Image Restoration and Standard Forms of Second Order Cone Programming (SOCP)

I'm studying the application of SOCP methods in Image restoration And I want to understand the difference between the two formulas of SOCP and how they are related. Standard form (1) :

min $$f^{t}x$$

subject to $$\left \| A_{i}x + b_{i} \right \|_{{2}} \leq c^{{i}^{t}}x + d_{i}$$

but also in some papers i found this standard form (2) :

min $$c{_{1}}^{t}x_{1} + c{_{2}}^{t}x_{2} +. . . +c{_{r}}^{t}x_{r}$$

$$A_{1}x_{1}+ A_{2}x_{2} +...+A_{r}x_{r}=b$$

• What are the major differences between the two?

• when and how should each one be applied?