A Fast Detection Method for Bottle Caps Surface Defect Based on Sparse Representation - Intelligent Computing for Sustainable Energy and Environment - page 78

Similarly, for the defect image on the cap surface, sparse decomposition matrix
D d ∈ M N×K

d

is written as

X d = D d α d , (3)

where α d is the defect sparse factor. D d is the defect dictionary. So, X can be
denoted as follow,

X = D t α t + D d α d . (4)

To seek a sparse representation over a combined dictionary containing both D t
and D d , the generic method use the L 0 norm as a definition of sparsity. Hence,
the following equations need to be solved

{ α t ,α d } =argmin α t 0 + α d 0

st. X = D t α t + D d α d . (5)

As well known, the problem formulated in Eq.(5) is non-convex and intractable.
Its complexity grows exponentially with the number of columns in the overall
dictionary. In order to obtain a tractable convex optimization solution, the basis
pursuit (BP) method suggests the replacement of the L 0 -norm with an L 1 -norm
[8]. Eq.(5) is re-written as a linear programming problem:

{ α t ,α d } =argmin α t 1 + α d 1 ,

st. X − D t α t − D d α d 2 ε. (6)

where the parameter ε stands for the residual, which is the tolerance between
the sample image and the template image. Although these methods above are
very effective for image separating, they are computationally complex and time-
consuming. One of the reasons is that the dictionaries are redundant and over-
complete, these make the dictionaries very large. The more atoms there are
in a dictionary, the better the searching precision is. But everything have two
sides, large dictionary increases computational complexity, and consumes large
amounts of time. Therefore, we must trade off between matching precision and
computational complexity. In this paper, a novel dictionary is proposed which is
compactly supported and simple. A simple dictionary may reduce the accuracy
of separation whilst satisfying the requirements of real-time process.

3 Feature Extract and Dictionary

The circular region of search and extraction is the key point to improve the over-
all speed of the detection algorithm. The projective transform are performed in
the region of interest(ROI). Apparently, this way can save a lot of computing time
since the ROI has a smaller size than that of the entire image. Fig. 1 shows the
extracting of the region of interesting. The left is the original images. The right
is the ROI. In order to carry out the rotation match between the template ROI
and the sample ROI, we propose a method of using the circular region projec-
tion histogram transform as the rotation invariant feature. Such, the 2D image