Author's personal copy
Moreover, most of these methods are computationally expensive,
thus limiting their utility in plant leaf image data sets.
1.2. Feasibility of plant leaf recognition based on manifold learning
Over the last decade, a large number of nonlinear manifold
learning algorithms have been proposed under the assumption
that the input data set lies on or near some low-dimensional man-
ifold embedded in a high-dimensional unorganized Euclidean
space. Among the most well known are isometric feature mapping
(ISOMAP) [14], locally linear embedding (LLE) [15–16], Laplacian
eigenmaps (LE) [17], Hessian-based locally linear embedding
(HLLE) [18], maximum variance unfolding (MVU) [19], local tan-
gent space alignment (LTSA) [20], and local spline embedding
(LSE) [21]. Each manifold learning algorithm attempts to preserve
a different geometrical property of the underlying manifold. Local
approaches such as LLE, HLLE, LE, LTSA, and LSE aim to preserve the
proximity relationship among the data, while global approaches
like ISOMAP aim to preserve the metrics at all scales. These nonlin-
ear methods yield impressive results on some benchmark artificial
and real world data sets due to their nonlinear nature, geometric
intuition, and computational feasibility.
An important advantage of manifold learning [14–21] com-
pared with conventional approaches concerns how the data are
treated mathematically. Conventional approaches typically pro-
duce a smaller data space from linear combinations of the original
data. One common example is PCA which seeks a low-dimensional
linear subspace spanned by the eigenvectors corresponding to the
largest eigenvalues of the covariance matrix of all samples. How-
ever, for plant leaf images, the assumption of global linearity is a
severe constraint since they are sensitive to period, location, and
illumination conditions and there is no reason to believe that the
leaf image data are linearly separable from each other. Manifold
learning approaches recognize this fact and allow the data to be
nonlinearly related, which results in the fact that manifold learning
approaches can much more accurately capture the proper informa-
tion relationships among the data thus allowing for accurate recog-
nition. Fig. 1 shows a simple example that 150 leaf images of two
kinds of plants are mapped into two-dimensional subspace by local
spline embedding (LSE). The size of each image is 32 32 pixels,
with 256 gray-levels per pixel. Thus, each leaf image is represented
by a point in the 1024-dimensional ambient space. The left and
right leaf images respectively correspond to the points with green
and cyan circles in the two-dimensional embedding subspace. As
can be seen, the leaf images are divided into two parts. The circles
and the asterisks represent leaf images of different classes. It can
be clearly seen that the sample points of each class exhibit a
sub-manifold distribution. The results demonstrate that LSE can
successfully find the discriminative directions, but the directions
are not optimal for leaf recognition task. This is because in trying
to preserve local structure in the embedding, the LSE implicitly
emphasizes the natural clusters in the data. More importantly,
LSE is capable of capturing the intrinsic leaf manifold structure to
some extent.
1.3. Contribution of the paper
In this study, we take an alternative view of manifold learning
to develop two orthogonal locally discriminant spline embedding
methods (OLDSE-I and OLDSE-II) for plant leaf recognition. The
goal of OLDSE-I or OLDSE-II is to map the plant leaf images into a
plant leaf subspace for analysis. Different from principal compo-
nent analysis (PCA) [22] and linear discriminant analysis (LDA)
[23] which can only deal with flat Euclidean structures of plant leaf
space, The OLDSE-I/OLDSE-II finds an embedding that not only
inherits the advantages of local spline embedding (LSE) [21] which
uses local neighborhoods as a representation of the local geometry
so as to preserve the local structure, but makes full use of class
information to improve discriminant power by introducing trans-
lation and rescaling models. In this way, a plant leaf subspace that
best detects the essential plant leaf manifold structure can be ob-
tained. It is worthwhile to highlight several aspects of the proposed
approaches here:
(1) An efficient subspace learning algorithm for plant recogni-
tion should be able to discover the nonlinear manifold struc-
ture of the leaf image space. Our proposed OLDSE-I and
OLDSE-II algorithms explicitly considers the intrinsic leaf
manifold structure which is modeled by an adjacency graph.
(2) OLDSE-I/OLDSE-II takes local structure and discriminant
information into consideration simultaneously and attempts
to manage the trade-off between LSE, which is based mainly
on preserving local geometry and maximum margin crite-
rion (MMC), which emphasizes discriminant power.
(3) OLDSE-I/OLDSE-II shares some similar properties to LSE,
such as a local neighborhood preserving character. However,
their objective functions are ultimately different. OLDSE-I/
OLDSE-II computes an explicit linear mapping from the
input space to the reduced space, while in LSE, the mapping
is implicit and it is not clear how new data samples can be
embedded.
(4) Although OLDSE-I and OLDSE-II seek to find a set of orthog-
onal basis functions and further improve their recognition
accuracy, they use different orthogonalization processes in
which their fundamental difference lies.
1.4. Paper organization
The rest of this paper is organized as follows: Section 2 briefly
describes the LSE algorithms. In Section 3, the OLDSE-I and
OLDSE-II algorithms are developed. A variety of the experimental
results are presented in Section 4. Finally, we provide some con-
cluding remarks and future work in Section 5.
2. A brief review of local spline embedding
Given a data set of n data points X ¼½x
1
; x
2
; ...; x
n
2R
Dn
,
the goal of dimensionality reduction is to project the high-
dimensional data into a low-dimensional feature space. Let us
denote the corresponding set of n points in the reduced space as
-5 -4 -3 -2 -1 0 1 2 3 4
-1
0
1
2
3
4
5
6
7
Fig. 1. Two-dimensional embedding of plant leaf images by LSE.
Y.-K. Lei et al. / Computer Vision and Image Understanding 119 (2014) 116–126
117
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