# Measures that can be used to truncate linear series of functions

Say we have the following (wavelet) series representation at some location and scale in a signal:

$$f = \sum_{k=1}^{K} c_k \psi_k$$

where $c_k$ is the coefficient, in order of magnitude ($|c_k| > |c_{k+1}|$) and $\psi_k$ is the wavelet, although it could be any $L_2$ function I guess.

Is there a principled way of truncating the series? E.g. could AIC be applied to say choose the first $N < K$ wavelets? If so how?

• Possibly $|c_k|\ge|c_{k+1}|$? – Laurent Duval Mar 20 '16 at 15:38

Best $N$-term approximation, with respect to some norm $\|f-f_N \|_p$, is quite involved (generically, NP-hard). A basic (suboptimal) approach keeps the $N$ biggest terms (esp. with an orthogonal basis)), which (since $|c_k|\ge|c_{k+1}|$) amounts to finding a proper threshold $\gamma$, and perfom hard-thresholding, up to $|c_N|\ge \gamma >|c_{N+1}|$. There are many strategies to pick an optimal threshold, with additional assumptions on the data, on the noise.
In general, you have to balance a number of terms (on even bits) with the accuracy. The $\ell_0$ constraint can be converted into other norm penalty. If you want to compress, denoise, you can use other sparsity measures like entropy-based method. Further, you have all sort of matching techniques. You can have a quick overview, for images, in Chapter 4: Redundancy and adaptivity.
• Thanks Laurent. I relation to the question, I was more thinking of the coefficients at a single, specific location independent of the rest of the image, so $K$ will be small, say only 5 or 6 terms. But I guess the principle is the same. Specifically, each wavelet represents a particular feature component, such as a line segment or an edge, that add together to make a complex feature, such as 'T' junction. So number of wavelets (components) determines the feature class. Currently I just use an arbitrary threshold, but I will try adding an $\ell_0$ penalty term like in 4.2.2 of your paper. – geometrikal Mar 20 '16 at 21:46