Image Processing Reference
In-Depth Information
Example 3.1. The 2D discrete color images (of the same size) are vector spaces if
wetreatthemasmatriceshavingelementsconsistingofarraysrepresentingthethree
color components,
R, G, B
. In the right-down direction, the digital pictures in Fig.
3.1 illustrate the frames of an image sequence except the last frame (bottom). The
last frame is the average of all frames observed by the camera (shown in the figure).
A
=(
A
1
+
A
2
+
···
A
15
)
/
15
(3.13)
where
A
i
is the
i
th frame. The frames are summed and finally multiplied with a
scalar (1/15), according to the addition and scaling rules of Eqs. (3.11)-(3.12)
In the observed sequence the camera is static (immobile) and the passing human
is present in every frame, and yet nearly no human is present in the averaged frame.
This is because at any given point, the color is nearly unchanged over the time be-
cause the camera is fixed. Though seldom, a change of the color does occur at that
point, due to the passage of the human. This happens in, roughly 1 out of 15 time
instants at the same pixel coordinate. Consequently, the time averages of color are
not sufficiently influenced by one or two outliers, to the effect that the means closely
approximate the background colors. Another effect of averaging is that the mean
representsthecolormoreaccuratelythananyoftheconstituentimages.Similaruses
of matrix averaging, sometimes combined with robust statistics (e.g., clipped aver-
age, median, etc.) are at the heart of many image processing applications, e.g noise
reduction [85, 192], super resolution, and the removal of moving objects, [9, 119].
Exercise 3.2.
Can we use the HSB space to represent the color arrays under the
rules of (3.11)-(3.12)?
HINT: Do the points on a circle represent a vector space? Is the numerical average
of two numbers representing hue always a hue (e.g., two complementary colors)?
3.3 Norms of Vectors and Distances Between Points
Adding or scaling vectors are tools that are not powerful enough for our needs. We
must be able to measure the “length” of the vectors as well. The length of a vector is
also known as the norm or the magnitude of a vector. The symbol for the norm of a
vector
u
is
.
In analogy with the length concept in
E
3
,
u
•
The norm should not be negative.
•
Only the null vector has the norm zero.
Despite its simplicity, the second requirement constitutes the backbone of nu-
merous impressive proofs in science and mathematics. It expresses when
only
the
knowledge of the norm is sufficient to identify the vector itself fully. The vector
space properties guarantee that the null vector is the only vector which enjoys the
privilege that the knowledge of its norm automatically determines the vector itself.
Paraphrasing this, we have the right to “throw away” the norm symbol only when we
are sure that the norm is zero.
