Help in problem formulation for estimation of image as a feature vector - SISO or MIMO FIR channel model?

Based on the paper Blind Image deconvolution:

A feature vector is a list of numbers used to represent an image. The feature vector for my case takes values as symbols $-1,1$.

An instance or an example of an image is represented as a feature vector of $d$ feature values $\mathbf{x} = [x_1,x_2,...,x_d]$. I have $N$ examples $$\{\mathbf{x}_i\}_{i=1}^N$$ in the form of a database. So, the database contains $N$ rows of feature vectors each of $d$ length. One such pattern (feature vector) is $$\mathbf{x} = [1, -1, -1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1]'$$ where $d = 30$ The database thus has $N$ rows of examples each of $d$ length : $$\mathbf{x_1} = x_{11},x_{12},....,x_{1d}$$ $$\mathbf{x_2} = x_{21},x_{22},....,x_{2d}$$ $$::$$ $$\mathbf{x_N} = x_{N1},x_{N2},....,x_{Nd}$$

I don't know how to use the above as input to the channel. In general,
the output is, $$\mathbf{y} = H^T x$$

Assuming that the impulse response is that of FIR, then I am facing difficulty in how to represent the channel. Should the source input be each component of the feature vector or an entire example?

• if the input in the form of the feature vector is transmitted via a channel whose impulse response is modeled as moving average (finite response, FIR) then inputs and how many outputs should there be? The output would be the convolution of the input and the impulse response. Would the channel be Single input-Single Output FIR or Multiple Input Multiple Output FIR ? The objective is to estimate the channel parameters and the input using estimation methods from the output observation only. Would the estimation be performed using one example or all the examples?

• Should there be $d$ channel coefficients? A mathematical representation of the system will really help to clear the concept. Thank you

• You say Each image is transmitted over a channel but you use $x$, which is the feature vector. Are you transmitting the image or the feature vector derived from the image? You say $\mathbf{y} = \hat{\mathbf{x}_i} = \mathbf{x}_i$ which is clearly impossible unless $H$ is the identity matrix. You say I want to reconstruct the original image but how can you do that if only the feature vectors are transmitted? – Peter K. Jul 11 '16 at 18:39
• @PeterK.: I have explained more clearly and also added a paper that is relevant to the problem. Thank you for pointing out the mistakes. – Srishti M Jul 11 '16 at 18:44
• That's somewhat clearer, but now I'm confused: the paper you link to is doing something different from what you describe, and it's still not clear to me whether you are trying to reproduce the image or the feature vector. You only describe the feature vector mathematically, so I assume that's what you are transmitting. – Peter K. Jul 11 '16 at 18:57
• I am transmitting the feature vector for $N$ images. The images are represented using the feature vector which I also called as the pattern. Similar to the paper, I want to estimate the input which is nothing but the pattern. What I do not understand is would the channel coefficients be estimated for each feature vector (each image) or for all the images together? In the former case, would the system be called Single Input Single Output (SISO)-FIR and for the latter multiple input multiple output (MIMO)-FIR? What is the time index and what would be the order etc – Srishti M Jul 11 '16 at 19:18
• OK, you are only looking at feature vectors so I have change the question over to ask about that. Nowhere can I see where you define notation for the actual image. I've reopened; let's see if someone answers. I'm confused now about the $y(i)$ summation, but let's see what others think. – Peter K. Jul 11 '16 at 19:30

This question is confusing, but I thought I'd try to make some sense of it.

Suppose you are transmitting one of your $\mathbf{x}_i$ vectors one component at a time. That is, the transmitted signal, $s[k]$, for the $i^{\rm th}$ feature vector is: $$s_i[k] = x_{ik}$$ If this signal is corrupted by a channel with impulse response $h[k]$ then the received signal, $y[k]$ will be $$y[k] = h[k] \star s_i[k] = \sum_{m=0}^{M-1} h[m] s_i[k-m] = \sum_{m=0}^{M-1} h[m] x_{im}$$ where $\star$ represents convolution.

If we use this formulation, then the answer to:

Should the source input be each component of the feature vector or an entire example?

is that it should be each component of the feature vector.

if the input in the form of the feature vector is transmitted via a channel whose impulse response is modeled as moving average (finite response, FIR) then inputs and how many outputs should there be?

There should be a sequence of outputs. If the FIR filter is length $M$, then the output will be of length $M+d-1$.

Would the channel be Single input-Single Output FIR or Multiple Input Multiple Output FIR ?

In the model above, it is a SISO system.

Would the estimation be performed using one example or all the examples?

If the channel does not change, then you might be better off using all the data you have available.

Should there be $d$ channel coefficients?

There is no point in tying the number of feature vector components with the length of the channel. In the above, the length of the channel is $M$.

• Thank you very much for taking the time out to make sense of what I asked. Just to be sure if I understood correctly, say each image $\mathbf{x}_i$ is characterized by 3 features - red,green, blue. Consider, the first image which is the first row in the database of images. Then $d=3$ and the input would be these 3 values and the output will contain $y = M+3-1$ values? – Srishti M Jul 12 '16 at 20:35
• @SrishtiM Yes, if you convolve an FIR of length $M$ with a signal of length $3$ then the result will be of length $M + 3 - 1$ – Peter K. Jul 12 '16 at 20:41
• Note that your problem is a little different from most deconvolution ones. You may be able to use the fact that the $\mathbf{x}_i$ components are only $\pm 1$ and use a POCS technique like the one used here. – Peter K. Jul 12 '16 at 20:45
• To summarize, when we deal with image feature vectors as input then each signal is considered to be a feature vector of an image? Also thank you for the link, I believe it is related to compressive sensing. I will study the information. – Srishti M Jul 12 '16 at 21:38