Scale Space Implementation

Scale Space Implementation

The linear scale-space representation of an N-dimensional continuous signal is obtained by convolving with an N-dimensional Gaussian kernel

\begin{align} L(x_1, x_2, \dots, x_N, t) = \int_{u_1=-\infty}^{\infty} \int_{u_2=-\infty}^{\infty} \dots \int_{u_N=-\infty}^{\infty} &f_C(x_1-u_1, x_2-u_2, \dots, x_N-u_N, t)\\ \, \cdot \, &g_N(u_1, u_2, \dots, u_N, t) \, du_1 \, du_2 \dots du_N \end{align}

However, for implementation, this definition is impractical, since it is continuous. When applying the scale space concept to a discrete signal, different approaches can be taken. This article is a brief summary of some of the most frequently used methods.

Read more about Scale Space Implementation:  Separability, The Sampled Gaussian Kernel, The Discrete Gaussian Kernel, Recursive Filters, Finite-impulse-response (FIR) Smoothers, Real-time Implementation Within Pyramids and Discrete Approximation of Scale-normalized Derivatives, Other Multi-scale Approaches, See Also

Famous quotes containing the words scale and/or space:

    “... the meanest life, the poorest existence, is attributed to God’s will, but as human beings become more affluent, as their living standard and style begin to ascend the material scale, God descends the scale of responsibility at a commensurate speed.”
    —Maya Angelou (b. 1928)

    “The true gardener then brushes over the ground with slow and gentle hand, to liberate a space for breath round some favourite; but he is not thinking about destruction except incidentally. It is only the amateur like myself who becomes obsessed and rejoices with a sadistic pleasure in weeds that are big and bad enough to pull, and at last, almost forgetting the flowers altogether, turns into a Reformer.”
    —Freya Stark (1893–1993)