Exploring Complex Wavelets and their Application to Optical Character Recognition Makarand Tapaswi MERIT Master under the guidance of Prof. Philippe Salembier

Structure • Introduction to Complex Wavelets – Motivation – Key Idea

• Implementations – Redundant Complex Wavelet Transform (Dual-Tree based) – Non-Redundant Complex Wavelet Transform (Projection based)

• Properties additional to Perfect Reconstruction – Shift Invariance – Better directionality – Others...

• Application to Optical Character Recognition – Simplification of Text – Comparison with Baseline (DCT) and Real Wavelets

• Conclusion June 15th 2010

Complex Wavelets and OCR

Complex Wavelets Motivation • Continuous Wavelet Transform [1] – – – –

Too many coefficients Shift invariant Poor directionality Missing phase information

• Discrete Wavelet Transform – – – –

Critically sampled, no redundancy Shift sensitive (due to sub-sampling) Poor and Limited directionality Missing “phase” information like that of the Fourier Transforms

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Complex Wavelets and OCR

Complex Wavelets Key Idea • Real signals [2] – Filtered using analytic filters or “analytic wavelets” – Transformed to complex domain and then filtered with DWT wavelets

• Analytic function: – – – –

Complex valued Real function projected onto real and imaginary subspaces Single Sided spectrum! But, cannot be applied to the finite support wavelets so approximate

• Properties – Of course retain perfect reconstruction – Additional features like shift invariance, more directionality June 15th 2010

Complex Wavelets and OCR

Analytic Filter • Complex extension of real signal where • Real 2-sided spectrum, now becomes 1-sided

• Similarly, decompose real filter coefficients to get a pair of real filter coefficients which together form “analytic filter” June 15th 2010

Complex Wavelets and OCR

Redundant CWT (Complex Wavelet Transform) • Dual-Tree based CWT • m-dimensional signal, 2m:1 redundancy • • • •

Two conventional DWT filter banks, working in parallel Interpreted as complex because they are in quadrature Overall filter (1-D case): 2 – trees Overall filter (2-D case): 4 – trees

• No more useful for signal compression June 15th 2010

Complex Wavelets and OCR

Dual Tree DWT - Structure

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Complex Wavelets and OCR

Kingsbury’s DTDWT • Observation: Shift invariance possible if not sub-sampled • In the case of dual-tree structure Delay the filter of one tree by 1 sample than the other tree • Main Design criteria – Perfect reconstruction – Shift invariance: modeled by alias free reconstruction

• Two possible design ideas [3] – Odd and Even alternating filter lengths – Quarter-Shift Filters (Q-shift) with all even filter lengths

• Single sided pass-band (analytic) helps in meeting design requirements June 15th 2010

Complex Wavelets and OCR

Kingsbury’s DTDWT • Odd and Even Alternating Filter Lengths – – – –

Design based on MMSE in the approximation 13/12 tap and 19/16 tap filters designed Sub-sampling structure not very symmetrical Slightly different frequency response!

• Q-Shift Filters – – – – –

All filters of even length (beyond level 1) No longer strictly linear phase Tree-a has ¼ group delay, then tree-b has ¾ delay (time reversed filter) Orthonormal filters can be designed Directly time-reversed version for synthesis filters

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Complex Wavelets and OCR

Selesnick’s DTDWT • Alternate design approach, however same structure used • Design constraints – Fixed length – Vanishing moment

• Design proves that for 2 orthogonal filters to form a Hilbert pair, their scaling filters should be offset by ½ sample! [4] • Some spectral factorization and Grobner bases technique used to obtain this ½ sample delay using FIR filters… June 15th 2010

Complex Wavelets and OCR

Non-Redundant CWT • Projection based methods – Real world signal mapped onto complex function space – Followed by any type of DWT on the complex mapping

• Projection based CWT with Controllable Redundancy – Allows shift invariance, and good directionality [5]

• Projection based CWT with Non-Redundancy – No more shift invariant, but retains good directionality June 15th 2010

Complex Wavelets and OCR

Properties of DTDWT Approx. Shift Invariance

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Complex Wavelets and OCR

Properties of DTDWT Approx. Shift Invariance

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Complex Wavelets and OCR

Properties of DTDWT Better directionality

From 00, 450?, 900 now to ±150, ±450, ±750 June 15th 2010

Complex Wavelets and OCR

Properties of DTDWT Better directionality

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Complex Wavelets and OCR

Properties of DTDWT • Now with Phase Information – Allows pointing to location

• Limited Redundancy – 1D signals: twice the samples – mD signals: 2m samples – Much less compared to Continuous WT or Wavelet Packets

• and finally Perfect Reconstruction!

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Complex Wavelets and OCR

Applications • Motion Estimation and Compensation – Phase shift of complex coefficients proportional to motion [6]

• De-noising and De-convolution – Better results with un-decimated DWT than standard DWT – Comparable results with DTCWT, with much lower computation

• Texture Analysis – Owing to much better directionality properties

• Watermarking – Perceptual masking, and robustness against denoising – Easier in the complex wavelet domain

• Optical Character Recognition ?? June 15th 2010

Complex Wavelets and OCR

Introduction Devanagari Script • Primary Languages: Sanskrit, Hindi, Marathi, … [7] • Vowels

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Complex Wavelets and OCR

Introduction Devanagari Script • Lots of consonants and crazy conjuncts!

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Complex Wavelets and OCR

Devanagari OCR Database Generation • Applied heuristics [8] on photographs of text from a book • Mainly vertical and horizontal projections of the bw images • Line splitting, Word splitting

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Complex Wavelets and OCR

Devanagari OCR Line Splitting • Applied heuristics on photographs of text from a book • Mainly vertical and horizontal projections of the bw images • Line splitting, Word splitting

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Complex Wavelets and OCR

Devanagari OCR Word Splitting • Word splitting; Character pick

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Complex Wavelets and OCR

Devanagari OCR Database Montage • Contains 32 base characters, unequal number of samples • Vowels, conjuncts all are ignored for simplicity • Used 50% for training, 50% for testing • Resized to 50x45 px • Total train set: 377 • Total test set: 367

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Complex Wavelets and OCR

Devanagari OCR Classification Methodology • Baseline – Compute DCT coefficients of complete image – Store 8x8 coefficients of the top-left corner – Euclidean distance based classification

• Real DWT – Perform wavelet decomposition – Pick the level 3, horizontal coefficients as feature – Euclidean distance based classification

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Complex Wavelets and OCR

Devanagari OCR Classification Methodology • DTDWT – Simplify the initial image • Thinning operation: bwmorph(im,'thin',Inf); • Edge detection: edge(im,'sobel');

– – – – –

Perform wavelet decomposition Pick the level 2, horizontal coefficients of both the trees Compute magnitude and phase as feature Euclidean distance based classification DTDWT Toolbox: from Brooklyn University [9]

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Complex Wavelets and OCR

Comparison DWT and DTDWT

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Complex Wavelets and OCR

Recognition Accuracy Standard DWT approach Wavelet Accuracy

– – – –

Haar

Daubechies

Coiflets

Biorthogonal

db1

db4

db7

coif1

bior1.1

bior4.4

96.7%

93.2%

95.4%

98.1%

96.7%

89.4%

Haar gives good accuracy due to short length All small length filters give higher accuracy (except db7 > db4) Coiflets: symmetric, N/3 vanishing moments, length N. coif1: length 6 Symmetry (linear phase) maybe useful for performance

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Complex Wavelets and OCR

Recognition Accuracy DTDWT approach Image Type Normal Image Thinned Image Edge Image

Parameters

Accuracy

Magnitude

82.29%

Phase

41.14%

Magnitude

71.66%

Phase

42.23%

Magnitude

92.64%

Phase

48.77%

Mag + Phase

60.76%

• Magnitude looks quite good • Lots of zeros though! • Phase is quite noisy to look at June 15th 2010

Complex Wavelets and OCR

Recognition Accuracy Final Comparison Procedure

Parameters

Recognition Accuracy

Baseline DCT

64 coefficients

96.73%

Real Wavelets

coif1, horizontal details, level3

98.09%

DTDWT Kingsbury 10-tap filter

edge image, horizontal details, level2

92.64%

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Complex Wavelets and OCR

Conclusion • Quite strange to see DTDWT perform lower than standard DWT • Maybe symmetry is a key issue, since DTDWT Kingsbury filters are approximately symmetric • Filter length could also be an issue – Haar, Coiflets were quite small – Kingsbury 10-tap (smallest) filter was used – Needed to use level 2 instead of level 3 due to smudging

• Better classifiers could perhaps improve performance – Aritificial Neural Networks may be used like others [10] – No machine learning used in this project because of insufficient data

• Handwritten text could be a future interesting step, specially for a script which is quite different for printed and handwritten June 15th 2010

Complex Wavelets and OCR

References [1] S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, 1999. [2] P. D. Shukla, “Complex wavelet transforms and their applications,” Master’s thesis, University of Strathclyde, 2003. [3] N. Kingsbury, “Complex wavelets for shift invariant analysis and filtering of signals,” Journal of Applied and Computational Harmonic Analysis, vol. 10(3), pp. 234–253, May 2001. [4] I. W. Selesnick, “The design of hilbert transform pairs of wavelet bases via the flat delay filter,” in Proc. of ICASSP, 2001. [5] F. Fernandes, “Directional, shift-insensitive, complex wavelet transforms with controllable redundancy,” Ph.D. dissertation, Rice University, 2002. [6] J. Magarey and N. Kingsbury, “Motion estimation using a complex-valued wavelet transform,” IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 1069 –1084, Apr 1998. [7] Devanagari OCR, Wikipedia Article. [8] V. Bansal and R. M. K. Sinha, “A Devanagari OCR and a brief review of OCR research for Indian scripts,” STRANS01, IIT Kanpur, India, 2001. [9] Dual-Tree DWT Toolbox http://taco.poly.edu/WaveletSoftware/dt1D.htm. [10] P. Zhang, T. Bui, and C. Suen, “Extraction of hybrid complex wavelet features for the verification of handwritten numerals,” pp. 347 – 352, 26-29 2004. June 15th 2010

Complex Wavelets and OCR

Thank You!