

Classification
example
Download DFT Plotter
DFT calculation 
The basic idea of this classification is to cluster vegetation
covers according to the shapes of their annual growing cycle or
rather, to their remote sensing surrogate, the NDVIcycle. The shapes of the cycles
can assist in identifying vegetation types and thus help conclude to
distinct vegetation
characteristics including hydrometeorological behavior, annuals
or perennials, seasonal photosynthetic activity, carbon storage/release
or to crop management. The described technique offers a series of advantages over conventional classifications like Kmeans or decision tree that are especially suited for the classification of NDVI timeseries:
Remark
Step 1: Fourier Filtering Since the clustering of NDVIcycles is based on their shape it is important to smooth/reduce noise features that are typically part of remotely sensed data sets as much as possible but without compromising diagnostic phenological characteristics. This is achieved by decomposing the complex NDVIcycle (green) into its individual frequencies using a Discrete Fourier Transform (DFT). After removal of the higher frequencies, that are assumed noise, the inverse DFT provides a smoothed data set. In 1year, 10days interval layerstacks (36 layers = 18 harmonics/frequencies) we typically use the first 5 frequencies (red) and discard frequencies 6 to 18.
To reduce data dimensionality, in the following, a simplified Fourier filtered vegetation cycle is used (frequencies 1 and 2) for explanation of the classification algorithm.
Remark (noise reduction):


Step 2: Selection of Reference Cycles The selection of reference sites (red dots) for our current classification of Central Asia is based on a color composite composed of Magnitudes 1 (red), 2 (green), and 3 (blue), providing an efficient reduction of multilayer cyclic NDVIdata (subset shown below). In general, variations in hue are indicative for different vegetation types, variations in saturation for changes in vegetation coverage. At this point only limited field data are available that confine to the Ferghana Valley and to the SyrDarya Province area.


Classified Area 
Step 3: Classification Our definition of shape is oriented at a vegetation type's phenology, as expressed in the remote sensing surrogate the NDVIcycle, and is invariant to distinct cycle modifications that may be imposed by influences including climate, soil, topography or human. In particular, the algorithm allows variations within one vegetation type with respect to its coverage/vigor, to its phenological timing, and to its background reflectance, thus creating a more consistent classification result.
Principles of classification The shape of a complex (NDVI) cycle is completely defined by the Fourier components magnitude and phase. This is shown at a simplified NDVIcycle created from 36 temporal intervals. The decomposition of the simplified complex NDVI cycle into its individual frequencies using a Discrete Fourier Transform creates two output harmonics of frequency 1 and frequency 2 (Figure 3). In this simplified example the higher harmonics (frequencies) have an amplitude of zero. Frequency 1 + Frequency 2  Cycle Mean = Complex Cycle Cycle Mean =
straight grey line (zero line for amplitude measurement) All shapes identical to the solid black cycle (complex cycle) must have


Figure 3  Figure 4  
Note: The amplitude ratio (Amplitude 1 / Amplitude 2 = 1.42) in both examples is the same


Figure 5  Figure 6  
Note: 1. Phase difference (Phase 1  Phase 2 = 7.9) is in both examples the same.
Invariance to phenological shifts
Invariance to coverage variation


Figure 7 

Invariance to background
Thresholds for Phase differences and Amplitude ratios
The tolerable distance is defined in percent of the Sum of Amplitudes. Point 2 in Figure 7 fulfills the Amplitude ratio requirement but would be rejected because it falls outside the coverage threshold (black circle). The amount of deviation from the ideal phase difference depends on a frequency's contribution to the shape of the complex cycle, which is a function of its amplitude. In the extreme case of a harmonic with an amplitude of zero, there are no restrictions on tolerable phase difference offset θ_{k}. A frequency with an amplitude of zero has no influence on the shape of the complex cycle. This is expressed in the following equation: equation 1
A_{k} is the kth amplitude out of n frequencies, the 'kth Harmonic' or kth frequency is the number of the harmonic. Since phase differences are measured against Phase 1, calculation of θ1 is irrelevant. Exponent X can assume any value and is used to manipulate tolerable phase differences. The calculation of each tolerable phase offset from phase 1 is done individually and independently from other phase offsets. Therefore, shape fidelity might be compromised where all phases are displaced to their individual maximum (this would be the case where simple box filters are used, Figure 8). The solution is a second order filter that allows only a single phase to be displaced to the maximum value while keeping the rest to a minimum (Figure 8). As indicated in Figure 8, the curvature of the filter can be manipulated to allow for more or less phase tolerances resulting in lower or higher shape fidelity. The 2nd order filter takes the form: equation 2


Figure 8 

with a_{max} calculated as equation 3
where c = maximum tolerance for phase 2 and x = maximum tolerance for phase 3. The corresponding bvalues are calculated from
equation 4
The graph cuts the yaxis/xaxis at the maximum tolerable phase difference for phase 2 (yaxis) and phase 3 (xaxis), as calculated from equation 1. a_{max} and a_{max} is where the maximum/minimum of the function falls onto the yaxis and xaxis respectively. At higher/lower avalues filtering becomes increasingly biased. At a = 0 the equation becomes a linear.


Literature:

