})$, and $U(\mathbf{x})$ with f[:Ix], f[:Qx], and f[:Ux] respectively, or the Fourier coefficients, $I(\mathbf{l})$, $Q(\mathbf{l})$, and $U(\mathbf{l})$ with f[:Il], f[:Ql], and f[:Ul],

mapdata = f[:Ix]
256×256 Matrix{Float32}:
  16.9223   16.7107     6.76394   …   30.5313     27.1077     20.8228
  20.2004   23.8824    19.2507        25.6273     24.9317     22.0486
  27.3758   27.577     25.0777        20.4196     27.0562     30.6022
  41.6781   32.1097    24.8412        14.4931     31.064      43.4462
  67.2678   52.0998    35.5184        24.4832     47.7734     65.5578
  91.8447   74.6301    54.6156    …   55.4197     77.4977     90.8392
  91.8751   77.1666    65.84          87.8228    102.607     101.58
  82.4671   68.3388    67.8973       105.495     115.112     102.72
  84.6062   70.5613    68.8783       114.516     120.247     106.039
  88.2165   81.1797    76.8356       109.163     110.216      98.2464
  93.1327   96.2596    91.8871    …   88.6865     89.7113     88.2815
 105.105   111.808    106.006         76.6157     80.6468     90.0507
 112.853   117.914    108.188         87.4246     90.5486     98.5647
   ⋮                              ⋱                            ⋮
 -30.8362  -13.8389   -17.8254       -87.1399    -88.2407    -64.3883
 -20.2782    1.37109   -4.691     …  -92.9386    -97.7261    -64.5651
 -18.8288    7.22943    1.90754      -79.1766    -92.7357    -64.3492
 -26.2707   -2.28238   -3.46554      -40.2546    -65.8453    -56.9451
 -26.401   -13.5927   -17.6139        -0.951288  -32.363     -39.8456
 -11.1627   -9.29941  -25.6078        18.9132     -9.90499   -19.0793
  11.4837    8.56296  -16.9111    …   20.6765      0.616802   -0.268501
  28.7133   25.1438     0.519607      14.4127      1.52029    11.4055
  30.8485   29.9892    10.5519        10.528      -4.42103     9.50083
  21.9322   22.1668     8.35608       14.467      -2.81549     4.73002
  17.5845   15.3108     3.43599       23.1523      9.95823    10.3446
  17.9051   13.8167     1.72432   …   30.8429     23.9065     19.3472

If you have your own map data in an array you'd like to load into a CMBLensing Field object, you can construct it as follows:

FlatMap(mapdata, θpix=3)
65536-element 256×256-pixel 3.0′-resolution LambertMap{Array{Float32, 2}}:
  16.922283
  20.20036
  27.375805
  41.67814
  67.26776
  91.84473
  91.875145
  82.46712
  84.606186
  88.216545
  93.132675
 105.10481
 112.852715
   ⋮
 -64.38826
 -64.56508
 -64.34915
 -56.945107
 -39.845634
 -19.079266
  -0.26850128
  11.405504
   9.500831
   4.7300243
  10.344635
  19.347223

For more info on Field objects, see Field Basics.

Inverse lensing

You can inverse lense a map with the \ operator (which does A \ b ≡ inv(A) * b):

LenseFlow(ϕ) \ f;

Note that this is true inverse lensing, rather than lensing by the negative deflection (which is often called "anti-lensing"). This means that lensing then inverse lensing a map should get us back the original map. Lets check that this is the case:

Ns = [7 10 20]
plot([f - (LenseFlow(ϕ,N) \ (LenseFlow(ϕ,N) * f)) for N in Ns],
    title=["ODE steps = $N" for N in Ns]);

png

A cool feature of LenseFlow is that inverse lensing is trivially done by running the LenseFlow ODE in reverse. Note that as we crank up the number of ODE steps above, we recover the original map to higher and higher precision.

Other lensing algorithms

We can also lense via:

  • PowerLens: the standard Taylor series expansion to any order:
\[ f(x+\nabla x) \approx f(x) + (\nabla f)(\nabla \phi) + \frac{1}{2} (\nabla \nabla f) (\nabla \phi)^2 + ... \]
  • TayLens (Næss&Louis 2013): like PowerLens, but first a nearest-pixel permute step, then a Taylor expansion around the now-smaller residual displacement
plot([(PowerLens(ϕ,2)*f - f̃) (Taylens(ϕ,2)*f - f̃)], 
    title=["PowerLens - LenseFlow" "TayLens - LenseFlow"]);

png

Benchmarking

LenseFlow is highly optimized code since it appears on the inner-most loop of our analysis algorithms. To benchmark LenseFlow, note that there is first a precomputation step, which caches some data in preparation for applying it to a field of a given type. This was done automatically when evaluating LenseFlow(ϕ) * f but we can benchmark it separately since in many cases this only needs to be done once for a given $\phi$, e.g. when Wiener filtering at fixed $\phi$,

using BenchmarkTools
@benchmark cache(LenseFlow(ϕ),f)
BenchmarkTools.Trial: 89 samples with 1 evaluation.
 Range (min … max):  55.463 ms … 60.109 ms  ┊ GC (min … max): 6.89% … 12.55%
 Time  (median):     56.076 ms              ┊ GC (median):    6.78%
 Time  (mean ± σ):   56.500 ms ±  1.301 ms  ┊ GC (mean ± σ):  7.59% ±  1.98%

      █▆ ▃▄▂▆                                                  
  ▃▄█▅██▃████▄▄▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▁▅▁▆▁▃▃ ▁
  55.5 ms         Histogram: frequency by time          60 ms <

 Memory estimate: 92.37 MiB, allocs estimate: 1653.

Once cached, it's faster and less memory intensive to repeatedly apply the operator:

@benchmark Lϕ * f setup=(Lϕ=cache(LenseFlow(ϕ),f))
BenchmarkTools.Trial: 53 samples with 1 evaluation.
 Range (min … max):  37.149 ms … 42.434 ms  ┊ GC (min … max): 0.00% … 8.82%
 Time  (median):     38.136 ms              ┊ GC (median):    0.00%
 Time  (mean ± σ):   39.086 ms ±  1.713 ms  ┊ GC (mean ± σ):  2.47% ± 3.89%

         ▃ █ ▁                                                 
  ▄▁▁▁▇▇▄█▆█▇█▁▄▁▄▇▁▁▁▁▄▄▁▁▄▁▁▁▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▇▁▆▁▄▁▄▄▆▄▁▄ ▁
  37.1 ms         Histogram: frequency by time        42.4 ms <

 Memory estimate: 16.14 MiB, allocs estimate: 441.

Note that this documentation is generated on limited-performance cloud servers. Actual benchmarks are likely much faster locally or on a cluster, and yet (much) faster on GPU.