WIP: reduced precision for lanczos_opencv

[bjarthur]

using Float32 for Lanczos is ~2x faster and uses ~1/2x as much memory as the current Float64.

this PR currently uses whatever precision was input as the precision the internal calculations are performed with. i could also imagine specifying the type used for internal computations in the type (e.g. struct Lanczos4OpenCV{T} <: AbstractLanczos end) to separate it from the input.

i'm also curious where there is a more clever way to cast l4_2d_cs at compile time so as not to incur runtime penalities.

let me know what you think and i'll add some tests and docs.

julia> using Interpolations, BenchmarkTools

julia> x=rand(1_000_000);

julia> @benchmark Interpolations._lanczos4_opencv.(x)
BenchmarkTools.Trial: 231 samples with 1 evaluation per sample.
 Range (min … max):  19.061 ms … 83.561 ms  ┊ GC (min … max): 5.34% … 74.95%
 Time  (median):     21.303 ms              ┊ GC (median):    2.90%
 Time  (mean ± σ):   21.654 ms ±  4.135 ms  ┊ GC (mean ± σ):  4.35% ±  5.10%

                       ▅█▄▃                                    
  ▂▁▂▁▁▁▁▁▁▁▁▁▁▁▁▃▃▃▃▅▇█████▄▄▄▃▃▃▃▂▂▁▁▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▂ ▃
  19.1 ms         Histogram: frequency by time        24.8 ms <

 Memory estimate: 61.05 MiB, allocs estimate: 4.

julia> x=rand(Float32, 1_000_000);

julia> @benchmark Interpolations._lanczos4_opencv.(x)
BenchmarkTools.Trial: 387 samples with 1 evaluation per sample.
 Range (min … max):  12.078 ms … 76.608 ms  ┊ GC (min … max): 0.00% … 83.87%
 Time  (median):     12.695 ms              ┊ GC (median):    3.05%
 Time  (mean ± σ):   12.928 ms ±  3.290 ms  ┊ GC (mean ± σ):  4.56% ±  4.80%

     ▂       ▄▅▇██▇▅▂                                          
  ▆▇▇██▄▆▆▄▇██████████▄█▄▇▄▆▁▄▁▇▄▄▁▄▄▄▇▄▆▁▁▄▁▁▁▁▆▁▁▁▁▁▁▄▄▁▁▁▄ ▇
  12.1 ms      Histogram: log(frequency) by time      14.5 ms <

 Memory estimate: 30.53 MiB, allocs estimate: 4.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 88.01%. Comparing base (7a2d581) to head (740fa93).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #635      +/-   ##
==========================================
- Coverage   88.10%   88.01%   -0.10%     
==========================================
  Files          29       29              
  Lines        1908     1910       +2     
==========================================
  Hits         1681     1681              
- Misses        227      229       +2     

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[bjarthur]

second commit makes it slightly faster:

julia> x=rand(Float32, 1_000_000);

julia> @benchmark Interpolations.value_weights.(Ref(Lanczos4OpenCV()), x)
BenchmarkTools.Trial: 429 samples with 1 evaluation per sample.
 Range (min … max):  10.796 ms … 75.222 ms  ┊ GC (min … max): 0.00% … 85.30%
 Time  (median):     11.479 ms              ┊ GC (median):    3.84%
 Time  (mean ± σ):   11.664 ms ±  3.107 ms  ┊ GC (mean ± σ):  5.43% ±  4.66%

               ▂█▆▃▄▄▅▇▃▁                                      
  ▃▃▃▃▃▃▁▃▃▃▄▅▆██████████▆▃▃▃▄▃▁▁▂▃▁▁▃▃▂▁▁▃▁▂▁▃▁▂▂▁▁▁▁▁▁▁▁▂▂▂ ▃
  10.8 ms         Histogram: frequency by time          13 ms <

 Memory estimate: 30.53 MiB, allocs estimate: 5.

[bjarthur]

the output is qualitatively different for Float32 compared to Float64. :(

specifically, when δx is 0f0 (a 32-bit float) in _lanczos4_opencv, then s0 and c0 are exactly identical, and so cs[4] becomes 0f0 because the numerator is zero. this is not a problem when δx is 0.0 (a 64-bit float), because due to a higher numerical precision somehow s0 and c0 are off by 1 eps, and so the numerator of csis not zero.

i don't know enough about lanczos resampling to fix this in the right way. what i do know though is that this is not a problem if opencv's lanczos implementation is followed more closely (see the last commit of this PR). specifically, iszero is replaced with abs(y) <= 1e-6.

so i'm putting this on the back burner for now. below is the output without the last commit.

@mkitti @timholy @mileslucas

julia> Interpolations.value_weights(Lanczos4OpenCV{Float64}(), 0.0)
(8.837979241208245e-19, -2.8122277740499295e-18, 7.95418131708742e-18, 1.0, -7.95418131708742e-18, 2.8122277740499295e-18, -8.837979241208245e-19, -7.80550006310146e-35)

julia> Interpolations.value_weights(Lanczos4OpenCV{Float32}(), 0f0)
(8.547569f6, -2.7198194f7, 7.692811f7, -0.0f0, -7.692811f7, 2.7198194f7, -8.547569f6, -0.0f0)