Let $f$ denote the measurement contribution function for paths of length $k\in\mathbb N$, i.e. $$f(x)=g(x_0\leftrightarrow x_1)W_{\text e}(x_1\to x_0)t_k(x_0,\ldots,x_k)L_{\text e}(x_k\to x_{k-1}),$$ where $$t_k(x)=\prod_{i=2}^kg(x_{i-1}\leftrightarrow x_i)f_{\text s}(x_i\to x_{i-1}\to x_{i-2}),$$ and $h_j$ denote the image reconstruction filter of the $j$th pixel so that the measurement $I_j$ of the $j$th pixel color is given by $$I_j=\int h_jf\:{\rm d}\mu.$$

What can we assume know about $h_j$ in general? One thing is that it will only depend on first two vertices of a path.

There are two properties which I really would like to being able to assume: (a) $c:=\sup_j\left\|h_j\right\|<\infty$ (b) $\int|h_jf-I_j|^2\:{\rm d}\mu\le\alpha\int|f_0|^2\:{\rm d}\mu$ for some $\alpha\ge0$, where $f_0:=f-\int f\:{\rm d}\mu$.

However, I know almost nothing about the mathematical properties of $h_j$ and so it's hard for me to verify if these properties hold. I guess that generally $h_j\ge0$, but I'm even unsure about that. Maybe even $h_j\in[0,1]$? It would be great if someone could answer if (a) and/or (b) hold or link me to a reference.

Let $f$ denote the measurement contribution function for paths of length $k\in\mathbb N$, i.e. $$f(x)=g(x_0\leftrightarrow x_1)W_{\text e}(x_1\to x_0)t_k(x_0,\ldots,x_k)L_{\text e}(x_k\to x_{k-1}),$$ where $$t_k(x)=\prod_{i=2}^kg(x_{i-1}\leftrightarrow x_i)f_{\text s}(x_i\to x_{i-1}\to x_{i-2}),$$ and $h_j$ denote the image reconstruction filter of the $j$th pixel so that the measurement $I_j$ of the $j$th pixel color is given by $$I_j=\int h_jf\:{\rm d}\mu.$$

What can we assume know about $h_j$ in general? One thing is that it will only depend on first two vertices of a path.

There are two properties which I really would like to being able to assume: (a) $c:=\sup_j\left\|h_j\right\|<\infty$ (b) $\int|g_j|^2\:{\rm d}\mu\le\alpha\int|f_0|^2\:{\rm d}\mu$ for some $\alpha\ge0$, where $g_j:=1_{\{\:h_j\:>\:0\:\}}\left(h_jf-\frac1{\mu\left(\left\{h_j>0\right\}\right)}I_j\right)$ and $f_0:=1_{\{\:f\:\ne\:0\:\}}\left(f-\frac1{\mu\left(\left\{f\ne0\right\}\right)}\int f\:{\rm d}\mu\right)$.

However, I know almost nothing about the mathematical properties of $h_j$ and so it's hard for me to verify if these properties hold. I guess that generally $h_j\ge0$, but I'm even unsure about that. Maybe even $h_j\in[0,1]$? It would be great if someone could answer if (a) and/or (b) hold or link me to a reference.

Let $f$ denote the measurement contribution function for paths of length $k\in\mathbb N$, i.e. $$f(x)=g(x_0\leftrightarrow x_1)W_{\text e}(x_1\to x_0)t_k(x_0,\ldots,x_k)L_{\text e}(x_k\to x_{k-1}),$$ where $$t_k(x)=\prod_{i=2}^kg(x_{i-1}\leftrightarrow x_i)f_{\text s}(x_i\to x_{i-1}\to x_{i-2}),$$ and $h_j$ denote the image reconstruction filter of the $j$th pixel so that the measurement $I_j$ of the $j$th pixel color is given by $$I_j=\int h_jf\:{\rm d}\mu.$$

What can we assume know about $h_j$ in general? One thing is that it will only depend on first two vertices of a path.

There are two properties which I really would like to being able to assume: (a) $c:=\sup_j\left\|h_j\right\|<\infty$ (b) $\int|h_jf-I_j|^2\:{\rm d}\mu\le\int|f_0|^2\:{\rm d}\mu$, where $f_0:=f-\int f\:{\rm d}\mu$.

However, I know almost nothing about the mathematical properties of $h_j$ and so it's hard for me to verify if these properties hold. I guess that generally $h_j\ge0$, but I'm even unsure about that. Maybe even $h_j\in[0,1]$? It would be great if someone could answer if (a) and/or (b) hold or link me to a reference.

Let $f$ denote the measurement contribution function for paths of length $k\in\mathbb N$, i.e. $$f(x)=g(x_0\leftrightarrow x_1)W_{\text e}(x_1\to x_0)t_k(x_0,\ldots,x_k)L_{\text e}(x_k\to x_{k-1}),$$ where $$t_k(x)=\prod_{i=2}^kg(x_{i-1}\leftrightarrow x_i)f_{\text s}(x_i\to x_{i-1}\to x_{i-2}),$$ and $h_j$ denote the image reconstruction filter of the $j$th pixel so that the measurement $I_j$ of the $j$th pixel color is given by $$I_j=\int h_jf\:{\rm d}\mu.$$

What can we assume know about $h_j$ in general? One thing is that it will only depend on first two vertices of a path.

There are two properties which I really would like to being able to assume: (a) $c:=\sup_j\left\|h_j\right\|<\infty$ (b) $\int|h_jf-I_j|^2\:{\rm d}\mu\le\alpha\int|f_0|^2\:{\rm d}\mu$ for some $\alpha\ge0$, where $f_0:=f-\int f\:{\rm d}\mu$.

However, I know almost nothing about the mathematical properties of $h_j$ and so it's hard for me to verify if these properties hold. I guess that generally $h_j\ge0$, but I'm even unsure about that. Maybe even $h_j\in[0,1]$? It would be great if someone could answer if (a) and/or (b) hold or link me to a reference.