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In the general case, let $x = (x_1,\dots,x_n)$ be a finite-length vector (in a finite dimensional space). The finite sequence of absolute values $|x_i|$ does attain its maximum (because the sequence is finite), denoted $M = \max_i |x_i|$. Let $m$ be the (exact) number of coordinates in $x = (x_1,\dots,x_n)$ whose absolute value is equal to $M$. Thus, $1\... 4 Two codewords$c_1$and$c_2$of length$n$, with elements in$\lbrace +1, -1 \rbrace$, and Hamming distance$d$, have a cross-correlation given by $$(n-d) -d = n-2d.$$ The reason is that there are$n-d$bits that are equal and their product is$1$, and$d$bits that are different and their product is$-1$. Note that: The larger the distance$d$, the ... 3 Well, as you can easily verify, these two criteria aren't the same if you define "minimum correlation" to mean that the absolute value of the correlation coefficient is minimized (i.e. 0): In$\mathbb F_2^N$, the vector that's the farthest away from any given vector$v$is its bit-wise inverse$\overline v$(using Hamming distance) Using your mapping, the ... 2 I'm answering the question the way I understood it - How can one find a similarity measure which isn't sensitive to scaling and shifting. An approach could be borrowed from the Computer Vision world by comparing Shift and Scale Invariant features between the two signals. I'm not sure it will work for measuring the quality of recovering signals but it ... 2 Well, you can have an approximation. Due to the forms in the image are non regular, completely plain, its hard if not impossible to know the camera distortion. But if you know the size of the wheels and also you have the measures of the stones you can approximate the distance of the cart to the stones by using this equation: $$D_w = \frac{S_w * f}{P_w},$$ ... 2 The essential idea is: There is nothing wrong with color information- it is just insufficient. So best thing is to combine multiple feature sets. You can try multiple features to resolve this ambiguity. As far as feature set is concerned you can use following: Color (something like MPEG7's dominant color) OR Color Historgram Texture (either in the ... 2 To start with, 2 images and 3 measurements are not exactly enough to ascertain any kind of statistical model which in terms can be used to define the optimal comparison metric. I think you could start to have a look at texture recognition papers for methods and clues. It is an active field. For what it is worth, I ran a couple of perceptual hashing ... 2 Generally, for non-linear systems the aren't any tools that are guaranteed to work. You need to know something about the nature of the box. If you can model it with a system with unknown parameters, "learning" by observing input-output relationships can help you estimate those parameters, but I doubt that you can blindly "learn" the system model, especially ... 2 ...is the quality measure SSIM between two images (one base-line, one distorted) directly correlated to the mutual information between the two [?] The short answer to this is "yes" but it tells you nothing. Because the question is a little bit ill posed. You have two signals, let them be signals or images, it doesn't matter, you have two observations. And ... 2 Well, You're asking on one of the most researched topics. There is no 1 "Right" answer. Many models have been suggested and many of them works pretty well. In the Non Local Means they use Euclidean Distance. Later revisions on the idea used Mahalanobis Distance. Others are using different norm, other use Learned Dictionaries. Some use features or ... 2 Ok, I read the question posted in your link, so forget about MSE or SSIM. What you can do is define a specific distance for the particular type of data you are using. For example, if you are using video data, depending on what you are most interested in, you can define distance measure based on the average shot length, keyframe colour histogram, average ... 1 The$L_p$norm is $$d_p(\mathbf{x}, \mathbf{y}) \triangleq \left( \sum\limits_{i=1}^{n} |x_i - y_i|^p \right)^{\frac{1}{p}}$$ there exists a positive value that is the maximum value: $$M \triangleq \max_{1 \le i \le n} |x_i - y_i|$$ now, suppose you divide both sides of the$L_p$norm definition by that positive value, $$\frac{d_p(\mathbf{x}, \... 1 If I consider that your colored lines are multivariate signals (in Red, Green, Blue), that might be related through local offsets and scalings (stretch and squeeze), I would suggest the method in Correlation based dynamic time warping of multivariate time series, Expert Systems with Applications, 2012. It combines Dynamic Time Warping (DTW), a widely used ... 1 Say the intensity signal at one of the color channels changes linearly from 1 to 0.5 relative to full scale when the wavelength changes by 5 nm near 580 nm, and you have a laser for illumination at that wavelength. For an object moving 100 km/hour or 28 m/s, the doppler shift is 28 m/s * 2 / speed of light = 0.0000002 times the wavelength, 580 nm * 0.0000002 ... 1 Practically very difficult isn't it ? Speed of ordinary objects in an office environment will be so slow compared to light speed that the resulting shift in the color (wavelength) would be exremely small; much less than the unavoidable noise floor of the typical sensor... 1 Without loss of generality, let's define your anchor positions to be on the x axis with x_0=0 and the gap between each position be g. Thus$$ x[n] = n \cdot g $$Obviously, this can be exactly interpolated with the equivalent linear function.$$ x(n) = n \cdot g$$Let$(x_u,y_u)$be the position of the user. Now the distance from any position ... 1 This measure really depends on how your images look like. A very basic thing would be the calculate the point-wise difference between both images and summing them up. The smaller this value is, the more equal they are. Possibly, you can perform some Gaussian Smoothing and contrast equalization before. But, that's really depending on the images. 1 Maybe not an exact answer, but I'll to give a direction. What you are using is essentially an RBF-Kernel. First, it has a ready interpretation as a similarity measure (satisfies Mercer's conditions). In essence, other such kernels can be used. For instance, an inner product (linear kernel) would define an angular distance between pixels/features. For a ... 1 If your box is (mostly) linear that is a very simple problem, if it mainly non-linear that it can arbitrarily complicated. If we assume linearity, than simple superposition holds. You can measure the transfer function from each input to the output (while the other inputs are zero) and then calculate the output as the sum of the individual input responses. ... 1 I'm not sure what is meant here by the "optimal" similarity metric for a given situation. The textbook you linked to references the following thesis:  P. A. Viola, Alignment by maximization of mutual information. Ph.D. thesis, Massachusetts Institute of Technology, 1995. ...but I can't find it anywhere online after a quick search. From my point of ... 1 First of all, the mean squared error should not have a square root. As the name implies, it is the mean of the squared difference. What you are referring to is the root mean squared error. Second, the this is a pixel-wise comparison between to images. Therefore the number of pixels is the dimension, and the sum is performed on it. There is even a comment on ... 1 With slight modification you might want to use RootSift: http://www.robots.ox.ac.uk/~vgg/publications/2012/Arandjelovic12/arandjelovic12.pdf Also the other steps in the paper will guide to improve the recall rate. Cheers, 1 You can use the Structural Similarity Index (SSIM). It's not a distance per se, but you can use it in this case. 1 An obvious choice would be$\frac{d_1}{d_1+d_2+\epsilon }$, where$\epsilon\$ is a small value to prevent division by zero. Using your images, I get: top = ColorNegate[Binarize[ImageApply[StandardDeviation, Import["http://i.stack.imgur.com/aFNfB.jpg"]]]]; bottom = ColorNegate[Binarize[ImageApply[StandardDeviation, Import["http://i....