A Theory For Multiresolution Signal Decomposition: The Wavelet Representation Stephane Mallat, IEEE Transactions on Pattern Analysis and Machine Intelligence, July 1989 Presented by: Randeep Singh Gakhal CMPT 820, Spring 2004

Outline  Introduction  History  Multiresolution Decomposition  Wavelet Representation  Extension to Images  Conclusions

Introduction  What is a wavelet?  It’s just a signal that acts like a variable strength magnifying glass.  What does wavelet analysis give us?  Localized analysis of a signal  For analysis of low frequencies, we want a large frequency resolution  For analysis of high frequencies, we want a small frequency resolution  Wavelets makes both possible.

History  Wavelets originated as work done by engineers, not mathematicians.  Mallat saw wavelets being used in many disciplines and tried to tie things together and formalize the science.  This paper was co-authored by Yves Meyer who let Mallat have all the credit as he was already a full professor.

Multiresolution Decomposition

 We want to decompose f(x): Finite energy and measurable  The approximation operator at resolution 2 j :  The operator is an orthogonal projection onto a vector space : The Multiresolution Operator

 Approximation is the most accurate possible at that resolution:  Lower resolution approximations can be extracted from higher resolution approximations: Properties of the Multiresolution Approximation I 1 2

 Operator at different resolutions:   an isomorphism: I 2 is the vector space of square summable sequences Given I, we can move freely between the two domains without loss of information Properties of the Multiresolution Approximation II 4 3

Properties of the Multiresolution Approximation III  Translation in the approximation domain:  Translation in the sample domain: where: 5 6

Properties of the Multiresolution Approximation IV  As the resolution increases (j  ∞ ), the approximation converges to the original:  And vice versa: is dense7 8

Multiresolution Transformation  Conditions 1-8 define the requirements for a vector space to be a multiresolution approximation  The operator projects the signal onto the vector space  Before we can compute the projection, we need an orthonormal basis for the vector space

For the multiresolution approximation for : Orthonormal Basis Theorem  a unique scaling function: such that is an orthonormal basis of

Projecting onto the Vector Space The discrete approximation: The continuous approximation:

Implementation of the Multiresolution Transform Let the signal that is of highest resolution be at 1 (j=0) and successively coarser approximations be at decreasing j (j<0). We can express the inner product at a resolution j based on the higher resolution j+1: where With this form, we can compute all discrete approximations (j<0) from the original (j=0). This is the Pyramid Transform

The Scaling Function Theorem The scaling function essentially characterizes the entire multiresolution approximation. The calculation of the discrete approximation does NOT require the scaling function explicitly – we use h(n). If H() satisfies: Then, we define the scaling function as: We can define the filter first and the scaling function is the result!!!

An Example of Multiresolution Decomposition Successive discrete (a) and continuous (b) approximations of a function f(x). (a)(b)

Wavelet Representation

Projecting onto the Orthogonal Complement Wavelet representation is derived from the detail signal. At resolution 2 j, detail signal is difference in information between approximation j and j+1. Detail signal is result of orthogonal projection of f(x) on orthogonal complement of in,. Analogous to case for multiresolution approximation in, we need an orthonormal basis for to express the detail signal.

The Orthogonal Complement Basis Theorem For the multiresolution vector space, scaling function, and conjugate filter, define: Then is an orthonormal basis of And is an orthonormal basis of

Projecting onto the Orthogonal Complement Vector Space The discrete detail signal: The continuous detail signal: Orthogonal wavelet representation consists of a signal at a coarse resolution and a succession of refinements consisting of difference signals:

Implementation of the Orthogonal Wavelet Representation We express the inner product at a resolution j based on the higher resolution j+1: where We can thus successively decompose the discrete representation to compute the orthogonal wavelet representation. This is pyramid transform is called the Fast Wavelet Transform (FWT)

Implementation of the Orthogonal Wavelet Representation Cascading algorithm to compute detail signals successively, generating the wavelet representation. The output becomes the input to calculating the next (and coarser) detail signal.

An Example of Wavelet Representation Successive continuous approximations (a) and discrete detail signals (b) for a function f(x). (a)(b)

Extension to Images

Two dimensional scaling function is separable: Images require a natural extension to our previously discussed multiresolution analysis to two dimensions. Orthonormal basis of is now: Multiresolution Image Decomposition So our basis can be expressed as:

At resolution 2 j, our discrete approximation has 2 j N pixels. The discrete characterization of our image is: Multiresolution Image Decomposition Continued An example of an original image (resolution=1) and three approximations at 1/2, 1/4, and 1/8.

For the two-dimensional scaling function with (x) the wavelet associated with (x), the three wavelets The Multidimensional Orthogonal Complement Basis Theorem define the orthonormal basis for and the orthonormal basis for

The Multideminsional Wavelet Representation We can now define our wavelet representation of the image: Or, in a filter form:

The Three Frequency Channels We can interpret the decomposition as a breakdown of the signal into spatially oriented frequency channels. Decomposition of frequency support Arrangement of wavelet representations

Applying the Decomposition (a) Original image (b) Wavelet representation (c) Black and white view of high amplitude coefficients

Conclusions and Future Work

Conclusions  We can express any signal as a series of multiresolution approximations.  Using wavelets, we can represent the signal as a coarse approximation and a series of difference signals without any loss.  The multiresolution representation is characterized by the scaling function. The scaling function gives us the wavelet function.  Applied to images, we can get frequency content along each of the dimensions and joint frequency content

References [1] S. Mallat, "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation", IEEE Trans. on Pattern Analysis and Machine Intelligence, 11(7): , [2] B. B. Hubbard, “The World According to Wavelets”, A.K. Peters, Ltd., 1998.