SENTINEL-1 SPECKLE FILTER: refined LEE

/*Copyright (c) 2021 SERVIR-Mekong
 
Permission is hereby granted, free of charge, to any person obtaining a copy
of the data and associated documentation files, to deal in the data
without restriction, including without limitation the rights to use, copy, modify,
merge, publish, distribute, sublicense, and/or sell copies, and to permit persons
to whom the data is furnished to do so, subject to the following conditions:
 
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
 
THE DATA IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.*/

// Algorithm adapted from https://groups.google.com/g/google-earth-engine-developers/c/ExepnAmP-hQ/m/7e5DnjXXAQAJ

// Import Sentinel-1 Collection 
var s1 =  ee.ImageCollection('COPERNICUS/S1_GRD')
			.filter(ee.Filter.listContains('transmitterReceiverPolarisation', 'VH'))
			.filter(ee.Filter.listContains('transmitterReceiverPolarisation', 'VV'))
			.filter(ee.Filter.eq('orbitProperties_pass', 'DESCENDING'))
			.filter(ee.Filter.eq('instrumentMode', 'IW'))
			.filterBounds(geometry)
			.filterDate("2020-10-01","2020-10-31");
			
var firstNoTerrainCorrection = ee.Image(s1.first());

Map.addLayer(firstNoTerrainCorrection,{min:-25,max:20},"no terrain correction");

s1 = s1.map(terrainCorrection);
var s1_lee_sigma = s1.map(refinedLee);

var firstTerrainCorrection = ee.Image(s1.first());
var s1_refinedLee  = ee.Image(s1_lee_sigma.first());


Map.addLayer(firstTerrainCorrection,{min:-25,max:20},"Terrain corrected");
Map.addLayer(s1_refinedLee,{min:-25,max:20},"lee sigma");


// Implementation by Andreas Vollrath (ESA), inspired by Johannes Reiche (Wageningen)
function terrainCorrection(image) { 
  var imgGeom = image.geometry();
  var srtm = ee.Image('USGS/SRTMGL1_003').clip(imgGeom); // 30m srtm 
  var sigma0Pow = ee.Image.constant(10).pow(image.divide(10.0));

  // Article ( numbers relate to chapters) 
  // 2.1.1 Radar geometry 
  var theta_i = image.select('angle');
  var phi_i = ee.Terrain.aspect(theta_i)
    .reduceRegion(ee.Reducer.mean(), theta_i.get('system:footprint'), 1000)
    .get('aspect');

  // 2.1.2 Terrain geometry
  var alpha_s = ee.Terrain.slope(srtm).select('slope');
  var phi_s = ee.Terrain.aspect(srtm).select('aspect');

  // 2.1.3 Model geometry
  // reduce to 3 angle
  var phi_r = ee.Image.constant(phi_i).subtract(phi_s);

  // convert all to radians
  var phi_rRad = phi_r.multiply(Math.PI / 180);
  var alpha_sRad = alpha_s.multiply(Math.PI / 180);
  var theta_iRad = theta_i.multiply(Math.PI / 180);
  var ninetyRad = ee.Image.constant(90).multiply(Math.PI / 180);

  // slope steepness in range (eq. 2)
  var alpha_r = (alpha_sRad.tan().multiply(phi_rRad.cos())).atan();

  // slope steepness in azimuth (eq 3)
  var alpha_az = (alpha_sRad.tan().multiply(phi_rRad.sin())).atan();

  // local incidence angle (eq. 4)
  var theta_lia = (alpha_az.cos().multiply((theta_iRad.subtract(alpha_r)).cos())).acos();
  var theta_liaDeg = theta_lia.multiply(180 / Math.PI);
  // 2.2 
  // Gamma_nought_flat
  var gamma0 = sigma0Pow.divide(theta_iRad.cos());
  var gamma0dB = ee.Image.constant(10).multiply(gamma0.log10());
  var ratio_1 = gamma0dB.select('VV').subtract(gamma0dB.select('VH'));

  // Volumetric Model
  var nominator = (ninetyRad.subtract(theta_iRad).add(alpha_r)).tan();
  var denominator = (ninetyRad.subtract(theta_iRad)).tan();
  var volModel = (nominator.divide(denominator)).abs();

  // apply model
  var gamma0_Volume = gamma0.divide(volModel);
  var gamma0_VolumeDB = ee.Image.constant(10).multiply(gamma0_Volume.log10());

  // we add a layover/shadow maskto the original implmentation
  // layover, where slope > radar viewing angle 
  var alpha_rDeg = alpha_r.multiply(180 / Math.PI);
  var layover = alpha_rDeg.lt(theta_i);

  // shadow where LIA > 90
  var shadow = theta_liaDeg.lt(85);

  // calculate the ratio for RGB vis
  var ratio = gamma0_VolumeDB.select('VV').subtract(gamma0_VolumeDB.select('VH'));

  var output = gamma0_VolumeDB.addBands(ratio).addBands(alpha_r).addBands(phi_s).addBands(theta_iRad)
    .addBands(layover).addBands(shadow).addBands(gamma0dB).addBands(ratio_1);

  return image.addBands(
    output.select(['VV', 'VH'], ['VV', 'VH']),
    null,
    true
  );
}


function powerToDb(img){
  return ee.Image(10).multiply(img.log10());
}

function dbToPower(img){
  return ee.Image(10).pow(img.divide(10));
}

// The RL speckle filter
function refinedLee(image) {
  
  var bandNames = image.bandNames();
  image = dbToPower(image);
  
  var result = ee.ImageCollection(bandNames.map(function(b){
    var img = image.select([b]);
    
    // img must be in natural units, i.e. not in dB!
    // Set up 3x3 kernels 
    var weights3 = ee.List.repeat(ee.List.repeat(1,3),3);
    var kernel3 = ee.Kernel.fixed(3,3, weights3, 1, 1, false);
  
    var mean3 = img.reduceNeighborhood(ee.Reducer.mean(), kernel3);
    var variance3 = img.reduceNeighborhood(ee.Reducer.variance(), kernel3);
  
    // Use a sample of the 3x3 windows inside a 7x7 windows to determine gradients and directions
    var sample_weights = ee.List([[0,0,0,0,0,0,0], [0,1,0,1,0,1,0],[0,0,0,0,0,0,0], [0,1,0,1,0,1,0], [0,0,0,0,0,0,0], [0,1,0,1,0,1,0],[0,0,0,0,0,0,0]]);
  
    var sample_kernel = ee.Kernel.fixed(7,7, sample_weights, 3,3, false);
  
    // Calculate mean and variance for the sampled windows and store as 9 bands
    var sample_mean = mean3.neighborhoodToBands(sample_kernel); 
    var sample_var = variance3.neighborhoodToBands(sample_kernel);
  
    // Determine the 4 gradients for the sampled windows
    var gradients = sample_mean.select(1).subtract(sample_mean.select(7)).abs();
    gradients = gradients.addBands(sample_mean.select(6).subtract(sample_mean.select(2)).abs());
    gradients = gradients.addBands(sample_mean.select(3).subtract(sample_mean.select(5)).abs());
    gradients = gradients.addBands(sample_mean.select(0).subtract(sample_mean.select(8)).abs());
  
    // And find the maximum gradient amongst gradient bands
    var max_gradient = gradients.reduce(ee.Reducer.max());
  
    // Create a mask for band pixels that are the maximum gradient
    var gradmask = gradients.eq(max_gradient);
  
    // duplicate gradmask bands: each gradient represents 2 directions
    gradmask = gradmask.addBands(gradmask);
  
    // Determine the 8 directions
    var directions = sample_mean.select(1).subtract(sample_mean.select(4)).gt(sample_mean.select(4).subtract(sample_mean.select(7))).multiply(1);
    directions = directions.addBands(sample_mean.select(6).subtract(sample_mean.select(4)).gt(sample_mean.select(4).subtract(sample_mean.select(2))).multiply(2));
    directions = directions.addBands(sample_mean.select(3).subtract(sample_mean.select(4)).gt(sample_mean.select(4).subtract(sample_mean.select(5))).multiply(3));
    directions = directions.addBands(sample_mean.select(0).subtract(sample_mean.select(4)).gt(sample_mean.select(4).subtract(sample_mean.select(8))).multiply(4));
    // The next 4 are the not() of the previous 4
    directions = directions.addBands(directions.select(0).not().multiply(5));
    directions = directions.addBands(directions.select(1).not().multiply(6));
    directions = directions.addBands(directions.select(2).not().multiply(7));
    directions = directions.addBands(directions.select(3).not().multiply(8));
  
    // Mask all values that are not 1-8
    directions = directions.updateMask(gradmask);
  
    // "collapse" the stack into a singe band image (due to masking, each pixel has just one value (1-8) in it's directional band, and is otherwise masked)
    directions = directions.reduce(ee.Reducer.sum());  
  
    //var pal = ['ffffff','ff0000','ffff00', '00ff00', '00ffff', '0000ff', 'ff00ff', '000000'];
    //Map.addLayer(directions.reduce(ee.Reducer.sum()), {min:1, max:8, palette: pal}, 'Directions', false);
  
    var sample_stats = sample_var.divide(sample_mean.multiply(sample_mean));
  
    // Calculate localNoiseVariance
    var sigmaV = sample_stats.toArray().arraySort().arraySlice(0,0,5).arrayReduce(ee.Reducer.mean(), [0]);
  
    // Set up the 7*7 kernels for directional statistics
    var rect_weights = ee.List.repeat(ee.List.repeat(0,7),3).cat(ee.List.repeat(ee.List.repeat(1,7),4));
  
    var diag_weights = ee.List([[1,0,0,0,0,0,0], [1,1,0,0,0,0,0], [1,1,1,0,0,0,0], 
      [1,1,1,1,0,0,0], [1,1,1,1,1,0,0], [1,1,1,1,1,1,0], [1,1,1,1,1,1,1]]);
  
    var rect_kernel = ee.Kernel.fixed(7,7, rect_weights, 3, 3, false);
    var diag_kernel = ee.Kernel.fixed(7,7, diag_weights, 3, 3, false);
  
    // Create stacks for mean and variance using the original kernels. Mask with relevant direction.
    var dir_mean = img.reduceNeighborhood(ee.Reducer.mean(), rect_kernel).updateMask(directions.eq(1));
    var dir_var = img.reduceNeighborhood(ee.Reducer.variance(), rect_kernel).updateMask(directions.eq(1));
  
    dir_mean = dir_mean.addBands(img.reduceNeighborhood(ee.Reducer.mean(), diag_kernel).updateMask(directions.eq(2)));
    dir_var = dir_var.addBands(img.reduceNeighborhood(ee.Reducer.variance(), diag_kernel).updateMask(directions.eq(2)));
  
    // and add the bands for rotated kernels
    for (var i=1; i<4; i++) {
      dir_mean = dir_mean.addBands(img.reduceNeighborhood(ee.Reducer.mean(), rect_kernel.rotate(i)).updateMask(directions.eq(2*i+1)));
      dir_var = dir_var.addBands(img.reduceNeighborhood(ee.Reducer.variance(), rect_kernel.rotate(i)).updateMask(directions.eq(2*i+1)));
      dir_mean = dir_mean.addBands(img.reduceNeighborhood(ee.Reducer.mean(), diag_kernel.rotate(i)).updateMask(directions.eq(2*i+2)));
      dir_var = dir_var.addBands(img.reduceNeighborhood(ee.Reducer.variance(), diag_kernel.rotate(i)).updateMask(directions.eq(2*i+2)));
    }
  
    // "collapse" the stack into a single band image (due to masking, each pixel has just one value in it's directional band, and is otherwise masked)
    dir_mean = dir_mean.reduce(ee.Reducer.sum());
    dir_var = dir_var.reduce(ee.Reducer.sum());
  
    // A finally generate the filtered value
    var varX = dir_var.subtract(dir_mean.multiply(dir_mean).multiply(sigmaV)).divide(sigmaV.add(1.0));
  
    var b = varX.divide(dir_var);
  
    return dir_mean.add(b.multiply(img.subtract(dir_mean)))
      .arrayProject([0])
      // Get a multi-band image bands.
      .arrayFlatten([['sum']])
      .float();
  })).toBands().rename(bandNames);
  return powerToDb(ee.Image(result));
}