Can I decode a C++ exception from a Windows SEH exception? (And if so, how?)

In the event of an unhandled C++ exception I want to print:

1. The message (what()) of the C++ exception
2. A stack trace.

In order to get the stack trace, I'm using SetUnhandledExceptionFilter in combination with the StackWalker library:

struct FooStackWalker : StackWalker
{
    virtual void OnCallstackEntry(CallstackEntryType, CallstackEntry &entry) override
    {
        std::cerr << entry.lineFileName << " (" << entry.lineNumber << "): " << entry.undFullName << std::endl;
    }
};

LONG WINAPI UnhandledExceptionHandler(LPEXCEPTION_POINTERS pointers)
{
    FooStackWalker walker;
    walker.ShowCallstack(::GetCurrentThread(), pointers->ContextRecord);
    ::TerminateProcess(::GetCurrentProcess(), 1);
}


int main()
{
    ::SetUnhandledExceptionFilter(UnhandledExceptionHandler);
}

I've gotten the stack trace to print just fine, but now getting what is difficult.

Is there some way I can decode the SEH exception as a C++ exception in order to call this member function before termination?