Anti-aliasing lines when using CATransform3DRotate

I copied the code from the answer to How do I apply a perspective transform to a UIView? in order to achieve the following image. Note, however, the jagged lines separating the levels. Is there away to perform this kind of perspective transformation in a way that anti-aliases these jagged lines?

Edit I've tried the solutions offered by DarkDust in the comments on this post. None seem to work. Here's my code:

levelView = [[UIControl alloc] initWithFrame:CGRectMake(100, 60, 160, 230)];

    [self addSubview:levelView];

    levelView.transform = CGAffineTransformScale(self.transform, .8, .8);



    CALayer *layer = levelView.layer;



    //DarkDust's 2nd comment

    layer.borderWidth = 1;

    layer.borderColor = [[UIColor clearColor] CGColor];



    CATransform3D rotationAndPerspectiveTransform = CATransform3DIdentity;

    rotationAndPerspectiveTransform.m34 = 1.0 / 1000;

    rotationAndPerspectiveTransform = CATransform3DRotate(rotationAndPerspectiveTransform, 45.0f * M_PI / 180.0f, 0.0f, 1.0f, 0.0f);

    layer.transform = rotationAndPerspectiveTransform;



    for (NSString *ID in [NSArray arrayWithObjects:@"Level 1", @"Level 2", @"Level 3", @"Level 4", @"Level 5", nil]) {



        //Note the offset of 5 from the superview in both X and Y directions. This addresses DarkDusk's first comment

        UIControl *ctrl = [[UIControl alloc] initWithFrame:CGRectMake(5, y + 5, levelView.frame.size.width, 220/5)];

        [levelView addSubview:ctrl];

        [ctrl addTarget:self action:@selector(languageSelected:) forControlEvents:UIControlEventTouchDown];

        [self styleEnabled:ctrl];            



        UILabel *label = [[UILabel alloc] initWithFrame:CGRectMake(13, 0, ctrl.frame.size.width - 13, ctrl.frame.size.height)];

        [ctrl addSubview:label];

        label.text = ID;

        label.font = [UIFont systemFontOfSize:20];

        label.textColor = [UIColor colorWithRed:1 green:1 blue:1 alpha:.8];

        label.backgroundColor = [UIColor clearColor];



        y += ctrl.frame.size.height;

    }

The level buttons you see in the image are all instances of UIControl , and they are the direct subviews of UIControl *levelView . levelView is the one that is getting transformed.

The other two of DarkDusk's comments refer specifically to UIImages, which I am not using. If they are still applicable and someone can explain how to implement them, I'll certainly give them a shot.