ROBOTICS Handbook of Computer Vision Algorithms in Image Algebra Part 11 ppsx

To specify template matching in image algebra, define an invariant pattern template t, corresponding to the pattern p centered at the origin, by setting The unnormalized correlation algo

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Handbook of Computer Vision Algorithms in Image Algebra

by Gerhard X Ritter; Joseph N Wilson

CRC Press, CRC Press LLC

ISBN: 0849326362 Pub Date: 05/01/96 Search this book:

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Image Algebra Formulation

The exact formulation of a discrete correlation of an M × N image with a pattern p of size (2m - 1)

× (2n - 1) centered at the origin is given by

For (x + k, y + l) X, one assumes that a(x + k, y + l) = 0 It is also assumed that the pattern size is generally

smaller than the sensed image size Figure 9.2.5 illustrates the correlation as expressed by Equation 9.2.1

Figure 9.2.5 Computation of the correlation value c(x, y) at a point (x, y) X.

To specify template matching in image algebra, define an invariant pattern template t, corresponding to the pattern p centered at the origin, by setting

The unnormalized correlation algorithm is then given by

The following simple computation shows that this agrees with the formulation given by Equation 9.2.1

By definition of the operation •, we have that

Title

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-Since t is translation invariant, t(x, y) (u, v) = t(0, 0)(u - x, v - y) Thus, Equation 9.2.2 can be written as

Now t(0, 0)(u - x, v - y) = 0 unless (u - x, v - y) S(t(0, 0)) or, equivalently, unless (m 1) d u x d m 1 and (n -1) d v - y d n - 1 Changing variables by letting k = u - x and l = v - y changes Equation 9.2.3 to

To compute the normalized correlation image c, let N denote the neighborhood function defined by N(y) =

S(ty) The normalized correlation image is then computed as

An alternate normalized correlation image is given by the statement

Note that £t(0, 0) is simply the sum of all pixel values of the pattern template at the origin

Comments and Observations

To be effective, pattern matching requires an accurate pattern Even if an accurate pattern exists, slight variations in the size, shape, orientation, and gray level values of the object of interest will adversely affect performance For this reason, pattern matching is usually limited to smaller local features which are more invariant to size and shape variations of an object

9.3 Pattern Matching in the Frequency Domain

The purpose of this section is to present several approaches to template matching in the spectral or Fourier domain Since convolutions and correlations in the spatial domain correspond to multiplications in the

spectral domain, it is often advantageous to perform template matching in the spectral domain This holds especially true for templates with large support as well as for various parallel and optical implementations of matched filters

It follows from the convolution theorem [3] that the spatial correlation a•t corresponds to multiplication in the

frequency domain In particular,

where â denotes the Fourier transform of a, denotes the complex conjugate of , and the inverse

Fourier transform Thus, simple pointwise multiplication of the image â with the image and Fourier

transforming the result implements the spatial correlation a •t.

One limitation of the matched filter given by Equation 9.3.1 is that the output of the filter depends primarily

on the gray values of the image a rather than on its spatial structures This can be observed when considering

the output image and its corresponding gray value surface shown in Figure 9.3.2 For example, the letter E in the input image (Figure 9.3.1) produced a high-energy output when correlated with the pattern letter B shown

in Figure 9.3.1 Additionally, the filter output is proportional to its autocorrelation, and the shape of the filter output around its maximum match is fairly broad Accurately locating this maximum can therefore be difficult

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The image is now given by , the Fourier transform of p The correlation image c can therefore be

obtained using the following algorithm:

Using the image p constructed in the above algorithm, the phase-only filter and the symmetric phase-only

filter have now the following simple formulation:

and

respectively

In order to achieve the phase-only matching component to the matched filter approach we needed to divide the complex image by the amplitude image Problems can occur if some pixel values of are equal to zero However, in the image algebra pseudocode of the various matched filters we assume that

, where denotes the pseudoinverse of A similar comment holds for the quotient

Some further improvements of the symmetric phase-only matched filter can be achieved by processing the spectral phases [6, 7, 8, 9]

9.4 Rotation Invariant Pattern Matching

In Section 9.2 we noted that pattern matching using simple pattern correlation will be adversely affected if the pattern in the image is different in size or orientation then the template pattern Rotation invariant pattern matching solves this problem for patterns varying in orientation The technique presented here is a digital adaptation of optical methods of rotation invariant pattern matching [10, 11, 12, 13, 14]

Computing the Fourier transform of images and ignoring the phase provides for a pattern matching approach

that is insensitive to position (Section 9.3) since a shift in a(x, y) does not affect |â(u, v)| This follows from

the Fourier transform pair relation

which implies that

where x0 = y0 = N/2 denote the midpoint coordinates of the N × N domain of â However, rotation of a(x, y) rotates |â(u, v)| by the same amount This rotational effect can be taken care of by transforming |â(u, v)| to

polar form (u, v) (r, ¸) A rotation of a(x, y) will then manifest itself as a shift in the angle ¸ After

determining this shift, the pattern template can be rotated through the angle ¸ and then used in one of the standard correlation schemes in order to find the location of the pattern in the image

The exact specification of this technique — which, in the digital domain, is by no means trivial — is provided

by the image algebra formulation below

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This step is vital for the success of the proposed method Image spectra decrease rather rapidly as a function

of increasing frequency, resulting in suppression of high-frequency terms Taking the logarithm of the Fourier spectrum increases the amplitude of the side lobes and thus provides for more accurate results when

employing the symmetric phase-only filter at a later stage of this algorithm

Step 5 Convert â and to continuous image.

The conversion of â and to continuous images is accomplished by using bilinear interpolation An image algebra formulation of bilinear interpolation can be found in [15] Note that because of Step 4, â and are real-valued images Thus, if âb and denote the interpolated images, then , where

That is, âb and are real-valued images over a point set X with real-valued coordinates.

Although nearest neighbor interpolation can be used, bilinear interpolation results in a more robust matching algorithm

Step 6 Convert to polar coordinates.

Define the point set

and a spatial function f : Y ’ X by

Next compute the polar images

Step 7 Apply the SPOMF algorithm (Section 9.3).

Since the spectral magnitude is a periodic function of À and ¸ ranges over the interval [-À = ¸0, ¸N = À], the output of the SPOMF algorithm will produce two peaks along the ¸ axis, ¸j and ¸k for some and

Due to the periodicity, |¸j| + |¸k | = À and, hence, k = -(j + N/2) One of these two angles

corresponds to the angle of rotation of the pattern in the image with respect to the template pattern The complementary angle corresponds to the same image pattern rotated 180 °

To find the location of the rotated pattern in the spatial domain image, one must rotate the pattern template (or input image) through the angle ¸j as well as the angle ¸k The two templates thus obtained can then be used in one of the previous correlation methods Pixels with the highest correlation values will correspond to the pattern location

The following example will help to further clarify the algorithm described above The pattern image p and input image a are shown in Figure 9.4.1 The exemplar pattern is a rectangle rotated through an angle of 15°

while the input image contains the pattern rotated through an angle of 70° Figure 9.4.2 shows the output of Step 4 and Figure 9.4.3 illustrates the conversion to polar coordinates of the images shown in Figure 9.4.2 The output of the SPOMF process (before thresholding) is shown in Figure 9.4.4 The two high peaks appear

on the ¸ axis (r = 0).

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Figure 9.4.1 The input image a is shown on the left and the pattern template p on the right.

The reason for choosing grid spacing in Step 6 is that the maximum value of r is

which prevents mapping the polar coordinates outside the set X Finer sampling grids

will further improve the accuracy of pattern detection; however, computational costs will increase

proportionally A major drawback of this method is that it works best only when a single object is present in the image, and when the image and template backgrounds are identical

Figure 9.4.2 The log of the spectra of â (left) and (right).

Figure 9.4.3 Rectangular to polar conversion of â (left) and (right).

Figure 9.4.4 SPOMF of image and pattern shown in Figure 9.4.3

9.5 Rotation and Scale Invariant Pattern Matching

In this section we discuss a method of pattern matching which is invariant with respect to both rotation and scale The two main components of this method are the Fourier transform and the Mellin transform Rotation invariance is achieved by using the approach described in Section 9.4 For scale invariance we employ the Mellin transform Since the Mellin transform of an image a is given by

it follows that if b(x, y) = a(±x, ±y), then

Therefore,

which shows that the Mellin transform is scale invariant

Implementation of the Mellin transform can be accomplished by use of the Fourier transform by rescaling the

input function Specifically, letting ³ = logx and ² = logy we have

Therefore,

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which is the desired result.

It follows that combining the Fourier and Mellin transform with a rectangular to polar conversion yields a rotation and scale invariant matching scheme The approach takes advantage of the individual invariance properties of these two transforms as summarized by the following four basic steps:

(1) Fourier transform

(2) Rectangular to polar conversion

(3) Logarithmic scaling of r

(4) SPOMF

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9.6 Line Detection Using the Hough Transform

The Hough transform is a mapping from into the function space of sinusoidal functions It was first formulated in 1962 by Hough [16] Since its early formulation, this transform has undergone intense

investigations which have resulted in several generalizations and a variety of applications in computer vision and image processing [1, 2, 17, 18, 19] In this section we present a method for finding straight lines using the Hough transform The input for the Hough transform is an image that has been preprocessed by some type of edge detector and thresholded (see Chapters 3 and 4) Specifically, the input should be a binary edge image

Figure 9.5.4 SPOMF of image and pattern shown in Figure 9.5.3

A straight “line” in the sense of the Hough algorithm is a colinear set of points Thus, the number of points in

a straight line could range from one to the number of pixels along the diagonal of the image The quality of a straight “line” is judged by the number of points in it It is assumed that the natural straight lines in an image correspond to digitized straight “lines” in the image with relatively large cardinality

A brute force approach to finding straight lines in a binary image with N feature pixels would be to examine

all possible straight lines between the feature pixels For each of the possible lines, N - 2

tests for colinearity must be performed Thus, the brute force approach has a computational complexity on the

order of N3 The Hough algorithm provides a method of reducing this computational cost

To begin the description of the Hough algorithm, we first define the Hough transform and examine some of

its properties The Hough transform is a mapping h from into the function space of sinusoidal functions defined by

To see how the Hough transform can be used to find straight lines in an image, a few observations need to be made

Any straight line l0 in the xy-plane corresponds to a point (Á0, ¸0) in the Á¸-plane, where ¸0 [0, À) and

Let n0 be the line normal to l0 that passes through the origin of the xy-plane The angle n0 makes with the positive x-axis is ¸0 The distance from (0, 0) to l0 along n0 is |Á0| Figure 9.6.1 below

illustrates the relation between l0, n0, ¸0, and Á0 Note that the x-axis in the figure corresponds to the point (0, 0), while the y-axis corresponds to the point (0, À/2).

Figure 9.6.1 Relation of rectangular to polar representation of a line

Suppose (x i , yi), 1 d i d n, are points in the xy-plane that lie along the straight line l0 (see Figure 9.6.1) The

line l0 has a representation (Á0, ¸0) in the Á¸-plane The Hough transform takes each of the points (x i , y i) to a

sinusoidal curve Á = x i cos(¸) + y i sin(¸) in the ¸Á-plane The property that the Hough algorithm relies on is that

each of the curves Á = x i cos(¸) + y i sin(¸) have a common point of intersection, namely (Á0, ¸0) Conversely,

the sinusoidal curve Á = x cos(¸) + y sin(¸) passes through the point (Á0, ¸0) in the Á¸-plane only if (x, y) lies

on the line (Á0, ¸0) in the xy-plane.

As an example, consider the points (1, 7), (3, 5), (5, 3), and (6, 2) in the xy-plane that lie along the line l with

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¸ and Á representation and , respectively.

Figure 9.6.2 shows these points and the line l0

Figure 9.6.2 Polar parameters associated with points lying on a line

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