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I am not familiar with brain images analysis but I am still going to try to help you here. If each of your independent spatial components are denoted as long vectors $x_i$ $\forall i$, then each of your original brain images can be expressed as a linear combination of your independent spatial components.

The fact that your spatial components $x_1$ and $x_2$ are independent means that you can't predict the value taken by $x_1$ at one pixel based on the value taken by $x_2$ at the same pixel. Thus, $p(x_1,x_2) = p(x_1)p(x_2)$.

If you want to further understand the theoretical basis of ICA and how the different algorithms performing ICA such as infomax or FastICA work, I recommend you this review from the creators of the FastICA algorithm.

Tell me if anything is unclear.

Also, ICA is usually formulated as: $x = As$ where $s$ are your source variables (or independent components), $x$ your observed variables (e.g. your brain pictures), and $A$ the mixing matrix containing coefficients necessary for the above mentioned linear combination.

I am not familiar with brain images analysis but I am still going to try to help you here. If each of your independent spatial components are denoted as long vectors $x_i$ $\forall i$, then each of your original brain images can be expressed as a linear combination of your independent spatial components.

The fact that your spatial components $x_1$ and $x_2$ are independent means that you can't predict the value taken by $x_1$ at one pixel based on the value taken by $x_2$ at the same pixel. Thus, $p(x_1,x_2) = p(x_1)p(x_2)$.

If you want to further understand the theoretical basis of ICA and how the different algorithms performing ICA such as infomax or FastICA work, I recommend you this review from the creators of the FastICA algorithm.

Tell me if anything is unclear.

Also, ICA is conventionnaly formulated as: $x = As$ where $s$ are your source variables (or independent components), $x$ your observed variables (e.g. your brain pictures), and $A$ the mixing matrix containing coefficients necessary for the above mentioned linear combination.

I am not familiar with brain images analysis but I am still going to try to help you here. If each of your independent spatial components are denoted as long vectors $x_i$ $\forall i$, then each of your original brain images can be expressed as a linear combination of your independent spatial components.

The fact that your spatial components $x_1$ and $x_2$ are independent means that you can't predict the value taken by $x_1$ at one pixel based on the value taken by $x_2$ at the same pixel. Thus, $p(x_1,x_2) = p(x_1)p(x_2)$.

If you want to further understand the theoretical basis of ICA and how the different algorithms performing ICA such as infomax or FastICA, I recommend you this review of the creators of the FastICA algorithm.

Tell me if anything is unclear.

Also, ICA is usually formulated as: $x = As$ where $s$ are your source variables (or independent components), $x$ your observed variables (e.g. your brain pictures), and $A$ the mixing matrix containing coefficients necessary for the above mentioned linear combination.

I am not familiar with brain images analysis but I am still going to try to help you here. If each of your independent spatial components are denoted as long vectors $x_i$ $\forall i$, then each of your original brain images can be expressed as a linear combination of your independent spatial components.

The fact that your spatial components $x_1$ and $x_2$ are independent means that you can't predict the value taken by $x_1$ at one pixel based on the value taken by $x_2$ at the same pixel. Thus, $p(x_1,x_2) = p(x_1)p(x_2)$.

If you want to further understand the theoretical basis of ICA and how the different algorithms performing ICA such as infomax or FastICA work, I recommend you this review from the creators of the FastICA algorithm.

Tell me if anything is unclear.

Also, ICA is usually formulated as: $x = As$ where $s$ are your source variables (or independent components), $x$ your observed variables (e.g. your brain pictures), and $A$ the mixing matrix containing coefficients necessary for the above mentioned linear combination.

I am not familiar with brain images analysis but I am still going to try to help you here. If each of your independent spatial components are denoted as long vectors $x_i$ $\forall i$, then each of your original brain images can be expressed as a linear combination of your independent spatial components. Or, $image_k = \sum\limits_{i=1}^n a_{ik}*x_i$ for any of your $k$ images, and for a total of $n \leq k $ independent spatial components. The fact that your spatial components $x_1$ and $x_2$ are independent means that you can't predict the value of the coefficient $a_{1}$ based on the value of another coefficient $a_{2}$ for any image $k$. The mathematical relationship between $x_1$ and $x_2$ is thus $p(a_1,a_2) = p(a_1)p(a_2)$.

If you want to further understand the theoretical basis of ICA and how the different algorithms performing ICA such as infomax or FastICA, I recommend you this review of the creators of the FastICA algorithm.

Tell me if anything is unclear.

Also, ICA is usually formulated as: $x = As$ where $s$ are your source variables (or independent components), $x$ your observed variables (e.g. your brain pictures), and $A$ the mixing matrix containing the above mentioned coefficients.

I am not familiar with brain images analysis but I am still going to try to help you here. If each of your independent spatial components are denoted as long vectors $x_i$ $\forall i$, then each of your original brain images can be expressed as a linear combination of your independent spatial components.

The fact that your spatial components $x_1$ and $x_2$ are independent means that you can't predict the value taken by $x_1$ at one pixel based on the value taken by $x_2$ at the same pixel. Thus, $p(x_1,x_2) = p(x_1)p(x_2)$.

If you want to further understand the theoretical basis of ICA and how the different algorithms performing ICA such as infomax or FastICA, I recommend you this review of the creators of the FastICA algorithm.

Tell me if anything is unclear.

Also, ICA is usually formulated as: $x = As$ where $s$ are your source variables (or independent components), $x$ your observed variables (e.g. your brain pictures), and $A$ the mixing matrix containing coefficients necessary for the above mentioned linear combination.