whittaker Smoothing algorithm for GEE

Smoothing of time series

Some great work that was presented in this paper:

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// helper function to convert qa bit image to flag
function extractBits(image, start, end, newName) {
    // Compute the bits we need to extract.
    var pattern = 0;
    for (var i = start; i <= end; i++) {
       pattern += Math.pow(2, i);
    }
    // Return a single band image of the extracted QA bits, giving the band
    // a new name.
    return image.select([0], [newName])
                  .bitwiseAnd(pattern)
                  .rightShift(start);
}

// function to get a Difference mattrix of specified order
// on the input matrix. takes matrix and order as parameters
function getDifferenceMatrix(inputMatrix, order){
    var rowCount = ee.Number(inputMatrix.length().get([0]));
    var left = inputMatrix.slice(0,0,rowCount.subtract(1));
    var right = inputMatrix.slice(0,1,rowCount);
    if (order > 1 ){
        return getDifferenceMatrix(left.subtract(right), order-1)}
    return left.subtract(right);
};

// unpacks an array image into images and bands
// takes an array image, list of image IDs and list of band names as arguments
function unpack(arrayImage, imageIds, bands){
    
    function iter(item, icoll){
        
        function innerIter(innerItem, innerList){
            return ee.List(innerList).add(ee.String(item).cat("_").cat(ee.String(innerItem)))}
        
        var temp = bands.iterate(innerIter, ee.List([]));
        return ee.ImageCollection(icoll)
            .merge(ee.ImageCollection(ee.Image(arrayImage).select(temp,bands).set("id",item)))}

    var imgcoll  = ee.ImageCollection(imageIds.iterate(iter, ee.ImageCollection([])));
    return imgcoll}



// Function to compute the inverse log ratio of a regression results to 
// transform back to percent units
function inverseLogRatio(image) {
  var bands = image.bandNames();
  var t = image.get("system:time_start");
  var ilrImage = ee.Image(100).divide(ee.Image(1).add(image.exp())).rename(bands);
  return ilrImage.set("system:time_start",t);
}

function whittakerSmoothing(imageCollection, isCompositional, lambda){
  // quick configs to set defaults
  if (isCompositional === undefined || isCompositional !==true) isCompositional = false;
  if (lambda === undefined ) lambda = 10;

  // procedure start  
  var ic = imageCollection.map(function(image){
     var t = image.get("system:time_start");
    return image.toFloat().set("system:time_start",t);
  });

  var dimension = ic.size();
  var identity_mat = ee.Array.identity(dimension);
  var difference_mat = getDifferenceMatrix(identity_mat,3);
  var difference_mat_transpose = difference_mat.transpose();
  var lamda_difference_mat = difference_mat_transpose.multiply(lambda);
  var res_mat = lamda_difference_mat.matrixMultiply(difference_mat);
  var hat_matrix = res_mat.add(identity_mat);

  
  // backing up original data
  var original = ic;

  // get original image properties
  var properties = ee.List(ic.iterate(function(image, list){
    return ee.List(list).add(image.toDictionary());
  },[]));
  
  var time = ee.List(ic.iterate(function(image, list){
    return ee.List(list).add(image.get("system:time_start"));
  },[]));
  
  // if data is compositional
  // calculate the logratio of an image between 0 and 100. First
  // clamps between delta and 100-delta, where delta is a small positive value.
  if (isCompositional){
    ic = ic.map(function(image){
      var t = image.get("system:time_start");
      var delta = 0.001;
      var bands = image.bandNames();
      image = image.clamp(delta,100-delta);
      image = (ee.Image.constant(100).subtract(image)).divide(image).log().rename(bands);
      return image.set("system:time_start",t);
    });
  }

  var arrayImage = original.toArray();
  var coeffimage = ee.Image(hat_matrix);
  var smoothImage = coeffimage.matrixSolve(arrayImage);
  
  var idlist = ee.List(ic.iterate(function(image, list){
    return ee.List(list).add(image.id());
  },[]));
  var bandlist = ee.Image(ic.first()).bandNames();
  
  var flatImage = smoothImage.arrayFlatten([idlist,bandlist]);
  var smoothCollection = ee.ImageCollection(unpack(flatImage, idlist, bandlist));
  
  if (isCompositional){
    smoothCollection = smoothCollection.map(inverseLogRatio);
  }
  // get new band names by adding suffix fitted
  var newBandNames = bandlist.map(function(band){return ee.String(band).cat("_fitted")});
  // rename the bands in smoothened images
  smoothCollection = smoothCollection.map(function(image){return ee.Image(image).rename(newBandNames)});
  
  // a really dumb way to loose the google earth engine generated ID so that the two
  // images can be combined for the chart
  var dumbimg = arrayImage.arrayFlatten([idlist,bandlist]);
  var dumbcoll = ee.ImageCollection(unpack(dumbimg,idlist, bandlist));
  var outCollection = dumbcoll.combine(smoothCollection);
  
  var outCollectionProp = outCollection.iterate(function(image,list){
      var t = image.get("system:time_start")
    return ee.List(list).add(image.set(properties.get(ee.List(list).size())));
  },[]);

  var outCollectionProp = outCollection.iterate(function(image,list){
    return ee.List(list).add(image.set("system:time_start",time.get(ee.List(list).size())));
  },[]);


  var residue_sq = smoothImage.subtract(arrayImage).pow(ee.Image(2)).divide(dimension);
  var rmse_array = residue_sq.arrayReduce(ee.Reducer.sum(),[0]).pow(ee.Image(1/2));
  
  var rmseImage = rmse_array.arrayFlatten([["rmse"],bandlist]);
  
  return [ee.ImageCollection.fromImages(outCollectionProp), rmseImage];
}



var ndvi =ee.ImageCollection("NOAA/VIIRS/001/VNP13A1").select('NDVI').filterDate("2019-01-01","2019-12-31");
// getting rid of masked pixels
ndvi = ndvi.map(function(img){return img.unmask(ndvi.mean())});

var ndvi =  whittakerSmoothing(ndvi)[0];


// add chart
print(ui.Chart.image.series(
  ndvi.select(['NDVI', 'NDVI_fitted']), geometry, ee.Reducer.mean(), 500)
    .setSeriesNames(['NDVI', 'NDVI_fitted'])
    .setOptions({
      title: 'smoothed',
      lineWidth: 1,
      pointSize: 3,
}));



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