Invariant pattern recognition using multiple filter image representations

COMPUTER

VISION,

GRAPHICS,

AND IMAGE

PROCESSING

45, 251-262

(1989)

NOTE

Invariant Pattern Recognition Using Multiple Filter Image Representations’: CHRISTOPH ZETZSCHE AND TERRY CAELLI Lehrstuhl

fir Nachrichtentechnik, Technische Universitiir, Institute for Medical Psychology, Ludwig-MaximiliansReceived

May

8,1987:

accepted

August

Munich, West Germany Universitiit, FRG

and

19,1988

In this paper we develop a 4-dimensional representation for patterns based on image decompositions via orientationand size-specific filters. By retaining image positional information, this encoding scheme reduces pattern rotations, translations, and scale changes to shifts in the filter outputs. The appropriate correlation processes for matching are discussed and the 0 1989 Academic Press. Inc. recognition system is illustrated by a number of examples.

1. INTRODUCTION

This paper is concerned with the old problem of how a visual system (in man or machine) may attain pattern recognition independent of the position (shift), size (scale), and orientation (rotation) of the pattern where the recognition process must have two characteristics. One, recognition must be unique insofar as the only pattern which matches a given pattern is the one specified (up to a scalar multiple); and two, the system must be able to determine the transformation states (position, orientation, and size) of the pattern relative to those specified, a priori. We call this problem the “invariance coding problem” and the aim of this paper is to evaluate one specific computational procedure for its solution. Over the past 20 years a number of partial solutions to this problem have been reported which attain at least one of the above goals, but not all. In particular, the more popular methods involve the use of the transformations of the image into another (complex) domain where the amplitude (power) spectrum is invariant to some of these operations. For example, the normal (Cartesian) Fourier power-spectrum is shift invariant but not rotation and size invariant [16]. Similarly, the circular harmonic decomposition of an image has a power spectrum which is rotation (but not shift) invariant [lo], while the full 2-dimensional (log-polar) decomposition has a power spectrum with both rotation and size, but not shift, invariant [5]. Further, schemes which use Fourier and Mellin transforms sequentially, attain their invariance by ignoring transform phase spectra and so do not guarantee uniqueness of the transform representation of the pattern (see [3], for example). It should be also noted that since such methods rely upon the power spectrum being invariant to the pattern’s geometric transformations, such schemes are not capable of registering the actual transformational states (rotations, size, or position). In spite of these limitations such techniques are used to enact aspects of invariant pattern recognition and *This work was supported in part by Canada under Grant A2568, and by the Reprint requests should be forwarded to University, Kingston, Ontario, K7L 3N6,

the Natural Sciences and Engineering Research Council of Deutsche Forschungsgemeinschaft under Grant Re 337/4/I. Professor Terry Caelli, Department of Psychology, Queen’s Canada. 251 0734-189X/89

$3.00

Copyright 0 1989 by Academic Press. Inc. All rights of reproduction in any form reserved.

252

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AND

CAELLI

are even proposed to represent the types of computations enacted in the vertebrate visual cortex (see [17]). All the above approaches have the common feature of preserving the image dimensionality in the transformation spaces. For example, the Fourier transform maps a (general) complex a-dimensional image into a complex 2-dimensional spectrum. Indeed, it is relatively clear that in order to attain invariant pattern recognition (with the properties defined above) with respect to four transformations (2 translations, rotations, and size changes) it is necessary to have a 4-dimensional pattern representation in which these transformations are linked directly with the representations’ coordinate system (see [4; 51). The necessity for representing the dimensionality of possible pattern transformations in the recognition system has been noted by Marko [12]. Jacobson and Wecksler [ll] argued for a 6-dimensional representation in order to capture the four degrees of freedom of planar transforms. We will show, however, that all essential properties can be achieved with a 4-dimensional representation. The principle behind such ideas is that in order to attain rotation, size (scale), and shift invariance, while preserving uniqueness of the pattern representation, it is necessary to generate an image-encoding scheme where all transformations are reduced to shifts, or translations, along the representation-h coordinate axes. This is thought to be one of the major reasons why the vertebrate visual system is organized in terms of cells which, at each retinotopic position, have a linear organization in terms of their orientation and sizes (scales) as already suggested by Schwartz [17] and Cavanagh [3]. This property has already been discussed by Palmer [14], Zetzsche and Elsner [20] and Ghinder [9]. However, to this stage a full implementation of the principle has not been considered for puttern recognition. This paper is concerned with developing such a computational scheme for invariant pattern recognition which enables recognition by matched filtering (cross correlation) methods. 2. THE

AUGMENTED

PATTERN

REPRESENTATION

SCHEME

The essential idea behind this encoding scheme is as follows. The 4-dimensional (4D) representation is encapsulated in a set of filtered images, each containing the original Cartesian coordinates of the image and indexed in terms of the orientation and scale-specific filters used to produce them. The result of this scheme is that a change in position, scale, or orientation of the input pattern has the effect of shifting the distribution of Jilter outputs within (position) and between (orientation and scale ) the columns. That is, the pattern’s response properties (“signature”) are distributed over the extended image-filter domain. Formally, we require a pattern representation 9(X, y, (J, 0) of the input image g(x, y), where translations of (x0, yO), rotations through angle 0, and scale changes of uO, collectively defined by the transformation T, alter the representation by shifts. That is,

where T is defined by: (2)

INVARIANT

PATTJiRN

RECOGNITION

253

SCHEME

In order to attain a 4-dimensional pattern representation which satisfies (l), we - - introduce a transform to “normalized” coordinates (x, y, x0, Jo):

This transform represents the original image g(x, y) as a set of image columns with - normalized coordinates (x, y), where each column is determined by 8, u. Here “a ” can assume any real value and is simply used to provide a logarithmic scaling of the scale (size) axis, that is, linearity in terms of the scaling component CT.The transform (3) leads to the 4-dimensional representation of the pattern, g(x, y), as B(Z,y,a,f3)

mg(a”(Zcosf3+Jsin8),ao(-Zsin@+ycos@)).

(4

That is, the standardized, or “normalized,” coordinate transformation (3) is necessary to produce a representation of the pattern g, defined by 9, such that (1) is satisfied (see Appendix for proof). Obviously this representational redundancy is determined by the degree to which the filters are disjoint or uncorrelated (see [8] for more details). The invariance properties, however, are independent of this redun- dancy factor. The filtered representation Q(x, y, CT,6) can be derived from the representation 9’ by convolution with the filter kernels, which in normalized - coordinates all have identical form h(x, y) for all u and 8 values, where

.qx,y,u,e)*h(x,j)

= Q<%.Ld>-

(5)

By definition, convolution does not influence linear shift properties as defined in (1). Therefore application of (2) to the filtered representation Q results in

Q(Z, J, (I, 6) -+ Q(Z + X0,j + Jo,0 + 00,fl + 00).

(6)

While the filter kernels are of identical form in the normalized representation, writing them with respect to (x, y) coordinates reveals their size and orientation specificity

hex, y; u, e) = h(a-yxcOse -ysino),a-“(xsin8

+ycose)).

(7)

As mentioned above, our proposed multiple representation scheme involves the decomposition of the input pattern (g(x, y)) in (4) via filters having specific orientation and scale (size) characteristics, akin to the receptive field profiles within the primary projection area of the vertebrate visual cortex which are selectively sensitive to bars and edges of specific sizes and orientations. The Fourier transform of such kernels, or point-spread functions, may be described as spectrally localized bivariate gaussian functions where the spectral position and bandwidth defines the orientation and scale selectivity range. Such filters, if adequately distributed over the spectrum, are sufficient to reconstruct the input image completely [S].

254

ZETZSCHE AND CAELLI

We have employed gaussian filters of the spectral form: H,,(u,

u) = exp ---7~. i

(u - UiJ2 + (u - cli,)2 ((2/3uJ2

:, -

(8)

where Ulj = f, cos f,

v,, = f, sin 8;

(9)

correspond to the (spectral) filter center of radial frequency f, and orientation 0,. The filters are symmetric (even, zero phase) and have an isotropic bandwidth of one octave as determined by (8). (u, u) correspond to the Fourier transform (spatial frequency) coordinates and ij indexes the filters for size (scale) and orientation, according to (9). This choice of gaussian filters is not related to the fact that such filters have minimum joint entropy, as proved by Gabor in 1947, or that they may or may not be the “best” approximation to receptive field profiles. Rather, they are simple algebraically since the Fourier transform preserves the gaussian form, they do not exhibit “ringing” and have simple representations of signal bandwidths. Indeed, our representation only requires that the filters’ point spread functions, or kernels, actually are tuned to orientation and scale. In all simulations we have used 128 x 128 (8-bit) pixel images. The gaussian filters were set at three different (radial) frequency centers of 4, 8, and 16 picture cycles (f, in (X), (9)) each of four orientations of 0, 45, 90, and 135” (dj in (8) (9)). The responses of these filters to the letter “R” are shown in Fig. 1 (top) in the (unnormalized) Cartesian coordinate system of the input image, as representing the cross correlation between each filter’s point spread function and the letter pattern. The bottom half of Fig. 1 (rows 6 to 8) show the same responses in normalized coordinates, employing (3), and extensions of the orientation responses to include detectors centered at 180, 225, 270, and 315”. These latter filter responses are simple reflections of the initial four detectors about the center of the normalized response images which are necessary to cover the whole 360” range of possible rotations (see Fig. 2). Figure 2 demonstrates how the representation actually functions. Here the two versions of the input letter differ in size by a factor of two and are 180” apart in orientation (rows 1, 5). According to (1) (3) and (6) the distribution of filter responses should only differ by shifts over the filter columns between the two states of the input letter. This can be readily seen by comparing the two sets of responses (of rows 2 to 5 to rows 6 to 8). First it should be noted that a change of size (scale) simply moves the responses one row between the larger and smaller versions of the letter. Compare rows 3 to 8, for example, where the responses are identical (up to rotations due to orientation differences). Similarly, the 180’ orientation dit%erences between the patterns is reflected in the shift in responses across rows, consistent with (1). For example, the normalized response of the l/O filter to the larger “R” (row 3, column 1) is repeated (up to the translation in the smaller letter) in the 2/180 filter response to the smaller letter (row 8, column 5). That is, the filter responses are all shifted over the filter columns in a way consistent with (1).

INVARIANT

PATTERN

RECOGNITION

SCHEME

255

FIG. 1. Upper left images show input pattern R and 12 versions filtered by gaussian modulated orientated gratings spectrally located according to (8). Here four different orientations (0,45,90,135”) and three different sized detectors (0, 1, 2 : e, in uma (7)) were used. Lower half show input pattern and 24 versions (3 sizes and 8 orientations) as shown above but plotted in (X, j)-normalized filter coordinates, according to Eq. (3).

3. THE

MATCHING

PROCESS

By developing a pattern representation scheme whereby rotations, scale changes, and translations are all simuftaneously encoded as shifts in the 4-dimensional representation space, we can enact cross correlation (or matched filtering) to accomplish the recognition process-since cross correlation is shift invariant [16]. In the following discussion we develop such correlational techniques when the representation is based upon a finite ensemble of filters. Consider the image (row 1, column 3) and test pattern (row 5, column 3) shown in Fig. 3. For illustration purposes we have used only one frequency band (scale) for the best pattern: in this case, the lowest bandwith frequency (a = 0 in (3)) having a center frequency of 4 picture cycles-producing the bandpass filtered template shown to the right of the test pattern via the composite of the four orientation-specific filters at this bandpass. Row 6 shows the normalized (see (3)) responses of these four

256

ZETZSCHE AND CAEILI

FIG. 2. Comparison of responses for two versions of the letter R differing in scale (by a factor of 2), orientation (by 180“) and position. Note the simple shift in the pattern of responses over the various columns. For example, response (l/O): row 3, column 1 corresponds to (2/180’): row 8, column 5, etc. (also note correspondence to Fig. 1.)

individual filters. Row 7 shows the corresponding pair-wise normalized coordinate outputs for the cross-correlations between 4 filter responses (a = 1, fI = 45,90, 135,180’) of the input image (top row)-as shown in row 3, columns 2 to 5 and the 4 test pattern filter responses (row 6). The correlation outputs are displayed in Cartesian coordinates in row 8, and summed to produce a total match in the last column of row 8. However, in order to register the transformation states one must examine the peaks in the individual outputs of the correlations. It can be seen that this subset of possible correlations is capable of detecting the pattern at half size, 45’ orientation and in a different position -as evidenced by the position of the correlation peak in row 8, column 6 of Fig. 3. Formally the cross-correlation images are determined by

INVARIANT

PATTERN RECOGNITION

SCHEME

257

FIG. 3. Correlation (or matching) process. Here we search for the presence of the pattern shown in column 3, of row 5 in the image shown in column 3 of row 1. For illustration purposes we use only one scale (u = 0) range to represent the pattern (row 5, column 4). The corresponding 4 orientation filter responses (u = 0,B = 0,45,!90,135”) are shown in normalized coordinates in row 6. These four responses are cross correlated with columns (images) 2 to 5 of row 3 which correspond to the responses of four orientation filters (u = 1,8 = 45,90,135,180”) to the image. These correlations are shown in row 7 in normalized coordinates, and row 8 (columns 2 to 5) in Cartesian coordinates. Column 6 in row 8 shows the total of columns (images) 2 to 5. Here the output intensity corresponds to the likelihood that the pattern was present at that position reduced in size by a factor of 2 and rotated by 45”.

where B, and 0, are the normalized 4-dimensional representations of the image and the pattern to be detected, * denotes cross correlation with respect to X and j, N-’ corresponds to the inverse of the normalized function (3), x0, y,,, a,,, 8, define the function (3), and x0, y,, a,,, 0, define the transformational state with respect to the basic pattern version Q,(x,- y,- (I, 0.). Figure 4 shows extended computations involving the detection of two test patterns in an image containing several rotated and/or reduced versions of same (column 3, rows 1 and 5), Though the complete 4-dimensional cross-correlation process requires, in principle, the correlation of all filter responses to the image with all filter responses to the

258

FIG. 4. 45,90,135,180” (right-most versions of Notice, in relative to

ZETZSCHE

AND

CAELLI

Here

we have implemented the matching process described in Fig. 3 for the a,, = 1, @,, = case over all (u, 0) conditions, using only one scale (U = 1) range for the input pattern images of rows 1 and 5). As shown in Fig. 3. these images correspond to the (x. .)‘) coordinate the summed outputs of the correlation processes enacted in the normalized coordinate system both examples (top four and bottom four rows) the positions of the cross-correlation peaks the size, position, and orientation of the patterns in the image.

test patterns, reductions can be made as a function of frequency bands which capture little of the spatial energy or add little to differentiating the signals from others. Again, we have restricted the test pattern representation to one significant frequency band, as evidenced in columns 4 and 5 of row 1 and row 5, respectively. That is, each of the images shown in rows 2 to 4 and 6 to 8 of Fig. 4 correspond to the accumulation of the cross correlations between four orientation excerpts of the target and the four orientation filtered versions of the test pattern and four corresponding filtered versions of the image, as in Fig. 3. Of particular importance is the observation that the peaks of each cross-correlation image occur in the correct spatial position and for the appropriate filter state (a, 19) corresponding to the size and orientation of the pattern being detected. Two important results emerge from these demonstrations. One, the process is capable of detecting differences between the two patterns independent of scale (size)

INVARIANT

FIG. 5. Performance,

PATTERN

RECOGNITION

SCHEME

259

Same as Fig. 4 but white noise with an SNR of -8 dB is added to the test pictures. though decreasing with the smaller versions, as has to be expected, is still remarkably good.

and orientation-evidenced by the difference in the peak intensities between the top and bottom outputs of Fig. 4. Two, the detection process is robust, as is demonstrated by the system performance under additive white noise. Figure 5 shows comparative performance under a SNR of - 8 dB (relative to the power of the whole input image). Clearly performance decreases in the upper frequency range with the smaller patterns. This has to be expected since white noise is defined by constant noise power within equal spectral bandwidths while for the filters used here, the spectral bandwidth increases with frequency. However, with the larger patterns there appears to be little decrement in performance compared to the case shown in Fig. 4. Discussion

There are many ways one can conceive of 4-dimensional pattern representations which satisfy (1). Being concerned with more efficient matching techniques based upon cross correlation we have used multiple filters to index the decompositions at

260

ZETZSCHE AND CAELLI

each image position in terms of orientation- and size-specific characteristicsappropriately normalized to capture the properties of (1). Another (continuous) way to conceive of such representations is that of a log-polar mapping of the image at each position of the image, which would produce n2 log-polar images for n x n pixel patterns. Again, rotations, size, and positional changes would translate the pattern in this 4-dimensional space. However. the matching process would be formidable and would involve 4-dimensional cross correlations within this highly redundant space of size n4. This is to be contrasted with the scheme suggested here. where the use of a quasi-orthogonal decomposition by a set of size- and orientation-specific filters enables us to create a compact and less redundant 4-dimensional representation while having the desired invariance properties. The computational effort of the suggested procedure is determined by the following. First, the effort needed for computation of the 4-dimensional image representation. Here the Gabor-type pyramid scheme as illustrated by Watson (1987) can be normalized to achieve this 4-dimensional representation. Further, the number of operations in the cross-correlational process is given by MN. where M and N are the number of samples needed for the 4-dimensional representation of the pattern and the image, respectively. It should be noted that, in the ideal limit, M and N can be made equal to m2 and n 2, the number of samples in the corresponding 2-dimensional pattern and image (as discussed above), respectively, by using appropriate sampling schemes [S]. This is to be contrasted with the n4 x n” = nh number of operations required for the direct log-polar mapping technique discussed above. While this ideal limit can be used for the pattern without any loss in the precision or correlation, the achievable N depends on additional factors. M. however, can be further reduced by taking into account that an orientation- and size-selective image decomposition converts the higher order statistical dependencies present in natural patterns into first-order ones [6, 18, 19, 211. As a consequence. a small fraction of the m2 samples will suffice to capture most of the signal energy. which is the determining factor for the cross-correlation output. Further, the number of filters required for a given pattern recognition task varies as a function of the pattern detail required for optimal classification and prior knowledge of the other possible patterns involved in the recognition environment. Indeed, one of us [l] has considered this issue in more detail: principles for generating the minimum number of filters to attain invariant pattern recognition. However, we further note that similar signal tasks that require N can be kept much smaller than the n6 needed in straightforward approaches like the direct log-polar representation described above. Obviously the characteristics inherent in the representation which converts dilations and rotations into translations (while preserving translations itself) can be used to extract new “invariant features” by applying translation invariant transforms, (e.g., autocorrelation) to the 4-dimensional representation. For example, the scheme suggested by Gltinder [9] can be viewed in this way. However, the problem with all these approaches is lack of uniqueness-a problem which is inherent in other invariance coding techniques based on the invariance of the image (Fourier) power spectra to shifts (using grating basic functions) or rotations and scale changes (using spiral-type basis functions for the associated Fourier series, IS]). Such problems are also inherent in R-transform [15] encoding models, complex log-polar mapping techniques [13, 21, and all models based only upon transform magnitude.

INVARIANT

PATTERN RECOGNITION

261

SCHEME

Indeed, Gardenier et al. [7] have recently demonstrated what specific classes of images are uniquely encoded by power spectra. In conclusion, then, we have explored an image representation scheme which maps shift, rotation, and size (scale) transformations into translations in a 4-dimensional space consisting of the (normalized) outputs of filters which are scaled logarithmically and are orientation selective. However, the uniqueness and accuracy of the matching process can be influenced by the sampling of the representation and by the number and tuning characteristics of the filters. This latter issue is critical since, in general, the full Cdimensional cross-correlation function is costly even with parallel architecture. For these reasons, further work on the minimum number of such filters and cross correlations required to attain a specified level of efficiency (matching threshold) is in progress. APPENDIX

For the input image g(x, JJ) we consider the 4D representation mapping N = g(x, Y) + .m, =

g(a5cos8

+

defined by the

v, u,e) a"jsint?,-a"Fsin8

+

a"jcos8).

(Al)

If the input image is shifted by (x0, J+,), scaled by uO, and rotated by e,, degrees, then we have g’(x, Y) = gb’,

Y’)

where

The resulting 4D representation 9(Z, j, u, @) is derived from g{ a”O[a”FcosB -aQ[a%cosf3

+ a”jsinB]cosB,

+ a’0[-a”%sine

+ a”jsinB]sin&

+ a”O[--aSsin

= g{a ~+~o(xc0s(e+ e,) +ysin(fI

+ a”jcosB]sinO,

+ x0,

+ d~cOse]cOseo+ yO}

+ e,)) + x0, a”+oO(-Zsin(8

+ e,)

+jCOS(e + e,)) + y,,}.

(A3)

Noting that

c0s2(e+ e,) + sin*(e + e,) = 1, the above function (image) g{ } becomes g{ aO+Oo((2 +

[a-(~+~O)(xoc0s(f3 + e,) - yosin(e + e,))])c0s(e + e,)

+ (j + [a-(a+uoO)( x,sin(B

a”+““( -(X +(j

+ e,) +y,c0s(e + B,))])sin(B

+ [a-(~+ao)(xocOs(fl+ e,) - y,sin(O + e,))])sin(e + [a-(a+oo)( x,sin(B

+ e,)), + e,)

+ e,) +yo~O~(e+ e,))])c0s(e + e,))}.

262 NOW,

ZETZSCHE

AND

CAELLI

for U’ = u + u,, B’ = 8 + 6,, X; = X + X0, j;’ = j + JO, and defining the

normalized shifts (X0, j,,) as ql(xo,

Yo, 0 + ql, 0 + 0,)

= a-(n+oO)(

X,COS(~ + e,) -y,sin(O

c e,))

Y,( x09 Yo, u + uo, 8 + e,) = ap(o+no)( x0 sin(8 + e,) + y,,cos(e + e,)) results in

.9(x7, y’, d, eq = s(x + x0, j + yo,u + uo,8 +

e,).

This demonstrates that such a 4-dimensional explicit representation translates shifts, rotations, and scale changes of the image into translations in the representational domain. REFERENCES

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On functional concepts for the explanation of visual pattern recognition, Human 5, 1986, 37-41. 10. Y. Hsu, H. Arsenault, and G. April, Rotation-invariant digital pattern recognition using circular harmonic expansion, Appl. Opt. 21, No. 22, 1982,4012-4015. 11. L. Jacobson and H. Wecksler, Invariant analogical image representation and pattern recognition, 9.

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