Probabilistic Model of Error in Fixed-Point Arithmetic Gaussian Pyramid

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1 Probabilistic Model o Error in Fixed-Point Arithmetic Gaussian Pyramid Antoine Méler John A. Ruiz-Hernandez James L. Crowley INRIA Grenoble - Rhône-Alpes 655 avenue de l Europe Saint Ismier Cedex France {Antoine.Meler,John-Alexander.Ruiz-Hernandez,James.Crowley}@inrialpes.r Abstract The hal-octave Gaussian pyramid is an important tool in computer vision and image processing. The existence o a ast algorithm with linear computational complexity makes it easible to implement the hal-octave Gaussian pyramid in embedded computing systems using only integer arithmetic. However, the use o repeated convolutions using integer coeicients imposes limits on the minimum number o bits that must be used or representing image data. Failure to respect this limits results in serious degradation o the signal to noise ratio o pyramid images. In this paper we present a theoretical analysis o the accumulated error due to repeated integer coeicient convolutions with the binomial kernel. We show that the error can be seen as a random variable and we deduce a probabilistic model that describes it. Experimental and theoretical results demonstrate that the linear complexity algorithm using integer coeicients can be made suitable or video rate computation o a hal-octave pyramid on embedded image acquisition devices. 1. Introduction Over the last years, the pyramid and Gaussian derivative eatures have emerged as a powerul source o image eatures or computer vision and pattern recognition. Gaussian derivative eatures have been used or 3D reconstruction, image stitching, object recognition [9] [1] and detection o aces [8] and other categories o image objects. Gaussian derivative eatures are the basis or SIFT descriptor [7], the histogram o gradients [4] and the second local order structure image solid [6]. Such eatures are also used in biological vision models [5]. An inconvenience o Gaussian derivative eatures is that most algorithms require relatively expensive loatingpoint computations with algorithmic complexities o N or worse. It is possible to compute such eatures with a linear complexity algorithm using only integer coeicient convolutions. However, the use o this algorithm requires careul attention to the signal noise introduced by roundo error during repeated convolutions with integer coeicient ilters. Crowley and Stern [3] have proposed a linear complexity algorithm or computing an exactly scale invariant Gaussian pyramid using a technique reerred to as Cascade convolution with resampling. This algorithm has an O(N) complexity where N is the number o pixels in the image and produces log n (N) resampled images in which each image has an identical impulse response at a logarithmically increasing set o scales. An integer coeicient version o this algorithm, the haloctave binomial pyramid was proposed by Crowley and Ri [] in 3. A pyramid computed with this algorithm can provide Gaussian derivative image eatures using only simple sums and dierences o pixels. As a result, this algorithm is suitable or calculation o Gaussian derivative eatures on embedded computing devices. However restrictions in the available memory on such devices can lead designers to implement the algorithm with as ew as 8 bits per pixel. When used with repeated convolution, this a restricted bit length introduces serious degradation in the signal to noise ratio o the resulting pyramid. Such errors can cause important loss o reliability and precision o image descriptors based on the pyramid. In this paper we present a probabilistic model or the error resulting rom the computation o the hal-octave Gaussian pyramid ixed point integer coeicients. This model describes the error generation and propagation at each pyramid level due to recursive convolutions with a binomial kernel. Our results show that the hal-octave pyramid can be accurately computed with short ixed-point numbers. This paper is organized as ollows. Section explains the hal-octave Gaussian pyramid algorithm. Our probabilistic error model is developed in section 3, ollowed experimental veriication in section 4. Section 5 concludes with discussion o the results and its application or embedded image processing.

2 Image σ = 8 σ = 4 D Binomial Convolutions σ = 4 D Binomial Convolutions σ = σ = Resampling Level 1 Pyramid Buer derivation Pyramid Derivative σ = D Binomial Convolutions l = l + 1 Resampling Level l Figure 1. The O(N) cascade convolution pyramid computation scheme. The Hal-Octave Gaussian Pyramid In this section we describe the calculation o the haloctave Gaussian pyramid and its derivatives. A linear complexity pyramid algorithm or the calculation o Gaussian derivatives has been known since the 198 s [1] (igure 1). This algorithm involves alternatively convolving with a Gaussian support, and resampling the resulting image. The result o this algorithm is a hal-octave Gaussian pyramid. An integer coeicient version o this algorithm [] has been demonstrated using repeated convolutions o the binomial kernels [1,, 1] and [1,, 1] t. Implementations that compute such pyramids on PAL sized images and bigger exist or the current generation o computer stations, and can be embedded in dedicated signal processor. The hal-octave Gaussian pyramid or an N N pixels image is composed o up to L = log (N) images. The eect o cascade convolution is to sum the variances o the ilters, so that the cumulative variance or the image l [[ 1, L ]] is σ l = l. That or, the convolution step involves convolutions by [1,, 1] in each direction. Interleaving resampling with convolutions decreases the number o image samples while expanding the distance between samples. This has the eect o dilating the Gaussian support without increasing the number o samples used or the Gaussian, eectively increasing the scale. The resampling rate is set to, resulting in a constant ratio o scale over sample distance. The resulting pyramid represents the original N N pixels image with a sequence o log (N) images at a geometric progression o scales each with twice as less samples as the previous, resulting in a total o N samples (igure ). The -dimensional derivative o order (d X, d Y ) is then obtained by convolving d X times the resulting pyramid with the derivative kernel [ 1,, 1], and d Y times with [ 1,, 1] t. Figure. The hal-octave pyramid algorithm result... xk 1 xk xk xk(i + 1, j)... xk 1 xk xk xk(i, j + 1) Figure 3. Convolution with the two base kernels (smoothing and dierentiation ). The n operator stands or right n bits shit with zero padding 3. Probabilistic Formulation 3.1. Random Error Model In this section, we present our model to analyze the noise eect o using ixed point coeicients or the pixel values in the pyramid. Let x k ( i, j) be the value o the pixel o coordinate k ater i smoothing convolutions and j dierentiations in directions X and Y. x k ( i, j) is the sum o an exact theoretical value x k ( i, j) and an error E k ( i, j). x k ( i, j) = x k ( i, j) + E k ( i, j) (1) We model the error E by a random variable. In order to simpliy the problem, the error part o contiguous pixels is supposed independent. This approximation is veriied to be very accurate in section 4. Thus, we will simply consider E where i and j are the number o smoothing and dierentiation convolutions respectively applied to one pixel, including both directions X and Y. We consider only two operations: convolution with the normalized kernels K s = [ 1 4, 1, 1 4 ] and with K d = [ 1,, 1 ] as schematized on igure 3. We also suppose that all the convolutions with K s are applied beore those with K d. Thus, we express E(i, j = ) as a unction o E(i 1, j), then E(i, j > ) as a unction o E(i, j 1).

3 3.. Error Generation and Propagation x is a ixed point number with n i bits o integer part and n bits o ractional part. Furthermore, x is supposed to be initially an integer (i.e. its n least signiicant bits are ). Because pyramid computation is recursive, the error o one value is composed o two parts. One is a new error, generated by the current computation, and the other, E prop, is accumulated during the previous computations and propagated to the new value. In this section, we determine, or a given number o ractional bits n, the expressions o the random variables En new and. Error generation is the error generated by the calculation o a new pyramid value. As shown in igure 3, a convolution with K s or K d starts with divisions by a number n shit, which, in ixed point arithmetic, corresponds to a bit shit with the loss n o the n shit least signiicant bits. Such a division can be modeled with U Vn (n shit ), the uniorm discrete random variable { } with the set o possible values V n (n shit ) = i n +n shit. In the case o values being i [, n shit 1] initially integers, this model is well-suited or i + j > n. Beore this condition is true, the divisions do not lead to a loss o inormation. Thus, the generated error is equal to: I i + j n : Otherwise: n = () { E new n (i, j = ) = U Vn () + U Vn (1) + U Vn () n (i, j > ) = U Vn (1) U Vn (1) Error propagation This paragraph considers the error that propagates rom a state to the next (i + 1, j) or (i, j + 1). We assume the number o integer bits, n i, is suicient to avoid overlow. In this case, the error propagated during a convolution with K s or K d is a random variable deined by recurrence: (3) 3.3. Total Accumulated Error The total error is the sum o the propagated part and the generated part: E n = + En new (5) We deduce rom the above model the error characteristics (standard deviation and variance) as a unction o the number o smoothing convolutions (i) and dierentiations (j). We irst express the error with recurrence ormulation, and then give the solution. Error evolution The values are initially integers, E n = or i + j n. Thus we express the expected value E[E n ] and the variance V ar[e n ] o the complete error random variable or i + j n. In this condition, and En new can be supposed independent. Smoothing convolutions are computed beore computing image derivatives rom image dierences. Thus we irst give the recurrence law or smoothing (j = ) and then or derivative computation (j > ). Smoothing: E[E n (i, j = )] = E + E En new = E [ E n (i 1, j) ] 1 n V ar[e n (i, j = )] = V ar + V ar En new Derivation: E[E n (i, j > )] = E = 3 8 V ar [ E n (i 1, j) ] + 5/3 (n +5) + E En new (6) (7) = (8) V ar[e n (i, j > )] = V ar + V ar En new = 1 V ar [ E n (i, j 1) ] + /3 (n +) (9) E prop n (, ) = (i, j = ) = En (i 1,j) 4 + En (i 1,j) + En (i 1,j) 4 (i, j > ) = En (i,j 1) En (i,j 1) (4) Error expression We can deduce the absolute expressions o E[E n ] and V ar[e n ] as a unction o i, j and n. E[E n ] = ( n i ) i j = otherwise { 1 n (1)

4 V ar[e n ] = 1 3 ) i n 1 ( 3 8 n + 1 +j+ ( 1 j ) (11) n + level 1 level 1, nd derivative Note that the error variance increases with i and j toward a limit: V ar[e n (, )] = V ar[e n (i, )] = 1/3 n + 4. Experiments and validation level level 6, nd derivative In this section we compare the error predicted by our model with the one measured by subtracting an image pyramid with varying precision ( n 1) and a 64 bits one. We perorm this analyse on images (Perlin noise) and photos randomly downloaded rom the internet Error Distribution To validate our model, we compare predicted and measured error. There is no simple mathematic expression or the distribution o the error as modeled in this paper. However we can estimate it as accurately as needed by running s o our random variable. As shown igure 4, our model accurately matches the error distribution measured in pyramids built rom highlyinormative images like Perlin noise. In the case o, an unpredicted null-error peak can be observed. The igure 5 demonstrates that in homogeneous areas (sky, unocused background, over/under-exposition,...), smoothing convolutions do not alter pixel values which thus stay integers. In such areas, our model or the loss o inormation as an addition by a random uniorm variable ails while the pyramid level is under the singular zone characteristic scale. Thus, our theoretical analysis provides inormation about the accumulated error in the worst case. One can also observe a larger spread in the measured distributions than in the theoretical one. This dierence is due the independence hypothesis which underestimate the variance by neglecting a positive covariance term. However, it appears that this term play a minor part. 4.. Error Parameters In this section, we graphically display the parameters o the random error variable. Due to the presence o homogeneous areas in photos, experimental standard deviation and variance are very datadependant. It is thus more interesting to visualize the theoretical values which correspond to the worst case (igure 6). It can be seen that the standard deviation convergence in the levels is quite ast. We thus can simply study the limit value (igure 7) Figure 4. Distribution o the error o the pyramid values with n = 6. abscissa: error value, ordinate: pixel rate with this error. (a ) (b ) Figure 5. Over-exposed photography and associated error image at pyramid level 3 with n = 6 (a ). The saturated area corresponds to a null-error area which is very poorly modeled by our random error variable. Perlin noise and its associated error image (b ). 5. Conclusion This paper has presented a model or the error due to recursive truncations by a random noise in computing a Gaussian pyramid using a linear complexity algorithm involving cascaded convolution with integer coeicient binomial ilters. This model predicts the additive image error as a unction o the number o ractional bits used, the scale o the pyramid and the order o the derivative. We have validated our model by comparing a o our random variable and measurements perormed in pyramids applied

5 Expected Value Standard Deviation n level Expected Value Standard Deviation n level Figure 6. Expected value and standard deviation o the error variable as a unction o the level and the number o ractional bits n beore dierentiations and ater dierentiations standard deviation s limit to images and photos. We have shown that the noise variance rapidly converges to a ixed value as a unction o the pyramid scale. However, the divergence is constant and grows rapidly or small numbers o ractional bits in the ixed point representation. This divergence becomes zero when the signal is dierentiated, thus making it irrelevant or many applications. We can conclude by observing that binomial convolution kernels tends to smooth the error noise, thus compensating or random noise added by integer truncation in cascade convolution. This makes the Gaussian binomial pyramid a good multi-scale image representation solution or use in embedded computer vision systems. Reerences [1] J. Crowley and A. Parker. A representation or shape based on peaks and ridges in thedierence o low-pass transorm. IEEE PAMI, 6(): , March [] J. Crowley and O. Ri. Fast computation o scale normalised gaussian receptive ields. pages , 3. 1, [3] J. Crowley and R. Stern. Fast computation o the dierence o low-pass transorm. PAMI, 6():1, March [4] N. Dalal and B. Triggs. Histograms o oriented gradients or human detection. IEEE CVPR, pages , June 5. 1 [5] M. A. Georgeson, K. A. May, T. C. Freeman, and G. S. Hesse. From ilters to eatures: scale-space analysis o edge and blur coding in human vision. Journal o vision, 7(13), 7. 1 [6] L. Griin. The second order local-image-structure solid. IEEE PAMI, 9(8): , 7. 1 [7] D. G. Lowe. Distinctive image eatures rom scale-invariant keypoints. International Journal o Computer Vision, 6:91 11, 4. 1 [8] J. A. Ruiz Hernandez, A. Lux, and J. L. Crowley. Face detection by cascade o gaussian derivatives classiiers calculated with a hal-octave pyramid. IEEE Conerence on Automatic Face and Gesture Recognition, Amsterdam, Sep 8. 1 [9] B. Schiele and J. Crowley. Recognition without correspondence using multidimensional receptive ield histograms. International Journal o Computer Vision, 36:31 5,. 1 [1] J. Yokono and T. Poggio. Oriented ilters or object recognition: an empirical study. In Proceedings o the IEEE FG4. Seoul, Korea, page 755, n Figure 7. Limit value o the error standard deviation as a unction o the number o ractional bits n.