^{1}

^{2}

^{*}

^{2}

^{th}Avenue South, Lethbridge, Alberta, Canada, T1J 0P3, E-Mail:

Reproduction is permitted for noncommercial purposes.

Spectral mixing is a problem inherent to remote sensing data and results in few image pixel spectra representing ″pure″ targets. Linear spectral mixture analysis is designed to address this problem and it assumes that the pixel-to-pixel variability in a scene results from varying proportions of spectral endmembers. In this paper we present a different endmember-search algorithm called the Successive Projection Algorithm (SPA). SPA builds on convex geometry and orthogonal projection common to other endmember search algorithms by including a constraint on the spatial adjacency of endmember candidate pixels. Consequently it can reduce the susceptibility to outlier pixels and generates realistic endmembers.This is demonstrated using two case studies (AVIRIS Cuprite cube and Probe-1 imagery for Baffin Island) where image endmembers can be validated with ground truth data. The SPA algorithm extracts endmembers from hyperspectral data without having to reduce the data dimensionality. It uses the spectral angle (alike IEA) and the spatial adjacency of pixels in the image to constrain the selection of candidate pixels representing an endmember. We designed SPA based on the observation that many targets have spatial continuity (e.g. bedrock lithologies) in imagery and thus a spatial constraint would be beneficial in the endmember search. An additional product of the SPA is data describing the change of the simplex volume ratio between successive iterations during the endmember extraction. It illustrates the influence of a new endmember on the data structure, and provides information on the convergence of the algorithm. It can provide a general guideline to constrain the total number of endmembers in a search.

Linear spectral mixture analysis (SMA) is based on the simple assumption that remotely sensed spectral measurements are mixed signatures that vary across the scene as the relative proportion of endmembers change. It is commonly used for the analysis of hyperspectral data [

Spectral endmembers can be derived from the imagery (image endmembers) or measurements in the laboratory/field (library endmembers). Library endmembers may not always be available, and if available, they are not necessarily acquired under the same conditions as airborne or satellite image data and may not be good representations of the image components. Thus there are advantages in being able to extract endmembers directly from imagery. The selection of image endmembers is typically achieved through the implicit (PPI, pixel purity index [

In this paper we present a different endmember-search algorithm called the Successive Projection Algorithm (SPA). SPA builds on the convex geometry endmember search algorithms described above by including a constraint on the spatial adjacency of endmember candidate pixels, whereby this approach can reduce the susceptibility to outlier pixels and generates realistic endmembers. This is demonstrated using two case studies where image endmembers can be validated with ground truth data. The spatial constraint was introduced based on success we have had with the spatial-spectral endmember extraction algorithm (SSEE) that makes use of the spectral and spatial characteristics of image pixels during the search for image endmembers [

In Section 2, we present the concept of convex geometry and its relevance for endmember selection, followed by a summary of current convex-based endmember-search algorithms. In Section 3, we describe SPA and its functionality. Section 4 describes the characteristics of the two test datasets (AVIRIS Cuprite cube and Probe-1 imagery for Baffin Island) that are used to evaluate the SPA algorithm. The experimental results are presented in Section 5, followed by a discussion.

Linear spectral mixture analysis (LSMA) assumes that the pixel-to-pixel variability in a scene results from varying abundances of spectral endmembers. It follows that the spectral response for each pixel is a linear combination of endmember spectra, weighted by their fractional abundances. Assuming that the number of endmembers and their spectral signatures are known, the fractional abundances of endmembers in a given pixel are typically determined from a least squares fit [

Let _{(i,j}_{)} denote the spectrum for the pixel in the image coordinates (

where _{k}_{i,j}_{(i,j)k}

Spectra can be represented as points in an _{m}_{1},_{2},⋯,_{m}_{m}_{2} −_{1}_{3}_{1}_{n}_{1}_{m}_{1},_{2},⋯,_{m}) are determined, their abundance can be estimated through the least squares method, which is equivalent to a projection on the simplex [

Using this framework, if all data points (pixels) are examined in

The set of endmembers determined from convex geometry has the following properties that are relevant to the SPA algorithm proposed in this paper:

The endmembers represent the pixels that contain the largest data “volume” [

A vector (pixel) with maximum Euclidean norm (magnitude) must be located at one of the vertices of the simplex [

For a given point in the simplex, a point with maximum distance must be a vertex of the simplex [

The affine transformation (e.g. orthogonal projection) of a simplex is also a simplex, and endmembers are still located in the vertices of the new simplex after this transformation [

Search algorithms based on convex geometry rely on the four properties listed above, but differ in their approach to locate the vertices of the simplex. Such methods include PPI, N-FINDER, IEA, VCA, Max_D, ORASIS, SMACC, ICE, MVC-NMF and SGA. N-FINDER finds the set of pixels that define the simplex with the maximum volume inscribed within the dataset. IEA uses a series of constrained unmixing and chooses as the endmembers those pixels that minimize the residual error in the unmixing images. VCA and Max_D exploit the orthogonal projection approach to iteratively find the vertices of the simplex. SMACC is another algorithm for endmember extraction, which uses a convex cone model (also known as Residual Minimization) and constrained oblique projection to derive endmembers [

In this study we propose a more robust approach that uses the spectral angle (alike IEA) and the spatial adjacency of pixels in the image to constrain the selection of candidate pixels representing an endmember. Two assumptions are made: 1) pixels that are spatially adjacent are more likely to have similar spectral properties and thus represent one endmember, and, 2) the probability that two adjacent pixels are both spurious is low. These assumptions are certainly reasonable if the target application is geological mapping because mappable units (e.g. bedrock lithologies) typically have spectral properties with spatial continuity in hyperspectral imagery.

The next section (e.g. 3.2.) describes how a vertex (e.g. an extreme pixel) is identified based on its spectral uniqueness in the simplex (the distinctness is measured in terms of the vector Euclidean norm or the distance of the pixel to the subspace defined by the previously selected endmembers). A meaningful endmember for this vertex is then the average of multiple candidate pixels that are spectrally distinct (e.g. they are located at or near one of the corners of the simplex) and are spatially adjacent. To find these candidate pixels we construct a pixel set, _{possible}

Then a subset,
_{possible}_{i},y_{i}_{xi,yi;},p⃗_{(xj,yj}_{)}) is the spectral angle between two spectra and is calculated as

The average vector of _{candidate}_{possible}_{possible}

SPA starts by identifying the two most distinct endmembers, _{1}_{2}

Values for the following three parameters must be set: the number of endmembers(

The vector norms of all pixels in the image are calculated to locate the pixel that has the largest norm. According to _{1}

The distances between all pixels and _{1}^{nd} endmember, _{2}, can then be estimated according to section 3.1.

An endmember matrix _{1}_{2}_{proj}_{(i,j)}__{proj and p⃗(i,j)} are the projected and original pixel vector at image location (^{+} is the pseudo inverse of

In the projected subspace (_{proj}_{proj}_{3}_{1}_{2}

The endmember matrix, _{1}_{2}_{3}

We calculate the change of the simplex volume with each subspace projection because it provides an insight on the convergence of the algorithm. The volume of the simplex can be calculated only when the simplex has more than 3 vertices. According to _{l}_{–1} and _{1},e⃗_{2}_{l}_{-1}, _{l}_{1},e⃗_{2}_{l}_{-1}}, the ratio of the volumes of _{l}_{–1} can be calculated as

As endmembers are selected (e.g. the value of _{l}_{l}

The SPA algorithm was evaluated using two hyperspectral datasets. The first one was collected over the Cuprite mining district, Nevada, in July 1995 wit the Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) as part of an AVIRIS Group Shoot [

This hyperspectral cube has 400 * 350 pixels, and 50 bands of short-wave infrared data (1.9 μm∼ 2.4 μm). The spatial and spectral resolutions are respectively 20m and 10 nm. The data were corrected to reflectance using the ATREM (ATmospheric REMoval) method [

The AVIRIS dataset was used to: 1) determine whether SPA can extract the 7 mineral endmembers documented by previous authors; 2) determine whether SPA converges; and 3) assess the merits of the spatial constraint in SPA. The SPA was applied to the Cuprite Cube with the following parameters: the total number of endmembers for this scene was set to 19 and the threshold values for

The test data from the Baffin island study area (

The airborne hyperspectral data (∼3.5 × 7 km) were acquired with the Probe I sensor, which comprises 128 channels from 446 - 2543 nm with an average band Full Width Half Maximum of ∼15 nm and a Ground Instantaneous Field of View of ∼7 m. (^{®} field spectrometer that has 2151 bands covering the 350 – 2500 nm spectral range. A total of 217 spectral measurements were acquired for 56 sites, some of which lie outside, but proximal to the study area, and are representative of the geology shown in

We chose to evaluate the performance of SPA with this test data owing to 1) excellent bedrock exposure and limited continuous vegetation; 2) the spectral diversity of the rock units and the relevance of some units to mining exploration (gaussan and peridotite); 3) the variable spatial distribution of the rock units spanning large continuous exposures to small sporadic outcroppings; and, 4) the availability of field spectra and spectra of rock samples for the validation of endmembers extracted from imagery. The extraction of geological endmembers from this imagery is more challenging than for the imagery of Cuprite. This can be attributed to the presence of snow, tundra vegetation and rock encrusting lichen, which lower the relative spectral contrast between geological endmembers.

We also compare the endmembers derived from SPA with that derived from IEA, given that IEA has been reported as the most robust convex-based algorithm [

We first examine the SPA endmembers in the context of the seven mineral PPI endmembers (zeolite, alunite, buddingtonite, calcite, kaolinite, silica and muscovite/illite) previously reported by Kruse and Huntington (1996)[

In

To illustrate the convergence of SPA we show the changes in the simplex volume between successive iterations (e.g. volume ratio, _{18}= 1.15) beyond which the volume ratio is less than 1.0. For this particular data set, the geological endmembers of interest are extracted before endmember 18 (

Out of thirty endmembers, twenty-one represent vegetation, water, snow and shadow (

Eight geological endmembers (SPA_3, SPA_6, SPA_12, SPA_16, SPA_22, SPA_23, SPA_28 and SPA_30) closely match field spectra (_{3}) feature near 2.30-2.35 μm, but they differ in overall spectral amplitude. For SPA_ 23 the carbonate feature is centered near 2.32 μm but for SPA_6 it lies near 2.34 μm. (

We failed to find field spectra that closely match the endmember SPA_11. The closest match is peridotite (

We also performed a comparison between endmembers of geological interest derived from IEA and SPA (

The change in the volume ratio (_{l}_{24}= 1.24) after which the volume ratio remains less than 1.0. However, we found that endmembers of geological interest were extracted after endmember 24. For example, peridotite, an important rock type for the mining exploration of nickel, is extracted as the 30^{th} endmember. The majority of the snow, water, shade and vegetation endmembers were derived before this point, as shown in Table-2.

The SPA algorithm extracts endmembers from hyperspectral data without having to reduce the data dimensionality. It uses the spectral angle (alike IEA) and the spatial adjacency of pixels in the image to constrain the selection of candidate pixels representing an endmember. We designed SPA based on the observation that many targets have spatial continuity (e.g. bedrock lithologies) in hyperspectral imagery and thus a spatial constraint would be beneficial in the endmember search. We assumed that pixels that are spatially adjacent are more likely to have similar spectral properties and thus represent one endmember, and, that the probability that two adjacent pixels are both spurious is low. Experiments on two datasets demonstrate that SPA can effectively extract endmembers while requiring minimal user interaction.

It should be pointed out that the procedure to identify the simplex vertices in SPA is similar to that for advanced convex-based endmember selection methods such as MAX-D, VCA and SGA. However, of the convex-based endmember-search algorithms discussed in this paper, only SPA makes use of both the spectral angle and spatial adjacency to determine which pixels should form one endmember. By using the average of multiple pixels as one endmember, the SPA-derived endmember spectra appear less noisy (e.g. smoother), which is helpful for the improvement of unmixing results [

An additional product of the SPA is data describing the change of the simplex volume ratio between successive iterations during the endmember extraction. It illustrates the influence of a new endmember on the data structure, and provides information on the convergence of the algorithm. Though the rate of convergence speed can vary with the complexity of the scene, the patterns are similar showing the largest changes in volume ratio at the beginning of the endmember extraction process, followed by progressively smaller changes and convergence towards a plateau. If the endmember search terminates before the convergence point (the volume ratio is close to 1.0), significant endmembers will be missed. However as seen in the Baffin island example, endmembers for targets of interest may also be found beyond the convergence point. Thus additional research is required to properly constrain the number of endmember for a given search and application.

Comparison of endmembers obtained from SPA and IEA showed that both algorithms generate similar results (

The computation load of endmember-search algorithms is an important issue for the automatic extraction of endmembers, given the increasing volumes of hyperspectral data available. We did not study the computational efficiency of SPA, but because SPA is fundamentally similar to VCA and Max_D, that reported to be of high computational efficiency [

There are a number of potential improvements to SPA that require further research namely: 1) a means for the automatic determination of the spectral angle threshold (

This research was funded by the GEOIDE (GEOmatics for Informed DEcisions) Network of Centres of Excellence of Canada and by the Natural Sciences and Engineering Research Council of Canada. We thank the Geological Survey of Canada for providing access to the Probe-1 data, the Canada Center for Remote Sensing for access to ISDAS and atmospheric correction of the Probe-1 data.

Regional geology of south-western Baffin Island and zoom of local geology of the study area (1:100 000) (modified from St-Onge et al., 1999).

Probe 1 hyperspectral data of the study area. Circles represent ground locations where field spectra and samples were collected.

Comparison between SPA endmember and PPI endmember(“true”endmember). The solid lines denotes PPI endmembers.

Comparison of SPA and SMACC endmembers. a) endmember representatives of the same target (zeolite), b) SMACC endmember capturing noise

Convergence of SPA for the Cuprite data. The arrow marks the last iteration where the simplex volume ratio for successive iterations exceeds 1.0.

Endmember spectra for snow, vegetation and lichen. The dashed lines are the SPA-endmembers, and the solid lines are the corresponding closely matched field spectra. The strong water absorption features near 1.4 and 1.9 um were discarded because of low signal.

Comparison of SPA geological endmembers and field spectra. The strong water absorption features near 1.4 and 1.9 um were discarded because of low signal.

Comparison between the SPA_11 endmember and field spectrum of peridotite. The circle marks the region where absorption feature is present on the field spectrum of peridotite but absent from the SPA_11.

The endmember found by IEA but not by SPA. The solid line is a field spectrum for metasediment, the dashed line is the IEA endmember.

Convergence of the SPA for the Baffin data The arrow marks the last iteration where the simplex volume ratio for successive iterations exceeds 1.0.

List of SPA endmembers derived from the Cuprite data and corresponding PPI endmember when applicable

SPA_1, SPA-17 | Silicate (bright) |

SPA_16 | Silica (dark) |

SPA_4 | Alunite 1 (2.16 μm) |

SPA_12, SPA_15 | Alunite 2(2.18 μm) |

SPA_8 | Buddingtonite |

SPA_3, SPA_14 | Kaolinite |

SPA_6 | Calcite |

SPA_5 | Zeolite |

SPA_7, SPA_10, SPA_11, SPA_13 | Muscovite/illite |

Na | |

Na |

Thisendmember is a rock/mineral.

These endmembers are for shade/shadow.

Summary of SPA endmembers derived for the Baffin Island site. The geological endmember are highlighted

1, 4, 8, 9, 13, 14, 15, 17, 19, 20, 24, 26, 27 | Snow |

2 | Water |

3 | Felsic (Granite/varnish) |

5 | Vegetation (wet) |

6 | Marble |

7 | Shade |

10, 18, 25, 29 | Lichen |

11 | Peridotite |

12 | Fe-metasediment |

16 | Clay-metasediment |

21 | Dry vegetation |

22 | Quartzite |

23 | Marble (low albedo) |

28 | Quartzite (low albedo) |

30 | Peridotite |

Comparison between IEA and SPA endmembers

Felsic (Granite/varnish) | IEA_3 | SPA_3 |

Marble | IEA_7, IEA_22 (Similar but differ in amplitude) | SPA_6, SPA_23 (Similar but differs in amplitude) |

Peridotite | IEA_14 | |

Fe-metasediment | IEA_19 | SPA_12 |

Clay-metasediment | IEA_13 | SPA_16 |

Metasediment | IEA_28 | Missing |

Quartzite | IEA_30 | SPA_22, SPA_28 (Similar but differs in amplitude) |

This endmember is labeled as peridotite based on