Dimensionality reduction is the mapping of data from a high dimensional space to a lower dimension space such that the result obtained by analyzing the reduced dataset is a good approximation to the result obtained by analyzing the original data set. There are several dimensionality reduction approaches which include Random Projections, Principal Component Analysis, the Variance approach, LSA-Transform, the Combined and Direct approaches, and the New Random Approach. In this paper, we propose three new techniques, each of which will be a modified version of the last three techniques mentioned above (the Combined and Direct approaches, and the New Random Approach). We shall implement each of the ten reduction techniques mentioned, after which we shall use these techniques to compress various pictures. Finally, we shall compare the ten reduction techniques implemented in this paper with each other by the extent to which they preserve images.
Index Terms— dimensionality reduction, image compression, principal component analysis
Given a collection of n data points (vectors) in high
dimensional space, it is often helpful to be able to
project it into a lower dimensional space without
suffering great distortion
There are many known methods of dimensionality reduction. These include Random Projection (RP), Singular Value Decomposition (SVD), Principal Component Analysis (PCA), Kernel Principal Component Analysis (KPCA), Discrete Cosine Transform (DCT) and Latent Semantic Analysis (LSA) (NR 2009). For each of these methods, each attribute in the reduced set is a linear combination of the attributes in the original data set.
Other dimensionality reduction methods, however,
reduce a dataset to a subset of the original attribute set.
These include the Combined Approach (CA), the Direct
Approach (DA), the Variance Approach (Var),
LSA-Transform, the New Top-Down Approach (NTDn),
the New Bottom-Up Approach (NBUp), the Weighted
Attribute Frequency Approach (WAF) and the Best
Clustering Performance Approach (BCP)
Dimensionality reduction has several advantages, the most important of which is the fact that with dimensionality reduction, we could drastically speed up the execution of an algorithm whose runtime depends exponentially on the dimensions of the working space. At the same time, the solution found by working in the low dimensional space is a good approximation to the solution in the original high dimensional space.
One application of dimensionality reduction is in the compression of image data. In this domain, digital images are stored as 2D matrices which represent the brightness of each pixel. Usually, the matrix representing an image can be quite large, and for this reason it could be very time consuming querying this matrix to find out any information about the features of the image. In this paper, dimensionality reduction techniques are used to reduce the matrix representation of an image. This makes it possible to query the reduced matrix to get any information about the original image. Besides, we can use these techniques to compress all the pictures we have in a given folder, or website, thus conserving memory.
The rest of the paper is organized as follows. In Section 2, we shall examine the different dimensionality reduction techniques that shall be used to reduce the images. In Section 3 we shall look at the effects of reducing images using each of these techniques, and in Section 4 we shall compare the ten reduction techniques implemented in this project with each other by the extent to which they preserve images, and by their speeds of execution. Then we shall conclude this paper in Section 5.
In this section, we shall examine the different reduction techniques that we will be using to reduce the images. They include the following:
In Random Projection, the original d-dimensional
data is projected to a k-dimensional (k << d) subspace
through the origin, using a random d x k matrix R whose
columns have unit lengths
nxk is the projection of the data onto a lower k-dimensional subspace.
The key idea of random mapping arises from the Johnson Lindenstrauss lemma (Johnson W. B.) which states that if points in a vector space are projected onto a randomly selected subspace of suitably high dimension, then the distances between the points are approximately preserved.
Given n data points in P as an n x p matrix X, we want to find the best q-dimensional approximation for the data (q << p). The PCA approach achieves this by first computing the Singular Value Decomposition of X. In other words, it finds matrices U, D and V such that X = UDVT where: U is an n x n orthogonal matrix (i.e. UTU = In) whose columns are the left singular vectors of X; V is a p x p orthogonal matrix (i.e. VTV = Ip) whose columns are the right singular vectors of X; D is an n x p diagonal matrix with diagonal elements d1 ≥ d2 ≥ d3 … ≥ dp ≥ 0 which are the singular values of X. Note that the bottom rows of D are zero rows. Define Uq to be the matrix whose columns are unit vectors corresponding to the q largest left singular values of X. Uq is a n x q matrix.
The transformed matrix is given by
XSVD = XTUq
With the Variance approach, to reduce a dataset D to a data set DR, we start with an empty set, I, and then add dimensions of D to this set in decreasing order of their variances. That means that a set I of r dimensions will contain the dimensions of top r variances. Intuitively, it easy to justify why dimensions of low variance are left out as they would fail to discriminate between the data. (Indeed, in an extreme case where all the values along a dimension are equal, the variance is 0, and hence this dimension cannot distinguish between data points). Thus, let
Ir = {i1, . . . , ir} ⊂ {1, . . . , n}, the collection of dimensions corresponding to the top r variances. That is i1 denotes the dimension of largest Thus the original sub-matrix of sixteen cells becomes reduced to a smaller matrix of four cells, which also represents one dark point on an image. After the execution of LSA-Transform, therefore, M and M1 become two matrices representing I, one a quarter the size of the other.
Like the two previous approaches, the Combined Approach is one approach which reduces a dataset D to a subset of the original attribute set.
To reduce a dataset Dnxp to a dataset containing k columns, the Combined Approach selects the combination of k attributes which best preserve the interpoint distances, and reduces the dataset to a dataset containing only those k attributes. To do so, it first determines the extent to which each attribute preserves variance, i2 the dimension of next larger variance, etc. The reduced data base is obtained by extracting the data corresponding to the selected dimensions. That is, project D on Ir to obtain:
DR = D(:, Ir), where DR has the same number of rows as D and r columns: the ith column of DR is the column of the original database with the ith largest variance.
LSA-Transform is probably the best technique for reducing image data. It makes use of the redundancy of the data in matrices that represent images, in practice. Specifically, if I is an image, and M is the matrix (of pixel brightness values) representing I, LSA-Transform simply selects only the even columns and rows of M to give M1. The simple explanation for this is as follows: one point on an image is usually represented by a whole rectangle of values in the corresponding matrix. For instance, a dark point maybe represented by the values: Each of these values, as we can see, is less than 95. Selecting only the even rows leaves us with: Similarly, selecting only the even columns leaves us with: the interpoint distances. In other words, for each attribute, x, in D, it computes gxm and gxM given by: compute the average distance preservation for this combination using the formulas above. gxm = min{ || f (u) f (v) ||2 }
|| u v ||2 gxM = max{ || f (u) f (v) ||2 }
|| u v ||2 where u and v are any two rows of D, and f(u) and f(v) are the corresponding rows in the dataset reduced to the single attribute x. The average distance preservation for the attribute x is then computed as:
gxmid = (gxm + gxM)/2 To reduce the dataset D from p columns to k columns, this approach then finds the combination of k attributes whose average value of gxmid is maximum.
As with the Combined Approach, to reduce a dataset Dnxp to a dataset containing k columns, the Direct Approach selects the combination of k attributes which best preserve the interpoint distances, and reduces the original dataset to a dataset containing only those k attributes. To do so, it first generates all possible combinations of k attributes from the original p attributes. Then, for each combination, C, it computes gcm and gcM given by: gcm = min{ || f (u) f (v) ||2 }
|| u v ||2 gcM = max{ || f (u) f (v) ||2 }
|| u v ||2 where u and v are any two rows of D, and f(u) and f(v) are the corresponding rows in the dataset reduced to the attributes in C. The average distance preservation for this combination of attributes is then computed as: gcmid = (gcm + gcM)/2 To reduce the dataset D from p attributes to k attributes, this approach then finds the combination of k attributes whose value of gcmid is maximum.
As we can see, the difference between the Combined and Direct Approaches is that for the Combined Approach, we first find the average distance preservation for each attribute, and then, for any combination of attributes, we compute its average distance preservation by finding the averages of the distance preservations of the individual attributes. With the Direct Approach, on the other hand, to find the average distance preservation for any combination of attributes, C, we reduce the original dataset directly to the dataset containing only the attributes in C, and then
This is a technique suggested by Nsang, Maikori, Oguntoyinbo and Yusuf in (NMOY 2015). With this technique, to reduce a data set D of dimensionality d to one of dimensionality k, a set Sk is formed consisting of k numbers selected at random from the set S given by:
S = {x ϵ N | 1 x d} Then, our reduced set, DR, will be given by:
DR = D(:, Sk) That is, DR is a data set having the same number of rows as D, and if Ai is the ith attribute of DR, then Ai is the jth attribute of D if j is the ith element of Sk..
As we saw in Section 2.5 above, the Combined Approach computes the average distance preservation of a combination of attributes by computing the average of their gxmid values. It’s very clear that for any given attribute, x, gxmid is only an estimate of its average distance preservation, since it is computed as the midpoint between gxm, the minimum distance preservation, and gxM, the maximum distance preservation.
The modified version of the Combined Approach improves on the original version by computing the average distance preservation of a combination of attributes as the average of the actual distance preservations of each attribute. If x is an attribute of a dataset D, the actual distance preservation of x is computed as: n n || f (u) f (v) ||2 g x u1 vu1 || u v ||2
nr where n is the number of rows of D, u and v are any two rows of D, and f(u) and f(v) are the corresponding rows in the dataset reduced to the single attribute x. The term nr in this equation is the number of pairs of rows of D computed as:
nr nC2 n(n21) .
Thus, for any combination of attributes C of D, the average distance preservation of C is given as: g x gC xC
nC where nC is the number of attributes in C. Therefore, to reduce a dataset D from p columns to k columns, the modified version of the Combined Approach finds the combination C of k attributes of D whose value of gC is maximum.
Like the Direct Approach, to reduce a dataset Dnxp to a dataset containing k columns, the modified version of the Direct Approach selects the combination of k attributes which best preserve the interpoint distances, and reduces the original dataset to a dataset containing only those k attributes. To do so, it first generates all possible combinations of k attributes from the original p attributes. However, for each combination, C, instead of estimating its average distance preservation using its gcmid value, it computes the actual average distance preservation of C using the following formula: n n || f (u) f (v) ||2 gC u1 vu1 || u v ||2
nr where n is the number of rows of D, u and v are any two rows of D, and f(u) and f(v) are the corresponding rows in the dataset D reduced to the attributes of C. Once again, the term nr in this equation is the number of pairs of rows of D computed as:
nr nC2 n(n21) .
Therefore, to reduce a dataset D from p columns to k columns, the modified version of the Direct Approach finds the combination C of k attributes of D whose value of gC is maximum.
This technique is suggested as an improvement of the New Random Approach discussed in Section 2.7 above. To reduce a dataset Dnxp from p attributes to k attributes using the modified version of the New Random Approach, we use the algorithm below. Clearly, the idea here is to generate a result which is less random (and thus more efficient) than the result of the New Random Approach. Note that m in the algorithm is the number of times the execution of the New Random Approach is repeated.
M = [] for i = 1 to m do Run the New Random Approach to generate k numbers at random in the range 1..p Store the list of numbers generated as the ith row of M end Generate the one-dimensional matrix M1 with p entries such that M1[p] holds the frequency of the number p in the matrix M Finally, generate the matrix Result which contains the k entries in M of highest frequency, arranged in ascending order Thus, if D is the original dataset, the result of reducing D using the modified version of the New Random Approach is given by:
DR = D(:, Result) Below is the result of a sample run of the program with D as given in Table 1 below (and with m = 4): i) After the first run of NRA:
M = [ 9 2 3 7 5
Result = [1 2 3 4 5 7 8] Thus the result of reducing the dataset D using the modified version of the New Random Approach is the dataset DR is given in Table 2 below.
In this section, we shall use each of the techniques examined in Section 2 above to reduce images, and the effects of each reduction will be presented.
To achieve this aim, we shall make use of the MATLAB functions imread which converts an image into a matrix, and imshow which converts a matrix representing pixel brightness values into the image. (In other words, the function of imread is the reverse of the function of imshow.)
Random Projection is useless in preserving images. The simple reason is that when we multiply the matrix representing an image by a random matrix R, the resulting matrix typically has values outside the range of pixel brightness values. In our experiment, this is the result we obtained:
Like RP, PCA is useless in preserving images. The following is the result obtained when we tried to reduce the result obtained using PCA: Apart from RP and PCA, all the other methods we implemented were reasonably efficient in preserving images. With the Variance method, the results obtained are displayed below:
As can be observed from the results above, some reduction methods (such as the Combined and Direct Approaches and their modified versions) maintain the sizes of the original image while others do not.
As mentioned above, there are two types of
reduction techniques: those in which each attribute of
the reduced set is a linear combination of the attributes
in the original data set; and those which reduce a
dataset to a subset of the original attribute set. Of the
ten techniques implemented in this paper, two of them
(RP and PCA) belong to the first category, and as we
have seen, they are both useless in preserving images.
This applies to almost every technique in this category.
Two exceptions in this regard include Two
Dimensional PCA and Discrete Cosine Transform
Of these eight techniques, as mentioned above, LSA-Transform is probably the best in preserving images. Apart from the fact that the quality of the reduced image is practically the same as the quality of the original image, its speed of execution is very high. All the other seven techniques also significantly maintain the quality of the original image especially when most of the attributes of the matrix representing the original image are maintained – for instance when the number of attributes of the matrix representing the reduced image is at least 90% of the number of attributes of the matrix representing the original image. However, if we reduce the matrix representing the original image to 60% say (as in this case in this paper), as we can see, some of these methods are more efficient than others in preserving the original image. From the best to worst (as we can see from the results above), we have the New Random Approach, the Modified New Random Approach, and the Variance Approach followed by the Direct and Combined Approaches and their modified versions. Interestingly, these last four approaches are also the least time efficient. As a matter of fact, these last four approaches could take many days to run!
Because the Combined and Direct approaches and their modified versions have high run-time complexities, they are only suitable for reducing small images. The Variance approach on the other hand has the lowest run-time complexity (apart from LSA-Transform, of course), which makes it the most suitable for reducing large images. Obviously, the New Random Approach and its modified version are also suitable for reducing large images.
In this paper, we have studied two categories of dimensionality reduction techniques: those in which each attribute in the reduced set is a linear combination of the attributes in the original set, and those which reduce a data set to a proper subset of the original attribute set. As we have realized, while most of the techniques in the first category are useless in preserving images, every technique in the second category can be used to preserve images. Image preservation is a very important application of dimensionality reduction as it enables us to conserve memory and to improve the speed of execution of any programs which use these images.
We also noticed that while LSA-Transform, the New Random Approach, the Modified New Random Approach, and the Variance Approach have low run-time complexities and can be used to reduce large images, the Direct and Combined Approaches and their modified versions have very large run-time complexities and can only be used to reduce small images.
We shall compare the extents to which each method discussed in this paper preserves the interpoint distances and k-means clustering of different datasets in another work. We shall also analyze the time complexity of each of the ten techniques implemented in this paper. We could not do these in this paper due to time and space constraints.